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Jan 15, 2014 - dynamical simulations to obtain analytical closed-form expressions for the shear modulus of fractal aggregates in shear flows. The prop...
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Analytical Model of Fractal Aggregate Stability and Restructuring in Shear Flows Breanndan O. Conchuir,† Yogesh M. Harshe,‡ Marco Lattuada,§ and Alessio Zaccone*,† †

Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, U.K. Department of Chemical Engineering, Delft University of Technology, 136 Julianalaan, 2628 BL Delft, The Netherlands § Adolphe Merkle Institute, University of Fribourg, Route de l’Ancienne Papeterie, PO Box 209, 1723 Marly 1, Switzerland ‡

ABSTRACT: We combine the analytical theory of nonaffine deformations of noncrystalline solids with numerical Stokesian dynamical simulations to obtain analytical closed-form expressions for the shear modulus of fractal aggregates in shear flows. The proposed framework also provides analytical predictions for the evolution of the fractal dimension df of the aggregate during the aggregation process. This leads to a lower bound on df below which aggregates are mechanically unstable (they possess floppy modes) and cannot survive without restructuring into more compact, higher-df configurations. In the limit of large aggregates, the predicted lower bound is df = 2.407. This result provides the long-sought explanation as to why all experimental and simulation studies in the past consistently reported df ≳ 2.4 for shear-induced colloidal aggregation. The analytical expressions derived here can be used within population balance calculations of colloidal aggregation in shear flows whereby until now the fractal dimension evolution was treated as a free parameter. These results may open up the possibility of developing new microscopic mechanical manipulation techniques to control nanoparticle and colloidal aggregates at the nanoscale.

I. INTRODUCTION The aggregation of colloidal particles and nanoparticles as well as large molecules in the liquid phase is very important for many industrial applications, especially in the processing and postprocessing of polymer materials, minerals, foods, cosmetics, paints, detergents, pharmaceuticals, etc.1 Under typical industrial conditions, colloidal suspensions are processed in reactors, stirred tanks, and pipes, where shear flow is ubiquitous.2 For this reason, it is imperative to achieve a good understanding of the aggregation behavior of colloidal particles under shear flow, because colloidal aggregates control the rheology of the suspension (e.g., the viscosity). In certain instances, aggregates are undesired byproducts if a stable monodispersed colloidal suspension is required as the final product. Further, uncontrolled aggregation may lead to jamming and fouling phenomena, with evident negative impacts on the industrial process.3 In other situations, instead, such as in the postprocessing of emulsion polymerization, aggregation is a key process to obtain the polymer material in solid form.4 Many studies in the past, experimental, numerical, and theoretical, have been devoted to the understanding and modeling of colloidal aggregation under shear and turbulent flows.5 More recently, progress has been made by M. Morbidelli and co-workers, in elucidating and quantitatively describing the mechanism of aggregation between two arbitrarily interacting colloidal particles in shear flow,6 and also in connection with the emergent macroscopic response of the suspension, where an explosive rise in the viscosity is triggered by the microscopic competition between shear stress and the electric double layer barrier between two particles.7,8 In parallel, progress has also been made in the characterization of the structure of colloidal aggregates generated under flow,9 and in the simulation of the aggregate structure with hydrodynamic interactions by means of Stokesian dynamics simulations.10 © 2014 American Chemical Society

Recently, a new approach to the mechanical stability and breakup of colloidal aggregates has been introduced, leading to a semianalytical framework which can predict the breakup rate as well as the stable fragment sizes by accounting for colloidal interactions, collective stress transmission inside the aggregate, hydrodynamic interactions (in a phenomenological way), fractal structure, and microscopic convective-diffusive dynamics of the bonded particles in the aggregate.11 Following this line of research, we construct a theoretical framework to investigate the stability and restructuring of aggregates in a flow field. Simulations and experiments (using both light scattering and microscopy techniques) have consistently observed that only aggregates with fractal dimension df > 2.4 survive in sheared environments.7−10,12−14 Nevertheless, a theoretical explanation for this phenomenon has not been found. Here we address this question by first performing a multiobject fitting of Stokesian dynamics simulation data for the mean coordination number z of fractal aggregates as a function of df and aggregate radius of gyration Rg. The obtained relationship is then implemented in the nonaffine deformation formalism leading to analytical expressions for the shear modulus G of an individual aggregate as a function of the structural parameters, assuming that both central-force and bending (tangential) interactions are present between the particles, which is also consistent with the simulations.10 Rich physical behavior is predicted due to the interplay between surface effects and structural evolution. This Special Issue: Massimo Morbidelli Festschrift Received: Revised: Accepted: Published: 9109

October 5, 2013 January 14, 2014 January 15, 2014 January 15, 2014 dx.doi.org/10.1021/ie4032605 | Ind. Eng. Chem. Res. 2014, 53, 9109−9119

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We extract the total average coordination number zt as a function of N from the Stokesian dynamics numerical simulations (see the Appendix for the details of the simulations). Aggregates with variable df were first generated with an algorithm described in ref 15, and the resulting structures were subsequently calibrated in the Stokesian dynamics simulations under low shear. To recover the mean internal coordination number zint, we divide the total number of internal bonds by the number of internal particles

framework leads to a complete map of aggregate stability which also allows us to disentangle the interplay between breakup and restructuring.

II. THEORY A. Fractal Dimension. Let us consider an aggregate of spherical particles. The general expression relating N, the number of particles in a colloidal cluster, and Rg, the aggregate radius of gyration, is ⎛ R g ⎞d f N = k 0⎜ ⎟ ⎝ a ⎠

z int =

⎛ R g ⎞5.65 − 3.66d f z int = 0.532 + 0.776df + 0.000192df 6.46d f ⎜ ⎟ ⎝ a ⎠ (5)

In our fitting we set an upper limit of zint = 8 beyond which this formula becomes unrealistic. The simulation data and their fitting are presented in Figure 1. It is seen that the mean internal coordination number is a

Figure 1. Plot of the mean internal coordination number zint (eq 5) as a function of the normalized aggregate radius of gyration Rg/a. From top to bottom: df = 3/2.7/2.4/2.1/1.8. Data points from our simulations are also displayed. The purple dashed line signifies the classic Maxwell constraint-counting rigidity-threshold for random networks.22 Note that eq 5 becomes unrealistic at small Rg/a. The lowest two curves are below the dashed threshold indicating that these clusters cannot survive in a shear flow.

decaying function of the linear aggregate size Rg/a, which is expected from the fractal geometry of clusters where the density gradually decays along the radial coordinate, with the cluster growth. What is perhaps more surprising, is that upon increasing df the decay becomes much steeper and confined to a relatively narrow linear size interval Rg/a = 10−15. This behavior can be ascribed to the fact that aggregates with df = 2.5−3 tend to have a dense core where particles are rather homogeneously distributed, and therefore the transition toward the aggregate surface where particles are much less coordinated (even upon neglecting the surface dangling particles), is much steeper. In other words, the density profile at the interface

(2)

Multiplying by the particle radius a we obtain the number of colloids in the outer layer of the aggregate Next,

Next

⎛ R g ⎞d f − 1 = k 0df ⎜ ⎟ ⎝ a ⎠

(4)

with Nint = N − Next denoting the number of internal particles. Analyzing our simulation data and applying standard nonlinear square fitting procedures, we obtain the following phenomenological relationship expressing zint as a function of df and Rg:

(1)

where a is the radius of the spherical particles, df is the cluster fractal dimension and k0 = 4.46df−2.08 is a prefactor of order unity.15 The mass fractal dimension df is also linked to the mechanical properties of the clusters. For self-similar fractals with df in the range 1.5−2.0, these properties can be described in terms of the chemical or backbone fractal dimension dB, that is the fractal dimension of the cluster subsection which participates in the transmission of mechanical stress.16,17 For DLCA aggregates with df ≃ 1.7 it is known that dB ≃ 1.1,18 which implies that stress transmission through the aggregate is nearly one-dimensional, and involves long particle chains. dB may also bear some connection to the rigidity percolation fractal dimension of amorphous solids.19 It disregards dangling chains which branch off from the stressed backbone as these bonds are generally not under stress. Hence, dB < df. In this study, as discussed in more detail below, we consider clusters under shear flow in which the hydrodynamic stress that each bond receives from the solvent is transmitted to every other bond in the structure, albeit with some localized dissipation.11 Therefore, the entire structure is stressed, including the dangling chains, and dB is close to the cluster fractal dimension df. This is justified by the fact that, in fluid-like colloidal suspensions under flow, all particles experience shear and hydrodynamic stresses to various extents, which has been shown in previous studies (see, e.g., ref 20). This is in contrast to the elastic deformation of a solid-like macroscopic gel of colloidal particles,16 where dangling chains, which do not receive any significant stress from the environment, do not participate in the stress transmission. B. Mean Internal Coordination Number. While many studies focus on the overall average coordination number,21 surface effects can greatly distort these figures. To decouple the internal dynamics from surface effects it is vital that an expression for the mean internal coordination number is found. By internal coordination number it is implied that bonds involving particles that are one-fold coordinated z = 1, on the surface of the aggregate are not considered in the counting. From eq 1 we can see that the rate of change of N as a function of the cluster size Rg is given by d −1 k 0df ⎛ R g ⎞ f dN = ⎜ ⎟ dR g a ⎝ a ⎠

z tN − Next Nint

(3) 9110

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15 where ⟨Fc⟩ is the average radial bond restoring energy (bond-stretching stiffness).27 A similar evaluation of the second term in eq 6 reveals an elastic energy of ⟨Fb⟩γ2/2 = (31/ 135)βR02γ2 per bond-bending constraint.28 The classical Phillips−Thorpe extension of Maxwell constraint-counting tells us that in a cluster of N particles there are ztN/2 nearest neighbor bonds, while there are (2zint − 3)Nint bond-bending constraints assuming that all internal particles have two or more neighbors and that all surface particles have only one (and thus encounter no bond-bending forces).22 It follows that the average total elastic energy per bond is given by ⟨Fco⟩ = Ωc⟨Fc⟩ + Ωb⟨Fb⟩, where Ωc = [ztN/2]/[(ztN/2) + (2zint − 3)Nint] and Ωb = [(2zint − 3)Nint]/[(ztN/2) + (2zint − 3)Nint] denote the relative bond-stretching and bond-bending contributions. The average nonaffine work W per degree of freedom cannot exceed the average potential energy of the bond, that is that W ≤ 3Nna⟨Fco⟩/2 where Nna is the number of nonaffine particles in the cluster. Assuming, variationally, that the system fully relaxes and the aggregate deforms along the path of minimum energy, this is equivalent to taking the equality of the above inequality for W. Thus the total free energy of deformation F is given by algebraically summing the negative energy W spent on sustaining the nonaffine relaxation, and the affine component Fa ,

aggregate/solvent is relatively sharp for dense aggregates. It could be useful to mention the analogy with atomic nuclei where the nucleon density is homogeneous, apart from a finite shell near the surface where the nucleon density decays quite sharply with a typical sigmoidal shape.23 The complex interplay between this surface effect, the magnitude of which decays itself subextensively with the aggregate size as ∼(Rg/a)−df+1, and fractal geometry, thus gives rise to the behavior observed in Figure 1 and to the mathematical dependence of zint upon Rg/a contained in eq 5. It is worth stressing that the fitting expression eq 5 becomes less accurate at df < 2.3, which is also visible in Figure.1. However, this problem is not of major concern here because, as we are going to show below, aggregates in shear flow are always above this df value. As we are going to see below, the df-mediated decay of the mean coordination number close to the aggregate surface plays an important role in controlling the restructuring behavior in shear of the aggregates, and it is an essential ingredient in deriving a stability map of colloidal aggregates in shear flow. C. Aggregate Shear Modulus. In all particle assemblies which lack both rotational and translation symmetry (noncrystalline aggregates), the particles undergo nonaffine displacements.24 This means that under an imposed macroscopic strain γ, the interparticle displacements are not just equal to affine displacements uA = R0γ, where R0 is the bond length at rest between two particles. Additional displacements uNA arise due to the constraint of energy minimization, to preserve mechanical equilibrium. This fact can be intuitively understood if one considers a particle bonded to its zint nearest neighbors. Upon applying the strain, all the neighbors tend to move affinely, and in so doing they transmit forces to the particle at the center. These forces would exactly cancel each other by the local rotational (point-group) symmetry of a crystalline lattice. In noncrystalline particle assemblies, the transmitted forces clearly do not balance, because the local mirror symmetry is broken. Since fractal aggregates are locally disordered, this mechanism applies fully to their deformation mechanics. Surface particles, with their one nearest-neighbor bond, lie between the limits of being fully affine and being fully nonaffine.25 In this section we will investigate the effect of affinity, or lack of it, at the surface of the cluster, on the shear modulus G of an individual aggregate. To answer these questions we model an aggregate held together in a spring-like ensemble of nearest neighbor bonds, with the total potential energy of the disordered lattice E given as E=

α 2

∑ (δrij)2 + ij

β 2

F = Fa − W =

(7)

The shear modulus G is found by differentiating the energy density by the strain twice. Thus G=

⎤ ⟨Fco⟩ ⎡ z tN + (2z int − 3)Nint − 3Nna ⎥ ⎢⎣ ⎦ V 2

(8)

If all the surface particles are nonaffine, then Nna = N and G = Gnas =

⎤ ⟨Fco⟩ ⎡ zt N + (2z int − 3)Nint − 3N ⎥ ⎢⎣ ⎦ V 2

(9)

If all the surface particles are affine, then Nna = Nint and G = Gas =

⎤ ⟨Fco⟩ ⎡ z tN + (2z int − 3)Nint − 3Nint ⎥ ⎢⎣ ⎦ V 2

(10)

If a surface particle had to be displaced by the flow in a purely affine way, it would ballistically follow the local shear flow field. The extent to which this is true of course depends on various factors, such as the Peclet number and the strength of the interparticle bonding. Hence, the fraction of affine surface particles varies in an unpredictable way.25 By investigating both limits of purely affine and purely nonaffine displacement of the surface particles, we can obtain an upper and a lower bound for the shear modulus. The rigidity, or marginal stability, threshold is the point beyond which the cluster develops a finite shear modulus, or conversely, the point at which G = 0, according to the Born rigidity criterion.29 By setting G = 0 the rigidity threshold of an aggregate is given by the following condition

∑ (δθijk)2 ijk

⎤ ⟨Fco⟩ ⎡ zt N + (2z int − 3)Nint − 3Nna ⎥γ 2 ⎢ ⎦ ⎣ 2 2

(6)

The first term is the energy cost to radially stretch the bond between neighbors i and j by a distance δrij, whereas the second term gives the energy cost in deviating the dihedral angle θijk between neighboring bonds from its equilibrium value. Here α denotes the second derivative of the two-body radial potential. This can be derived from a linear combination of an attractive van der Waals potential, a repulsive screened-electrostatic potential, and a short-ranged repulsive Born potential, such as in the standard DLVO theory of colloidal stability.26 β or the tangential (bond-bending) spring constant is typically β/α ≈ 0.2, for example, for covalently bonded disordered solids.21 According to the nonaffine deformation formalism, the average radial elastic energy per bond equals ⟨Fc⟩γ2/2 = αR02γ2/

z tN + (2z int − 3)Nint − 3Nna = 0 2

(11)

This relation sets a condition on the mean coordination number as a function of the aggregate size where surface effects play an important role. The aggregate below this threshold has 9111

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a finite number of floppy modes, that is to say, the aggregate has large, collective deformation modes that can be excited with zero external energy. Thus, below this threshold the aggregate is no longer rigid and can restructure without any energy cost. These modes emanate from unconstrained degrees of freedom. The fraction f of these floppy modes is given by f=

−(z tN /2) − (2z int − 3)Nint + 3Nna 3Nna

(12)

Since all parameters in this expression depend on the aggregate finite size and structure, it can be anticipated that the mechanical stability and restructuring of the aggregates in shear flow are a function of the aggregate finite size (surface effects) and structure (through df).

III. DISCUSSION A. Mean Internal Coordination Number. In Figure 1 we plot the mean internal coordination number, from Stokesian dynamics simulations, as a function of the normalized radius of gyration Rg/a for a variety of different cluster fractal dimensions df. We include the classical Maxwell constraint counting rigidity threshold of zint = 2.4 so we can estimate at first glance which aggregate structures can survive without either breaking or restructuring in a shear flow. We can see that only very small clusters can survive if df is small, which anticipates that these aggregates will need to restructure into higher-df configurations in order to grow. In addition, only dense clusters with large values of df can sustain growth to substantial sizes. This feature correlates with the density of particles n, decaying from the center of mass to the cluster surface as a power law with n = N/ V∝(Rg/a)df−3. Thus, larger clusters are expected to have lower zint than smaller clusters, which is a direct consequence of fractal growth with df < d, with d = 3 in our case. This effect is encoded in the last term on the right hand side of eq 5. Furthermore, the mean internal coordination number appears to be a monotonically increasing function of the fractal dimension, which is also in agreement with standard fractal growth.30 The steeper decay of zint with the linear size upon increasing df has been already discussed in the previous section and interpreted as due to nonfractal surface effects, because of the sharp density decay close to the aggregate surface, upon approaching the aggregate−solvent interface. In Figure 2 we plot the evolution of zint as a function of df. We can see that for large Rg/a, the final term in eq 5 becomes negligible and zint becomes a linear increasing function of df. The last term in the right hand side of eq 5 emanates from the fractal power-law, and clearly it has a greater effect on the mean internal coordination number for smaller rather than larger aggregates. It is important to stress that this term contains the effect of fractal growth on mechanical stability and aggregate restructuring. B. Shear Modulus. In this section we shall investigate the aggregate shear modulus in the limits where the displacements of particles on the outermost shell are either all affine or all nonaffine. We will refer to the two models as the affine-surface model and the nonaffine-surface model, respectively. The behavior of a realistic aggregate is found between these two limits, although at high shear rates the purely affine-surface model is probably more realistic, as in that limit the outermost particles would tend to follow the strain imposed by the flow in an affine way. First we look at the influence the normalized

Figure 2. Plot of the mean internal coordination number zint as a function of the fractal dimension df. From top to bottom: Rg/a = 10/ 11/12/13/14/16/18/20/22.

radius of gyration Rg/a has on the shear modulus G. For the nonaffine surface model we have Gnas ∝

⎤ 1 ⎡ z tN + (2z int − 3)Nint − 3N ⎥ 3⎢ ⎦ Rg ⎣ 2

(13)

Now from eq 3 we see that ⎡ ⎛ R g ⎞−1⎤ Nint = N − Next = N ⎢1 − df ⎜ ⎟ ⎥ ⎢⎣ ⎝ a ⎠ ⎥⎦

(14)

Therefore manipulating eq 4, and noting the fact that k0 changes more slowly as a function of df, and rearranging, we recover ⎡

Gnas ∝

5z R gd f − 3⎢ int ⎢⎣ 2

−1 ⎛ 5z int − 7 ⎞ ⎛ R g ⎞ ⎤⎥ ⎟ −6− d⎜ ⎟ ⎝ ⎠ f ⎝ a ⎠ ⎥⎦ 2 ⎜

(15)

The third term in the square brackets emerges from surface effects while the first two are consistent with the infinite random network model.32 By analyzing Figure 3 and eq 15, we see that for small clusters, as we increment Rg, both the particle density and the mean internal coordination number zint decrease and so the shear modulus also decreases. Since both the first and the third terms in the square brackets do contribute, this decrease does not follow a simple power law. Once Rg reaches a critical value, zint becomes size independent. Hence beyond this critical value, only the surface term in the square brackets is affected by any further increment of Rg. Consequently, over a limited range, an increase in Rg diminishes the negative surface term to such an extent that, despite the declining aggregate density, the shear modulus Gnas actually rises with Rg. This behavior gives rise to a minimum in Gnas as a function of the aggregate size which is visible in Figure 3. It may be useful to recall, here, that the surface term is negative for the nonaffine surface model because the surface particles contribute a much decreased coordination number, while they still undergo nonaffine relaxations which cost energy. For higher Rg, the surface contribution becomes negligible and Gnas decreases as a linear function of the particle density n ∝ d −3 Rgf . As we will show below, this nonmonotonic dependence is a peculiarity of the nonaffine surface model, because the surface term is negative here. It does not arise in the affine surface 9112

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coordination number in the regime of smaller clusters, followed by a more gradual decay for larger clusters where the stabilizing effect of the surface gets smeared. Also in this case, the initial steeper decay, at high df, of the modulus upon increasing the linear size is due to the steep decay of density close to the surface which in turn causes the steep decay of zint that we already discussed with reference to Figure 1. Now we are going to investigate the impact the fractal dimension has on the aggregate shear modulus. Analyzing Figure 5 and eq 15, we can see that the increase in Gnas as a

Figure 3. Plot of the shear modulus as a function of the normalized radius of gyration Rg/a, in the limit of fully nonaffine cluster. From top to bottom df = 3/2.9/2.8/2.7/2.6/2.5/2.4/2.3/2.2. Here β is set as 0.2α. α is calculated for a = 45 nm, σhc = 2.08 × 10−10 m (hard-core radius in the Born hard-core short-range repulsion31), T = 298.15 K and A = 1.33 × 10−20 J (Hamaker constant). These same parameter values will be used in plots for the remainder of the paper.

model where the surface term always contributes positively, as the surface particles do not undergo nonaffine relaxation thus “saving” energy. In considering Figure 3, it is important to recall that Gnas never truly vanishes as we extrapolate to infinite size, unless zint falls below the critical rigidity threshold zint = 2.4. The case for Gas proceeds in a similar way. Applying the same treatment, we obtain −1 ⎡ 5z ⎛ 5z − 13 ⎞ ⎛ R g ⎞ ⎤⎥ ⎟d ⎜ Gas ∝ R gd f − 3⎢ int − 6 − ⎜ int ⎟ ⎝ ⎠ f ⎝ a ⎠ ⎥⎦ ⎢⎣ 2 2

Figure 5. Plot of the shear modulus as a function of the fractal dimension in the limit of the fully nonaffine cluster. From top to bottom Rg/a = 5/8/11/14/17/20/23/26/29.

(16)

function of df is driven by corresponding increments in both n and zint. In particular, the interplay between the rising overall density and the surface contribution gives rise to two distinct regimes. In the first steeper increasing regime at smaller df, the increase with df results from a combination of the overall density factor Rdgf−3 and zint. Upon approaching the limit df = 3, the increasing contribution due to the surface term which is proportional to df becomes comparatively more important and melds into a less steep increase. Similar qualitative features can be identified in the case of a fully affine outer shell in Figure 6, although here the evolution of the curves as a function of the aggregate size is more gradual, due to the modest magnitude of the surface contribution discussed above. C. Behavior of Isolated Aggregates. All instances investigated so far have the mass of the aggregate (that is, the number of particles making an aggregate) increase as we increase the dependent variables (the abscissas) of the above plots, such as Rg or df. In other words, in all of the plots considered above, the number N of particles forming an aggregate always increases along the abscissa of a given plot. This would physically represent the most common and interesting situation where the aggregate size grows due to aggregation processes. However, there are also instances in which further aggregation and growth are suppressed, for example, upon diluting the colloidal suspension with a large amount of dispersing solvent, and one is interested in the behavior of the preformed isolated aggregates at the time at which further aggregation is suppressed. To study this situation in terms of the shear modulus dependence upon df within our framework, it suffices to express everything as a function of N by means of the relation

The important qualitative difference with respect to the nonaffine surface model lies in the fully monotonic trend with the absence of the regime of increasing shear modulus, where the decay in the surface term dominates over the decreasing number density. This is not possible with fully affine bonds on the surface because the surface term is always positive, i.e., −(5zint − 13) > 0, due to the stabilizing effect of affine displacements on the surface with the associated saving of energy. As a result, the slope of the curves in Figure 4 can be explained by a declining particle density and mean internal

Figure 4. Plot of the shear modulus as a function of the normalized radius of gyration Rg/a, in the limit of fully affine surface particles. From top to bottom df = 3/2.9/2.8/2.7/2.6/2.5/2.4/2.3/2.2. 9113

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⎡

1 − 3/ d f ⎢ 5z int

Gnas ∝ N

⎢⎣ 2

−1/ d f ⎤ ⎛ 5z int − 7 ⎞ ⎛ N ⎞ ⎥ ⎜ ⎟ −6− d⎜ ⎟ ⎝ ⎠ f ⎝ k0 ⎠ ⎥⎦ 2

(19)

Therefore as df increases, the modulus increases due to the increase of the prefactor N1−3/df. Furthermore, since the surface term is negative, the rise approaches the limit of a power law at higher df and for curves at smaller N due to the increased importance of the surface term, similar to Figure 5. The variation of the shear modulus of clusters with fully affine outer shell (affine-surface model) as a function of the fractal dimension, is qualitatively similar to the case of the fully nonaffine surface model above, and is expressed by the following relationship, −1/ d f ⎤ ⎡ ⎛ 5z − 13 ⎞ ⎛ N ⎞ 5z ⎥ ⎟d ⎜ Gas ∝ N1 − 3/ d f ⎢ int − 6 − ⎜ int ⎟ ⎝ ⎠ f ⎝ k0 ⎠ ⎢⎣ 2 ⎥⎦ 2

Figure 6. Plot of the shear modulus as a function of the fractal dimension in the limit of fully affine surface particles. From top to bottom Rg/a = 5/8/11/14/17/20/23/26/29/100.

(20)

R g /a = (N /k 0)1/ d f

The only difference of note is the sign of the contribution of the surface term, which is a positive contribution, whereas it is negative for the nonaffine-surface model, and therefore it leads here to quantitatively larger shear moduli in Figure 9 than in Figure 8.

(17)

With this relation, we can rewrite eq 5 in the form z int = 0.532 + 0.776df + 0.000192df

6.462d f

⎛ N ⎞5.654/ d f − 3.665 ⎜ ⎟ ⎝ k0 ⎠

(18)

In this way we can calculate the mean internal coordination number and the shear modulus as a function of df under the constraint N = const. In Figure 7 we plot the evolution of the

Figure 8. Plot of the shear modulus as a function of the fractal dimension under constant N, in the limit of the fully nonaffine cluster. From top to bottom N = 500/625/70/875/1000.

D. Aggregate Stability Diagram and Restructuring under Shear. The rigidity threshold for an infinite random network occurs at z = 2.4.22 Things however can be very different for finite-size aggregates where the density-decay close to the surface greatly alters the behavior with respect to an infinite-size amorphous solid. The contribution of the outermost shell to the overall aggregate shear modulus may be either positive or negative, depending on whether the surface particles displacements are fully affine or fully nonaffine, respectively. In both cases, however, the surface effects are more important in the regime of smaller clusters, where they cause a steeper dependence of the shear modulus on the structural parameters Rg/a and df. These effects get smeared out, instead, in the regime of large aggregates, where one recovers the limiting behavior of bulk amorphous solids in the Rg/a → ∞ limit. We shall now be able to predict stability diagrams for fractal aggregates in shear as a function of the geometric parameter df and of the aggregate size, the latter either in terms of Rg/a or N.

Figure 7. Plot of the mean internal coordination number zint as a function of the fractal dimension df with constant N. From top to bottom: N = 1000/875/750/625/500.

internal coordination number upon varying df at N = const. We can see that the smaller N is, the smaller the average density n becomes, and hence also zint decreases; this plot shares the same qualitative features with Figure 2. Investigating the shear modulus of the nonaffine-surface model with N = const, we rearrange eq 13 and apply eq 17 to retrieve the relation 9114

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Figure 9. Plot of the shear modulus as a function of the fractal dimension under constant N, in the limit of fully affine surface particles. From top to bottom N = 500/625/70/875/1000.

It is important to recall that the aggregate size represents here the aggregation coordinate, in view of its intimate connection with the aggregation time (the average aggregate size is typically an increasing function of the aggregation time). The rigidity threshold, according to rigidity percolation theory, is obtained by setting G = 0.22 Using eq 15, for the nonaffine-surface model, we obtain the following condition for the rigidity threshold −1 ⎛ 5z − 7 ⎞ ⎛ R g ⎞ 5z int ⎟d ⎜ − 6 − ⎜ int =0 ⎟ ⎝ ⎠ f⎝ a ⎠ 2 2

Figure 10. Plot of the fraction of floppy modes f as a function of the fractal dimension df and the normalized radius of gyration Rg/a, in the limit of the fully nonaffine cluster. From top to bottom f = 0/0.02/ 0.04/0.06/0.08/0.1/0.12. In the white region the cluster is rigid, and the number of floppy modes is zero. In the shaded region the cluster is unstable, and the number of floppy modes is finite.

(21)

Upon rearranging we get a self-consistent expression for the critical values of df which separates unstable aggregates, which possess floppy modes (such that f > 0) and immediately restructure toward larger df, from stable aggregates with f = 0 which can survive without restructuring, df =

R g ⎡ 5z int(df ) − 12 ⎤ ⎥ ⎢ a ⎣ 5z int(df ) − 7 ⎦

(22)

Here the notation zint(df) should remind the reader that zint depends on df through eq 5. The same treatment applied to the affine-surface model leads to the following expression df =

R g ⎡ 5z int(df ) − 12 ⎤ ⎥ ⎢ a ⎣ 5z int(df ) − 13 ⎦

(23)

In Figure 10 and Figure 11 we plot the rigidity maps, and several curves for different fractions of floppy modes in the unstable region, for both the nonaffine-surface model and affine-surface model, in terms of the evolution of df with the aggregate size. The shaded region in the diagram represents the unstable region where aggregates are floppy and undergo restructuring or breakup until they enter the stable region. The f = 0 curves in the above plot separates the stable region where aggregates can survive without undergoing immediate restructuring, from the unstable region where large deformations can be excited with zero energy. Therefore, the aggregate under shear will restructure by increasing its df until it reaches the stable region of the diagram. In other words, the f = 0 curve in this diagram predicts the minimum value of df which can be observed in shear-induced aggregation and how this evolves with the aggregation process, that is, as the aggregate grows further.

Figure 11. Plot of the fraction of floppy modes f as a function of the fractal dimension df and the normalized radius of gyration Rg/a, in the limit of fully affine surface particles. From top to bottom f = 0/0.02/ 0.04/0.06/0.08/0.1/0.12. In the white region the cluster is rigid and the number of floppy modes is zero. In the shaded region the cluster is unstable and the number of floppy modes is finite.

The f = 0 curve reaches a plateau in the limit of an infinite aggregate size or equivalently, at the steady-state of an aggregation process where no further growth occurs, and the final average aggregate size is large, Rg/a ≫ 1. In this limit, both the last nonlinear term on the right hand side of eq 5 and the 9115

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Using the results of the present work, in combination with our previous theory of breakup rates,11 we can now disentangle the interplay between restructuring and breakup. We can predict the regions in parameter space where clusters are floppy (and restructure into higher df structures) but do not breakup, as well as those regions where clusters are no longer floppy but instantaneously break up into smaller fragments without restructuring. Also, we can predict those regions in parameter space where aggregates are stable with respect to both restructuring and fast-breakup events, although they are still subject to stochastic breakup events on longer time scales. In Figures 12 and 13 we have incorporated breakup stability curves into the stability maps of Figure 10 and 11, to identify

third (surface) term on the left hand side of eq 21 can be neglected, and eq 21 can be rewritten in this limit as 5(0.532 + 0.776df ) −6=0 2

(24)

Hence we obtain

lim df = 2.407

N →∞

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This is a key result of this work. It stipulates that in the steadystate shear-induced aggregation of colloids, no large aggregates with df < 2.4 can ever survive under shear without restructuring into a more compact morphology with df > 2.4. Remarkably, this is the first theoretical argument which can consistently explain why all experimental and computational studies of colloidal aggregation in shear have consistently reported aggregate structures at steady-state with df > 2.4.7−10,12−14 With the affine-surface model, which is probably a more realistic description of shear-induced colloidal aggregation, the same limiting df is obtained, which is expected because, in the infinitely large aggregate limit, differences related to the surface behavior, such as the affine versus nonaffine surface, obviously must vanish. It is also interesting, to discuss the evolution of the minimally stable df as a function of the growth process, with respect to Figures 10−13. In all cases, it can be seen that df increases steeply in the initial phase of the growth, which reflects the need for the aggregate to restructure toward denser configurations with higher zint in order to compensate the steep decay of zint with the growing linear size in the range Rg/a = 10−15, that was already discussed in Figure 1 and attributed to the importance of the rapidly decaying coordination number close to the aggregate surface. Upon growing further, these surface effects get smeared out and the initially steep rise melds smoothly into a plateau. Before reaching the plateau, in the nonaffine-surface model a pronounced overshoot is observed, which is absent in the affine-surface model. This overshoot, as already was the case for the minimum in Gnas in Figure 3, is due to the competition between the initial need to raise df in order to compensate the initial steep decay of zint with the growing linear size, and the decreasing impact of deformation energy lost to nonaffinity on the surface as the aggregate grows. Since, however, this curve represents a lower bound to df, the decreasing part of the overshoot might not be observable in practice. E. Interplay of Cluster Restructuring and Breakup. In a typical shear-induced aggregation experiment, a colloidal suspension or sol is subjected to shear flow at t = 0 and the particles start to aggregate into growing clusters, and of course also clusters of any size can aggregate with each other, under the action of shear forces.33 In another type of experiment, preformed clusters suspended in the liquid can be subjected to shear to study their restructuring and breakup behavior in the absence of further growth (if one makes sure that aggregation between clusters is hindered by some colloidal stabilization mechanism).9,12 In these experiments, clearly there is an interplay between the shear-induced restructuring process, leading to increasingly higher df values, and cluster breakup processes, whose rate and extent crucially depend on df.11 Cluster-breakup studies are typically done with clusters having a steady-state df value reached after the restructuring process,9 and they allowed us to rationalize the dependence of the breakup rate on df.

Figure 12. Plot of the rigidity threshold and shear-induced breakup curves as a function of the fractal dimension df and the normalized radius of gyration Rg/a, in the limit of fully affine surface particles. From left to right the shear rate γ̇ equals 107/106/105/104/103 s−1.

Figure 13. Plot of the rigidity threshold and shear-induced breakup curves as a function of the fractal dimension df and the normalized radius of gyration Rg/a, in the limit of fully nonaffine cluster. From left to right the shear rate γ̇ equals 107/106/105/104/103 s−1.

the parameter space where mechanically stable clusters can be formed in a shear flow without undergoing either restructuring or fast breakup. The additional solid lines refer to maximum stable aggregate size values, for different df and shear rates values, calculated using our previous theory, namely using eq 11 of ref 11, where the potential barrier to breakup is set to zero to solve for the maximum stable size Rg/a under a given shear rate and df, and for a standard DLVO potential. Details of this calculation can be found in ref 11. Importantly, to the left of 9116

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developed, steady-state, colloidal aggregates always display a fractal dimension of above 2.4. Since it is known that the final fractal dimension reached is independent of the shear rate, it is reasonable to approximate that the fractal dimension evolves with the lower bound calculated here, although this aspect has to be clarified in depth by future studies. Thus, combining with previously derived expressions for the flow-induced aggregation and breakup rates, a population balance equation can be solved to predict the evolution of aggregates in shear over time. Finally, we combined our model for restructuring with our previously published microscopic theory of aggregate breakup reported in Conchuir and Zaccone.11 This calculation leads to new aggregate-stability maps which allow us to discriminate regions in parameters space (df, size, shear rate) where aggregates are fully stable to both restructuring and breakup, from regions where aggregates restructure only, but do not break up, and from other regions where breakup is faster than restructuring. These results are in good overall qualitative agreement with experiments of ref 8. We believe that this is the first analytical framework which allows one to disentangle the interplay between restructuring and breakup in a typical shearinduced colloidal aggregation experiment. As such this approach will prove useful to experimentalists working in the field.

these lines breakup occurs with an energy barrier, hence it is an activated process and it occurs with a finite lag-time. To the right of these lines, instead, breakup is fast or instantaneous because it occurs with zero energy barrier. In Figures 12 and 13, stable aggregates (with zero floppy modes) are confined to the white region, where the cluster is rigid and will not restructure, and to the left of the breakup curves beyond which they would break apart instantaneously. Applying varying shear rates, we can see that, early in its growth evolution, a cluster will deform and restructure rather than break apart. This allows for an initial stage of rapid cluster growth as found in time-resolved colloidal aggregation experiments.33 As the cluster develops and grows further with an increasingly higher df, the requirement for significant restructuring and densification becomes less stringent, especially for fully nonaffine-surface clusters. Eventually the cluster evolves to a narrow range of maximum stable sizes bounded by the breakup curve. The greater the external shear rate that is applied, the smaller this maximum stable cluster size will be, and the shorter the time is at which it will be reached in a typical aggregation experiment.

IV. COMPARISON WITH EXPERIMENTS Our theoretical analysis provides a lower bound to the fractal dimension of fully developed colloidal aggregates in shear flows and clarifies that only aggregates with df > 2.4 can be observed in that limit. Aggregates with smaller fractal dimension, instead, would be floppy and thus possess a finite fraction of soft modes, that is, collective sets of interparticle displacements which can be excited with zero external energy. Since under shear flow the external energy which is inserted into the system is finite, these collective soft modes can be excited leading to large deformations and restructuring of the aggregate into more compact configurations with higher df. This result is consistent with experimental simulation data in the literature.7−10,12−14 Also, the evolution of df during the aggregation process predicted by our model, has been observed in ref 8.

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APPENDIX

A. Stokesian Dynamic Simulations

Aggregate structures with variable df were first generated using the algorithm described in ref 15. The resulting aggregates were then implemented in the Stokesian dynamics simulations at low shear, and the final structures were recorded from which zt was measured. We have performed simulations to study the dynamics of single colloidal fractal aggregates using Stokesian dynamics34 to estimate the hydrodynamic interactions between particles, coupled to attractive van der Waals and repulsive short-range Born interactions from the DLVO theory,26,35 and tangential contact forces through the discrete element method (DEM).36 The Stokesian dynamics model in resistance matrix form is expressed as follows:37

V. CONCLUSION We have developed a first-principles analytical theory of fractal aggregate stability and restructuring in shear flows. First, Stokesian Dynamics simulation data were analyzed to extract a phenomenological expression for the aggregate mean internal coordination number. Then, utilizing nonaffine elastic response theory, an analytical equation for the shear modulus of a fractal aggregate was obtained. We investigated this quantity with fully affine and fully nonaffine boundary conditions imposed on the aggregate surface, through the intrinsic structural parameter, the fractal dimension, and the aggregate size. It is shown that the nonaffine displacement of particles on the surface creates a minimum in the evolution of the shear modulus, as the aggregate grows. In addition, the rapid decay in the density profile at the interface hinders the rise of the shear modulus during the restructuring process. Then the rigidity threshold for aggregates, below which unstable deformation modes trigger large scale restructuring under shear, was derived. This represents the minimum fractal dimension an aggregate must reconfigure to, throughout its evolution, in order to survive. We find that while surface effects dominate for smaller clusters, a limit of df = 2.407 is recovered for large aggregates. This microscopic, analytical theory provides the first physical explanation to the well-documented observation that fully

⎡ Fn̅ ⎤ ⎡ Un̅ − U̅ ∞ ⎤ ⎢ ⎥ ⎥ ⎢ ⎢Tn̅ ⎥ = R·⎢ Ω̅ − Ω̅ ∞⎥ n ⎢ ⎥ ⎥ ⎢ ⎢⎣ Sn ⎥⎦ ⎢⎣ −E ∞ ⎥⎦

(26)

Here R stands for the grand resistance matrix, which is a 11N × 11N symmetric, positive definite matrix. The resistance matrix should include both multibody far-field and lubrication interactions. This has been achieved by approximating R in the following manner: lub R = (M ∞)−1 + R 2B − R∞ 2B

(27)

where Rlub 2B is the resistance matrix constructed by combining in a pair-wise fashion the two-bodies lubrication contributions for ∞ is the resistance matrix all pairs of particles, while R2B constructed by combining in a pair-wise fashion the two-bodies far-field resistance interactions for all pairs of particles The grand resistance matrix in submatrix form can also be expressed as follows: 9117

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Industrial & Engineering Chemistry Research ⎡ R FU R FΩ R FE ⎤ ⎢ ⎥ R = ⎢ R TU R TΩ R TE ⎥ ⎢ ⎥ ⎣ R SU R SΩ R SE ⎦

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ACKNOWLEDGMENTS We would like to thank the Winton Programme (B.Ó .C.), the Ernest Oppenheimer Fellowship (A.Z.), and the Swiss National Science Foundation (M.L.) for supporting this work.

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We have developed a cluster library using different Monte Carlo algorithms,15,38 consisting of clusters composed of different numbers of uniform sized spheres, spanning a broad range of fractal dimension values. We found the range of applied shear rates of 25 000 to 100 000 1/s suitable to study in reasonable time the complete breakup dynamics of aggregates with a particle radius of 1 micrometer. Every simulation started by randomly choosing one cluster from our cluster library with a defined cluster mass and fractal dimension. To start with an energetically stable structure, each cluster was equilibrated. Since cluster structures were obtained from Monte Carlo simulations, the distances between particles at contact are defined within a small tolerance, but without accounting for inter-particle interactions. However, this tolerance, which could even result in a small overlapping between two particles in contact, has a strong effect on the inter-particle interactions. The equilibration has been carried out under the same conditions as the simulations, except for the absence of flow, until inter-particle distances were unchanged. Only minimal particle displacements after equilibration were observed, without any effect on the overall structure of the original cluster. After the equilibration step we started with the actual simulation, wherein by knowing the relative positions of all particles the grand resistance matrix and the interparticle interactions could be computed. The interparticle forces were then summed and added to the flow field at the particle center, to obtain the net force on each particle, which would ultimately govern whether or not a given bond would break. Thus the particle velocities can be computed from the equation, −1

∞

Un̅ − U̅ ∞ = R FU[F ̅ + R FE: E ]

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The translational velocity is then integrated using the RungeKutta 4th order method to estimate consequent displacements of each particle. The critical distance up to which two particles were still considered to be connected has been set to 2 nm (for particles with a radius of 1 micrometer), which is where the value of total potential dropped to 1/4th and the attractive force to 2% of their values at the minimum of the potential well. Beyond this distance interparticles interactions are negligible and the particles are no longer bound to one another. At each time step the interparticle distance between each pair of particles was calculated, enabling us to modify a matrix of particle contacts, in which we record the formation of new bonds as well as the maintaining or breaking apart of old ones. In addition, the number of fragments produced and the size of each fragment were calculated after each time increment. All the simulations were performed until a steady-state was reached by monitoring the evolution of the average fragment radius of gyration normalized by the radius of the particle. For each cluster fractal dimension 10 different cluster realizations were simulated to obtain statistically representative results.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 9118

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