Modeling the Mechanism of Coagulum Formation in Dispersions

Transcription

1 ubs.acs.org/langmuir Terms of Use Modeling the Mechanism of Coagulum Formation in Disersions Martin Kroua, Michal Vonka, and Juraj Kosek* Deartment of Chemical Engineering, Institute of Chemical Technology Prague, Technicka 5, 1668 Prague 6, Czech Reublic *W Web-Enhanced Feature *S Suorting Information Downloaded via on January 7, 019 at 04:54:49 (UTC). See htts://ubs.acs.org/sharingguidelines for otions on how to legitimately share ublished articles. ABSTRACT: The stability of colloidal disersions is of crucial imortance because the roerties of disersions are strongly affected by the degree of coagulation. Whereas the coagulation kinetics for quiescent (i.e., nonstirred) and diluted systems is wellestablished, the behavior of concentrated disersions subjected to shear is still not fully understood. We emloy the discrete element method (DEM) for the simulation of coagulation of concentrated colloidal disersions. Normal forces between interacting articles are described by a combination of the Derjaguin, Landau, Verwey, and Overbeek (DLVO) and Johnson, Kendall, and Roberts (JKR) theories. We show that, in accordance with the exectations, the coagulation behavior deends strongly on the article volume fraction, the surface otential, and the shear rate. Moreover, we demonstrate that the doublet formation rate is insufficient for the descrition of the coagulation kinetics and that the detailed DEM model is able to exlain the autocatalytic nature of the coagulation of stabilized disersions subjected to shear. With no adjustable arameters we are able to rovide semiquantitative redictions of the coagulation behavior in the high-shear regions for a broad range of article volume fractions. The results obtained using the DEM model can rovide valuable guidelines for the oeration of industrial disersion rocesses. INTRODUCTION The roduction of many olymers often involves stages in which the monomer, the olymer, or the intermediate roduct is disersed into small articles. This is the case of all olymers roduced by susension or emulsion olymerization. These techniques are undoubtedly advantageous because they rovide much easier control of the olymerization and also rocess safety in terms of a thermal runaway. However, the fact that the system is in the form of a disersion brings along some difficulties. These are mainly related to the colloidal stability and thus the rocess of coagulation and fouling (which deending on the situation can be desired or not). Colloidal systems contain usually two immiscible substances, one forming the continuous hase and the other the disersed hase. These systems are usually thermodynamically unstable, but so-called kinetic stability can be achieved by several means, such as the electrostatic reulsion due to the charge on the surface of colloidal articles, which creates an electrostatic barrier reventing the articles from coagulating. Although this method sufficiently stabilizes colloidal systems under quiescent conditions (i.e., with no stirring), for systems subjected to shear one has to carefully check whether the energy barrier between articles is sufficiently high to revent coagulation. The roblem of colloidal stability under high-shear conditions is even more comlex because collisions of articles are strongly affected by the flow attern, which might vary significantly throughout the rocess, esecially for turbulent flows. The behavior of the colloidal mixture is influenced by forces acting on each single article. These may arise from the interaction of the article with the surrounding fluid, interactions among articles, or article-wall interactions. For the latter two tyes of interactions, there are several models describing the forces and torques. The normal article-article interactions reresent the basis of the descrition and can generally be divided into two arts: (i) noncontact forces and (ii) contact forces and torques. The most widely used aroach for the descrition of noncontact forces of small sheres in electrolytes is the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory. 1 It combines the attractive van der Waals and reulsive electrostatic forces. On the other hand, the contact of adhesive elastic sheres is commonly described by the Johnson, Kendall, and Roberts (JKR) theory roosed by Johnson et al. in The roblem of the descrition of the normal interaction between two articles arises when one wants to satisfactorily describe the interaction force for the entire range of searation distances. In 1998, Hong 3 resented a way to overcome this issue by taking the maximum adhesion force from the JKR theory as a limiting value for the DLVO theory, which would otherwise rovide infinitely high adhesion force for very small searation distances. We adoted this idea and further develoed it to obtain the continuous force distance rofile for the whole range of searation distances. Mathematical modeling is a way of getting better insight into the dynamics of disersion systems. Because of the discrete nature of colloids, it is convenient to use the discrete element method (DEM), which is emloyed also in this work. The main Received: January 9, 014 Revised: February 18, 014 Published: February 4, American Chemical Society 693 dx.doi.org/10.101/la500101x Langmuir 014, 30,

2 advantage of the DEM lies in the direct incororation of the interarticle forces and torques into the model. In 007, Marshall used DEM models to simulate aerosol articles in a channel flow. 4 He concluded that for this nonstabilized system, the channel fouling is strongly deendent on the adhesion forces coming from both article-article and article-wall interactions. This means that the channel with more adhesive articles is more likely to end u with large agglomerates in the flow as well as clusters attached to the wall. Shear-induced aggregation of charge-stabilized colloids can be described as an activated rocess. 5 The activation energy is given by the balance between the height of the otential barrier (U max ) and the energy of the articles obtained from the flow. The ratio of the energy of the articles from shear to the thermal energy is described by the Pećlet number (Pe 3πR 3 ηg/(k B T), where R, η, G, k B, and T are the article radius, fluid dynamic viscosity, shear rate, Boltzmann constant, and absolute temerature, resectively). For the system with an energy barrier between the articles, the critical Peclet number can be exressed as 5,6 Umax Pec αkt B (1) where α is the flow-secific coefficient. The quantity Pe c can be viewed as the energy of the advective motion necessary to overcome the energy barrier between the articles. For values of Pe below the critical value Pe c, the system should be stable, while for suercritical values aggregation occurs. It is therefore the interlay between the stabilization and the shearing that defines the behavior of the system. The above-mentioned aroach has one limitation. It is not able to cature the effect of larger numbers of articles contributing to the energy of the collision. In other words, for more crowded systems, the critical Pećlet number should deend on the volume fraction of articles (ϕ). This henomenon was investigated by Zaccone et al. 7 in 010, who used the generalized Kramer s rate theory. This statistical aroach is able to describe the deendence of the characteristic coagulation time (t c ) on the volume fraction of the articles. This deendence is added to the model by means of the susension viscosity (η s ), which is a function of ϕ, and this is known as the effective medium aroach. The rate constant of doublet formation (k 1,1 ) in the turbulent regime was measured exerimentally by Sugimoto et al. 8 in 014. Comared with our simulations they reorted higher values of the doublet formation rate, but the conditions of the measurements were very different. In 013, Soos et al. 9 investigated the scaling of the maximum stable aggregate size (R g ) with the maximum shear rate (G max ). From the results of breakage exeriments in a stirred tank and contracting nozzle the value of R g was determined, while the values of G max for the resective geometries were obtained using comutational fluid dynamics simulations. Soos et al. reorted that R g follows the same scaling with the maximum shear rate indeendent of whether the flow attern is laminar (contracting nozzle) or turbulent (stirred tank). Another imortant result of this work is the finding that the maximum value of the hydrodynamic stress (related to G max ) is resonsible for the breakage of the aggregates (i.e., not the usually used average value). In this aer, we roose a new aroach for the modeling of colloidal stability. Using the DEM we can incororate all of the necessary forces and torques. Furthermore, because of the dynamic nature of the model we do not have to restrict ourselves to the statistical descrition. Our model directly accounts for several henomena, such as the crowding or stabilization of the articles, without simlifications that are often used. 5,7,10 Having articles immersed in water hase, we investigate a different case than Marshall 4 in this aer. The electrostatic double layer (EDL) is formed on the surface of the articles as a result of their charge, and this imoses an energy barrier. Also, the shear forces are more ronounced because of the higher dynamic viscosity of water comared with air. Moreover, articles in emulsion olymerization are soft because of their swelling by (co)monomers. Therefore, we utilize the descrition of the interarticle forces to account for all of these henomena. The disadvantage of our aroach lies in the necessity of comuting with a small integration ste, which imoses high demand for comutational ower. The comutational feasibility can be imroved by otimization of the algorithms and roer design of the simulations. The main result resented in this aer is the demonstration of the strong deendence of the coagulation kinetics on the article volume fraction, the surface otential, and the shear rate. We also show that the doublet formation rate is an imortant but not sufficient measure of the coagulation rate, since at some oint larger aggregates are formed and the rocess becomes autocatalytic. All of these results are obtained with no fitting arameters (i.e., material roerties and hysical constants are the only inuts into the model). STRUCTURE OF THE MATHEMATICAL MODEL The DEM is alicable for the descrition of either naturally discrete articles or continuous materials discretized into discrete elements. Once defined, the discrete elements (labeled i) are characterized by their masses (m i ), ositions (x i ), velocities (v i ), and rotation rates (Ω i ). In this work, for the sake of simlicity, the elements are assumed to be sherical with density ρ and radius R, which are the same for all of the discrete elements. The article trajectories are governed by the well-known Newton s equation of motion: d xi Fi dt mi () where F i reresents the sum of all forces acting on the discrete element i. The balance of angular momentum (L i ) for each article can be exressed in the following form: dω i M i dt Ii (3) where Ω i L i /I i is the rotation rate of article i, M i is the sum of all of the torques acting on article i, and I i is the article momentum of inertia. For a homogeneous solid shere, we have I i (/5)m i R (article moment of inertia around its center) and m i 4/3πR 3 ρ. Since we erform simulations in two satial dimensions, it is sufficient to introduce the rotation rate Ω i of article i only for rotation in the lane where the article motion occurs. The resulting direction of Ω i is then erendicular to this lane. Forces and torques acting on the article i can originate from (i) the interaction of the article with the surrounding fluid (suerscrit F), (ii) the article-article interactions (suerscrit -), or (iii) the article-wall interactions (suerscrit - w). These contributions are assumed to be additive, and the 694 dx.doi.org/10.101/la500101x Langmuir 014, 30,

3 resulting force (F i ) and torque (M i ) can be exressed as follows: F w i i i i F Mi Mi + M i. F F + F + F The moment of the article-wall interactions is not taken into account since a strong noncontact force is emloyed, thus avoiding the solid-body contact of the articles (cf. Interactions between the Particle and the Wall below). This defines (when initial conditions are sulied) a set of second-order ordinary differential equations that can be numerically integrated in time to obtain the dynamic behavior of the system. We used the solver LSODE from ODEPACK 11 to solve the set of differential equations. This solver uses the multiste Adams method with an adatable time ste. The external article trajectory samling time ste used for the recording of simulations was Δt s. However, the real time ste of the integration method was usually smaller, esecially when the roblem became stiff (during the solidbody contact of the articles). Structure of the Comutational Domain. The satially two-dimensional (D) comutational domain was chosen to be rectangular with dimensions L L/, and for most of the comutations we used L 1.5 μm. The velocity rofile in the domain was assumed to be linear. In our setu, the lower late stayed still and the uer late moved with velocity V. Therefore the resulting shear rate rofile was constant in the system, corresonding to the model of simle shear (the scheme of the comutational domain is shown in Figure S1 in the Suorting Information). The osition of zero velocity was at y L/ + R, because the force acting on the article was comuted from the osition of the article center, and thus, a article attached to the wall was not moved in the translational way by the flow but was still exosed to the torque caused by the flow. Particles were initially randomly laced into the domain in a way that avoided solid-body contact. The initial velocities and rotation rates of the articles were set to corresond to the fluid velocity and rotation at the location of the article. FORCES AND TORQUES IN COLLOIDS Colloidal systems are affected by a wide range of force and torque interactions. In the following text, we introduce only those relevant for our secific system. At first we describe the interaction of the articles with the surrounding fluid and then we introduce the equations for the article-article and articlewall interactions. Interactions of the Particles with the Flow. Particles immersed in the flowing fluid exerience a drag force and are also rotated by the fluid. For the colloidal articles investigated in the current aer, the Reynolds number is always much smaller than 1 because of their small inertia. Therefore, we can describe their interaction with the flow by the simle Stokes law. The drag force (F F i ) is then F F i F i i F 6 πη R ( v v ) where η F is the fluid dynamic viscosity, R is the radius of the article, and v i F and v i are the velocities of the fluid and the article, resectively. (4) (5) If the fluid surrounding the article rotates locally with angular velocity ω, the corresonding fluid torque (M i F )on article i is comuted as follows: 1 F 1 M πη ( R ) ω Ω i i F 3 (6) The velocity and angular velocity of the fluid are comuted from the linear velocity rofile (cf. Figure S1 in the Suorting Information). This rofile is a result of the system geometry, and the articles are assumed not to alter the rofile. Interactions between the Particles. The most imortant inut for the DEM model is the descrition of the normal forces between the articles. It is necessary to describe the interaction otential energy (or force) for the entire range of searation distances between articles. The most common descrition of the noncontact forces is the DLVO theory, while the contact forces can be described by the Born reulsion for hard articles 13 or the JKR theory for soft articles. 4 The most usual combinations are hard articles with noncontact forces and soft articles without noncontact forces. Our system is secific in the way that we deal with soft (swollen) adhesive articles immersed in water. Therefore, the model of normal interaction must be able to describe (i) the energy barrier between the articles due to the resence of the EDL, (ii) the attraction (adhesion) for short searation distances and during the solid-body contact, and (iii) the elastic reulsion of the soft articles. These henomena are searately described by the DLVO theory (noncontact interactions) and the JKR theory (contact mechanics), but their combination is not straightforward. 3 In the following art, we first introduce and then combine these two theories. Finally, the descrition of the tangential interactions is given. DLVO Theory. The DLVO theory is usually exressed in the form of interaction otential energy (U). The corresonding interaction force (F) is the negative of the derivative of U with resect to searation distance (h), that is, F du/dh. For the van der Waals attractive otential energy (U vdw ) between two sherical articles of the same radius (R ), we used the exression 1 U vdw A H 6 h R + 4hR + + R ln 1 + h R R ( h + R ) where the searation distance h is given by h u R, where u is the distance between the centers of the two articles. The quantity A H is the Hamaker constant, which characterizes the material of the articles and the surrounding medium. The electrostatic reulsive otential energy (U ele ) of two colloidal articles can be exressed as 1 ele 0 r 0 U πε ε R ψ ln[1 + ex( κh)] where ε 0 is the vacuum ermittivity, ε r is the relative ermittivity of the medium, ψ 0 is the surface otential of the articles (assumed to be constant in the model), and κ is the recirocal of the Debye length. The DLVO otential energy (U DLVO ) is obtained simly as the sum of the two reviously mentioned otential energies: (7) (8) 695 dx.doi.org/10.101/la500101x Langmuir 014, 30,

4 Figure 1. (a) Potential energy U and (b) force F as a function of the searation distance h. The values of the arameters are R 50 nm, A H J, ψ 0 40 mv, κ 1 1 nm, γ 3 mj m, E 40 MPa, and ν 0.. UDLVO UvdW + Uele (9) An examle of the DLVO otential energy curve and the scaling of the height of the otential barrier (U max ) with the model arameters are shown in Figures S and S3, resectively, in the Suorting Information. JKR Theory. The JKR theory describes the interaction of adhesive, elastic solid bodies. The adhesive force in JKR theory is assumed to act only within the flattened contact region. The equations for JKR theory used in this work were roosed by Johnson et al. in 1971 and later slightly modified by Marshall in The equations are valid generally for two sheres i j with radii R and R and the so-called effective radius (R) defined as R R i R j i /(R + R j ). For contact of a shere with a j lane, the radius R becomes infinite, and R R i. When no external force acts on the articles and the force equilibrium is reached, we can define the equilibrium radius of the contact area (a 0 )as a 0 1/3 πγ 9 R E (10) where γ denotes the surface energy (to be discussed later) and E is the effective Young s modulus, given by 1 ν ν i j E E E i j (11) where E i and E j are the Young s moduli and ν i and ν j are the Poisson s ratios of the two interacting bodies. The equation for the (nonequilibrium) radius of the contact area (a) is imlicit: 1 1/3 δ a h 6 C a0 4 a 3 a 0 1/ (1) where δ C is the critical overla, given by the following equation: a0 δ C 1/3 (6) R (13) Finally, the magnitude of the normal force between two colliding articles (F ne ) can be obtained from the exression F F ne a a 4 4 a a C / where F C is the magnitude of the critical force, given by (14) FC 3πγR (15) F C corresonds to the maximal adhesive force that can occur between articles described by JKR theory. A tyical force distance curve for the JKR theory is shown in Figure S3 in the Suorting Information. The recise determination of F C is one of the imortant tasks in order to obtain reliable results from the model. The maximum adhesion force (cf. eq 15) is directly roortional to the surface energy of the articles (γ), which is exerimentally accessible and does not deend on the article size. We tried to obtain γ from exerimental studies that used an atomic force microscoe (AFM) for the direct measurement of the forcedistance curves using a so-called colloidal robe. 14 In these AFM measurements, a colloidal article is attached to the ti of the AFM cantilever, and the force between the ti article and the substrate is measured. In 00, Hodges et al. 15 measured the ull-off force between olystyrene articles in water as a function of the size of the articles. Unfortunately, these kinds of measurements usually rovide quite scattered results, and the construction of any trends from the ublished data is very difficult and somewhat ambiguous. 15 The theoretical value of γ for olystyrene in air is redicted to be γ theor 30mJm. 16 However, the resence of water as the surrounding fluid is known to reduce the surface energy by aroximately 1 order of magnitude.,15 Therefore, in this work we used the value γ 3mJm, which should corresond to smooth olystyrene articles immersed in water. The effect of surface roughness is known to be more imortant for large articles. 15 Thus, for our case of very small articles (R 50 nm), we assumed the article surface to be smooth. Another ossibility for obtaining the value of γ would be to use the value of the DLVO otential at a certain cutoff distance (D). 17,18 The limitation of this aroach lies in the determination of the cutoff distance, which is influenced by many factors, and the resulting value of γ is sensitive to small variations in D. For this reason, the value of γ described in the revious aragrah was chosen. Connection of the DLVO Theory and the JKR Theory. The revious aragrahs describe well the noncontact and contact interactions in searate theories. A roblem arises when one wants to describe the interaction force or otential energy for the entire interval of searation distances. The JKR theory redicts the maximum adhesion force between articles (F C )to occur at the limit radius of the contact area, 17 a lim 0.63a 0 (i.e., at the corresonding limit of the searation distance h; cf. eq 1). The value of F C can be used to constrain the DLVO theory, which is the aroach used by Hong in However, the searation distance h for which the maximum adhesive force F C occurs is redicted differently in the DLVO 696 dx.doi.org/10.101/la500101x Langmuir 014, 30,

5 and JKR theories. This inconsistency is caused by different assumtions taken into account for the derivation of the resective theories. The easiest way to overcome this roblem without introducing a discontinuity into the force-distance rofile is to cut the JKR curve at the oint where it crosses the DLVO curve. However, in that case we lose the determination of F C, which is an imortant characteristics of the system. On the other hand, when we are interested in the agglomeration behavior of the system, the exact osition (unlike the value) of the maximum adhesion force is not of crucial imortance. 14 Therefore, keeing in mind a certain arbitrariness, we decided to overcome the different redictions of the JKR and DLVO theories by shifting the JKR curve in the direction of the x (searation distance) axis in such a way that the maximum adhesion force occurs exactly at the crossing with the DLVO curve. The resulting otential energy and force rofiles are shown in Figure 1. We can see that the otential energy curve describes, aart from the otential barrier already mentioned before, a dee otential well, the so-called rimary minimum. Two aroaching articles at first exerience weak adhesion due to the secondary otential minimum (not aarent in Figure 1) and then reulsion due to the resence of EDL. If their energy is high enough, they overcome the barrier and are brought together by the strong adhesion. When the otential minimum is reached, they are revented from further aroach by elastic reulsion. This behavior, reresented in terms of force, is described by the force-distance curve in Figure 1b. Dissiation Force. The interaction between the articles is rarely urely elastic because some energy dissiation is likely to occur during the collision. 1 This feature can be introduced into the DEM by the addition of a dissiation force (F N d ). Let us first define v i c as the surface velocity of article i at the contact oint with article j: c i i i i v v + Ω r (16) where denotes the cross roduct of the vectors and r i R i n is the vector ointing from the center of the article i to the contact oint (assuming that n is the unit normal vector ointing from the center of article i to the center of article j). For article j, the corresonding vector is r j R j n. Then the relative velocity of the surfaces of articles i and j at the contact oint (v ij r )isdefined as follows: r c c ij i j v v v Now we can roceed to define the dissiation force as 1 F N d r N ij η v n (17) (18) in which η N is the daming coefficient, which can be written as η 1/ α ( mk ) N f N (19) where m is the mass of the article, k N is the normal stiffness coefficient (k N 4/3E R), and α f is the coefficient of normal friction. Tangential Interactions. There are several different tyes of tangential interactions of the articles. Since we restrict ourselves to two satial dimensions, the twisting of articles is not of concern because it involves rotation of the articles in directions irrelevant for a satially D model. In this work, we emloyed models for the resistance to sliding and rolling of articles. The formulas for these models are given in the Suorting Information. Interactions between the Particle and the Wall. The descrition of the article-wall interactions is comlex and involves most of the interaction tyes and theories mentioned before in the section about the article-article interactions. In this aer, we focused on the behavior of the colloidal articles in the bulk. Therefore, we decided to minimize the influence of the wall on the system behavior. Colloidal articles immersed in solvent are known to exerience the so-called hydration (or solvation) force. 17 This force arises from the interactions of the solvent molecules in the so-called hydration shell, but its nature is still not comletely revealed. 17 Nevertheless, it causes strong shortrange reulsion between the involved bodies. In terms of the interaction otential energy (U h ), this reulsion can be henomenologically described using the following exonential deendence: 19 U πrfδ ex( H/ δ ) h (0) where H is the article-wall searation distance, F 0 is the hydration force constant, and δ 0 is the characteristic decay length. We adoted the values of the constants used by Wu et al., 19 as they satisfactorily describe the behavior of the force for colloidal articles. The values of the constants used in our model for the article-wall interactions were F N m and δ m. COAGULATION KINETICS The rocess of coagulation can be characterized by various quantities, among which the most imortant are the characteristic time of coagulation (t c ) and the rate at which doublets are formed (r 1,1 ). To determine r 1,1, we define the number of rimary articles contained in doublets as the quantity n,. The rate of doublet formation can be obtained from the temoral develoment of n, by taking its first derivative with resect to time [i.e., r 1,1 1/(dn, /dt) because we defined n, as the number of rimary articles in doublets], and the doublet formation rate constant (k 1,1 ) can be obtained as follows: k 1,1 VT N r 1,1 (1) where V T is the total volume of the system and N is the number of articles. To eliminate the influence of the initial ositions of the articles, we decided to evaluate k 1,1 at the oint where the growth rate of n, reaches its maximum [i.e., the doublet formation rate used for the evaluation of k 1,1 was taken to be r max 1,1 1/ max(dn, /dt)]. It is worth noting that the determination of V T in our system is not straightforward. Since the articles are allowed to move only in two satial dimensions, the aroriate quantity to describe the system would be its total area (A T ). However, as we want to be consistent with the common units of k 1,1 (m 3 s 1 ), we introduce the following transformation. In three dimensions, the concentration of the articles can be characterized by the volume fraction of the articles, while in two dimensions the corresonding quantity is the area fraction of the articles. If we assume that these two are equal (i.e., the ratio between solid and voids is the same for D and 3D), we can obtain the V T from the following relation: V T VA T A () 697 dx.doi.org/10.101/la500101x Langmuir 014, 30,

6 where V 4/3πR 3 is the volume of the article and A πr is the rojected area of the article. RESULTS AND DISCUSSION The stability of colloidal systems is controlled by the interlay between several influencing factors, of which the most imortant are (i) the height of the otential barrier (U max ); (ii) the energy inuts, due to either Brownian motion or agitation of the system; and (iii) the article volume fraction. These factors determine the rocess of coagulation, which is the main toic of this aer. First we resent the results for article-article interactions that illustrate the caabilities of the model. Then we show the results of larger simulations dealing with the coagulation kinetics. Particle-Particle Interactions. To introduce the model behavior, we briefly resent the results of two-article simulations. If not stated otherwise, we used the values of arameters listed in Table 1. Table 1. Default Values of Model Parameters Used in the Simulations quantity value name R 50 nm article radius A H J Hamaker constant η F Pa s fluid dynamic viscosity T K absolute temerature E 40 MPa Young s modulus ν 0. Poisson s ratio ρ 1000 kg m 3 article density μ eff 0.5 effective friction coefficient ε r 80 relative ermittivity of water at T K ψ 0 30 mv surface otential of the articles κ 1 1 nm Debye length θ crit π/4 critical rolling angle γ 3mJm surface energy of the articles The roblem of the stability of colloidal systems always (even for nonsimlified systems) can be reduced to the interactions of rimary articles. For dilute systems (ϕ 1%), this assumtion is further develoed, and only two-article collisions are usually assumed to affect the system stability. 5 Two asects are imortant for the resulting stability: (i) the collision frequency and (ii) the collision efficiency. The first quantity can be statistically determined from the Smoluchowski equation, 0 at least for dilute systems. The collision efficiency for two-article collisions refers to whether the collision results in coagulation or in a rebound and can be exressed in terms of the critical Pećlet number (Pe c ). To determine Pe c for a system with two articles, we erformed simulations of their direct collision. For this simlified system, Pe is defined as follows: Pe r 1/ m( v n) kt B (3) which means the energy of the collision in the normal direction in units of k B T. The increase in Pe c with increasing article radius is shown in Figure. The critical value searates the region where articles rebound (Pe < Pe c ) from that where they coagulate (Pe > Pe c ). We have also lotted the values of Pe c redicted by eq 1, where the flow-secific coefficient α is equal to unity for this case (i.e., no flow in the system). We can see that the results of the dynamic model corresond almost Figure. Deendence of the critical Pećlet number (Pe c ) on the article radius for different values of the surface otential (ψ 0 ). The lines reresent the rediction of eq 1, and the oints are results of the two-article dynamic simulations. erfectly with the theoretical rediction, confirming the accuracy of our calculations. Dynamics of Coagulation. The model used in this work is based on airwise interactions between two articles (i.e., threearticle forces and torques are not taken into account). These airwise interactions can be summed u, thus allowing systems with a large number of articles to be studied. One of the big advantages of the DEM model is the ossibility directly observing the henomena taking lace in the dynamic system. We erformed simulations of the system described in the revious sections for a broad range of volume fractions of the articles (ϕ 0.19; 0.49 ). For these very concentrated mixtures, we resent the results of their dynamic behavior when exosed to the shear flow. Snashots from the simulation of the system consisting of 4000 rimary articles (ϕ 0.4) exosed to a shear rate (G) of s 1 at different hases of the aggregation rocess are shown in Figure 3. We can see that during the initial lag hase, only a small number of small aggregates is formed. The fast aggregation rocess starts when a larger aggregate is formed in the system. The large aggregate then very quickly coagulates with the remaining rimary articles to form dendritic structures. One large interconnected aggregate is resent at the end of the coagulation rocess. In Figure 4a we show the develoment of the number of rimary articles in aggregates (n ) with time. We can see that the curve exhibits a sigmoidal shae with a very slow increase at the beginning. After the lag hase, the onset of coagulum formation occurs, and n linearly increases with time. Finally, n reaches a lateau where all of the articles are contained in the aggregate. This shae (with certain variation) is tyical for all of the simulations we erformed with shear rates in the range from to s 1. The rocess of coagulation is often characterized by the rate at which doublets are formed. This is usually considered as the measure of the overall rate of coagulation. In Figure 4b we resent the number of articles contained in doublets (n, )asa function of time. The total number of articles in aggregates (n ) is also shown for comarison. For the urose of the logarithmic lot, these quantities are lotted as n, + 1 and n + 1. Initially some doublets are formed, but as the coagulation roceeds their number remains more or less constant, and in the later stages n, gradually decreases and reaches a value of zero at the end. Coagulation of rimary articles is an imortant characteristic of the system since doublet formation is the dominant 698 dx.doi.org/10.101/la500101x Langmuir 014, 30,

7 Langmuir Figure 3. Snashots from the simulation for the system with article volume fraction (ϕ) of 0.4 under a shear rate (G) of s 1. A movie showing this simulation is available in the HTML version of this aer. Figure 4. (a) Number of rimary articles in aggregates (n) vs time and (b) numbers of rimary articles in doublets (n + 1) and in all aggregates (n + 1) vs time for the system with ϕ 0.4 and G s 1. shear flow, we obtain the following relation (a result due to Smoluchowski): rocess in the initial hases of the coagulation.8 The rate constant for doublet formation (k1,1) is strongly deendent on both the shear rate G and the volume fraction of articles ϕ, as is aarent from the results of simulations resented in Figure 5. The rate constant k1,1 can be obtained from theory. Considering two equal-sized nonstabilized articles in simle t k1,1 3R 3G 3 (4) which for our system results in a kt1,1 value of m3 s 1 (rate constant accounting only for the frequency of the collisions). With the DEM model, we obtained k1,1 values aroximately 1 3 orders of magnitude lower (cf. Figure 5). Since the articles are stabilized by the EDL in our model, slower coagulation than for the nonstabilized case is exected. This serves as another justification of the results obtained with our model (also see Figure S5 in the Suorting Information). Considering the very broad range of k1,1 values occurring in real systems and the difficulties in their determination, the values obtained from our simulations can be considered as meaningful. It is aarent that the aggregation behavior of the stabilized system exosed to shear is an autocatalytic rocess, which was also observed exerimentally.5,7,1 This means that after the initial lag hase, the system coagulates quickly until all of the rimary articles are contained in aggregates. This henomenon can be exlained both in terms of the frequency and the Figure 5. Rate constant of doublet formation (k1,1) as a function of shear rate G for different article volume fractions ϕ. 699 dx.doi.org/10.101/la500101x Langmuir 014, 30,

8 Figure 6. Characteristic coagulation time (t c ) as a function of the alied shear rate G for different article volume fractions ϕ: (a) logarithmic lot; (b) double logarithmic lot (the lines are ower-law fits). efficiency of the collisions. The exlanation of the first one comes naturally from eq 4, if we substitute the radius R with the mean radius of two different sheres [(R i + R j )/]. Then, as one of the radii increases, the resulting coagulation rate t constant k 1,1 grows with the third ower of the substituted radius. The efficiency is increased as a result of the higher energy of the collisions. In other words, as a result of the greater momentum of the larger cluster, the resulting Pećlet number of the collision increases, while the critical Pećlet number remains unaltered (since the same interactions between rimary articles exist). This leads to an easier crossing of the otential barrier for the collision of a rimary article with a large cluster. Another imortant quantity for the characterization of the coagulation rocess is the characteristic coagulation time (t c ). The coagulation dynamics is tyically connected with a shar increase in the viscosity of the whole system, which makes viscosity (or imeller torque) measurements a convenient tool for the observation of such systems. In 006, Guery et al. 1 roosed a method to determine t c from rheological data. It is based on finding the linear trends in the evolution of viscosity with time in both the lag and growth hases. The intersection of the obtained lines then determines the value of t c. Since the temoral develoment of the number of articles in aggregates (n ) in our simulations exhibits a trend very similar to the trend of viscosity measured by Guery et al. 1 and Zaccone et al., 7 we erformed the aforementioned rocedure for the determination of t c (cf. Figure 4a). The results of this analysis are shown in Figure 6 as the decreasing deendence of t c on the shear rate G in logarithmic and double logarithmic lots. The inverse trend can be observed for the deendence of t c on ϕ (i.e., more concentrated systems coagulate faster). These trends are in agreement with the exerimental measurements of Zaccone et al., 7 who roosed an exonential scaling of t c with G. However, it is aarent from Figure 6b that for a broader range of G the deendence rather follows a ower-law scaling. It is interesting to comare the characteristic time of coagulation obtained from the global time develoment of n with the characteristic time of doublet formation (t c,1,1 ). Since the formation of doublets follows second-order kinetics, the characteristic time of this rocess can be obtained from the following relation: 1 tc,1,1 ck1,1 (5) where c is the initial number concentration of the articles. From the comarison of t c with t c,1,1 shown in Figure 7, it is Figure 7. Characteristic times of coagulation (t c, oen symbols) and doublet formation (t c,1,1, solid symbols) as functions of the shear rate G for different article volume fractions ϕ. aarent that the characteristic time of doublet formation is much larger than the characteristic time for the whole coagulation rocess. This is the case because the rocess of coagulation is driven by doublet formation only in the early stage; once a larger aggregate is formed, it grows and the coagulation roceeds much faster. The data from Figure 6 can also be lotted as the deendence of the characteristic coagulation time t c on the volume fraction of articles ϕ. We resent this grah in Figure 8 as an exonential deendence of t c on ϕ, which varies in the range between 0.19 and Through this interval, t c decreases almost by orders of magnitude. Similar behavior has been observed exerimentally but for a narrower range of article volume fractions. 7 This suggests a very strong concentration Figure 8. Characteristic time of coagulation t c as a function of article volume fraction ϕ for different shear rates (G/s 1 ). 700 dx.doi.org/10.101/la500101x Langmuir 014, 30,

9 deendence of the behavior of stabilized colloidal systems subjected to shear, which is a very imortant observation for ractical usage. Another imortant issue is the influence of the surface otential (ψ 0 ) on the behavior of the system. We erformed simulations for the values of ψ 0 /mv 50; 30, and the results of this arametric study are shown in Figure 9. It is Figure 9. Characteristic time of coagulation t c as a function of surface otential ψ 0 for different shear rates (G/s 1 ). aarent that t c decreases exonentially with decreasing absolute value of ψ 0, as exected because the value of the surface otential is known to influence the behavior of the colloids very strongly. 1 Variation of the surface energy (γ) roduced no significant effect on the doublet formation rate and characteristic coagulation time. It has a serious effect on the shae of the aggregates, but this toic is out of the scoe of this aer. A negligible effect on the resented results was observed also for the variation of the arameters of the hydration force (articlewall interaction). All of the resented results for the coagulation dynamics deal with systems subjected to high shear rates (G/s ; ). The modeled values of G are higher than in industrial rocesses, but the choice of large shear rates was necessary for the following reasons. The resented DEM model is based on the interarticle interactions, and the formulations of the forces and torques acting on the articles require only hysical constants and material roerties as arameters. The formulations are based on the hysical laws describing the henomena of double-layer reulsion, solid-body contact, etc. Such a detailed model, however, requires a very small integration time ste (Δt s). At the same time, a high number of rimary articles is needed to obtain statistically relevant results (e.g., for the doublet formation rate). The accessible time scale of the whole rocess is therefore limited by the comutational feasibility of the calculations. To be able to cature the dynamics of the coagulation rocess within the time scale of our model, we had to consider systems subjected to high shear rates (that are still exerimentally accessible 6 ). Most imortantly, even with the above-mentioned limitations, our model is able to cature the trends of the coagulation in diserse systems and to rovide semiquantitative redictions with no adjustable arameters. A ossible aroach for accessing the low-shear-rate region with our model could emloy an initial configuration with seeds (doublets or trilets) and consequently model the fast-growth stage of coagulation. Unfortunately, in that case we could easily lose the crucial information about the characteristic time of the overall coagulation rocess. CONCLUSIONS To the best of our knowledge, we have resented here the first discrete element method (DEM)-based model of concentrated, charge-stabilized diserse systems under shear flow in the liquid hase. Parametric studies of the coagulation considered the influence of the most imortant arameters, namely, the article volume fraction (ϕ), the surface otential (ψ 0 ), and the shear rate (G). We quantified the deendence of the coagulation behavior on all of these arameters. A general trend that a more concentrated and less stabilized system subjected to higher shear is more likely to undergo coagulation is in agreement with exerimental and theoretical studies. We found that the rate of doublet formation, a commonly used measure of the coagulation rate, is an imortant characteristic of the rocess. Moreover, the DEM model emloyed in this study enabled us to investigate the behavior of the coagulation more in detail, since it was ossible to track each single collision and determine the events leading to the coagulation. From these results it is aarent that the autocatalytic nature of the rocess is the key feature for its understanding. In articular, we have shown that the characteristic time of doublet formation is substantially higher (roughly by orders of magnitude) than the characteristic time of the coagulation itself. This suggests that for the correct descrition of the coagulation kinetics we have to consider the whole system with its comlexity, and the DEM model (even with its simlifications) is a great tool for this urose. It is worth noting that the model used in this work is comletely based on hysical laws and mechanisms and contains no adjustable arameters. The lack of exerimental data in the field of the comlex fluid rheology does not allow a thorough validation of the quantitative redictions of our DEM model. Nevertheless, we have demonstrated that the model is able to semiquantitatively redict the behavior of concentrated diserse systems subjected to shear. These results are valuable for the oeration of industrial rocesses as well as rocess models, where they can serve as constraints for the arametric sace. An imortant drawback of our model is its high demand for comutational ower and time. In a future aer, the influence of the article-wall interaction on the rocess of fouling will be studied in detail with the introduced model. Also, the exerimentally observed aroachretraction hysteresis during solid-body contact of the articles (resulting from AFM measurements) will be incororated into the model. ASSOCIATED CONTENT *S Suorting Information Scheme of the comutational domain and detailed information on the article-article interactions. This material is available free of charge via the Internet at htt://ubs.acs.org. *W Web-Enhanced Feature A movie showing the simulation for the system with a article volume fraction (ϕ) of 0.4 under a shear rate (G) of s 1 is available in the HTML version of this aer. AUTHOR INFORMATION Corresonding Author * Juraj.Kosek@vscht.cz. 701 dx.doi.org/10.101/la500101x Langmuir 014, 30,

10 Notes The authors declare no cometing financial interest. ACKNOWLEDGMENTS The authors are grateful for suort rovided by EC FP7 Project COOPOL (NMP-SL ). This work was suorted by the Czech Science Foundation (Project GAP106/ 11/1069). REFERENCES (1) Hunter, R. J. Foundations of Colloid Science, nd ed.; Oxford University Press: Oxford, U.K., 001. () Johnson, K. L.; Kendall, K.; Roberts, A. D. Surface Energy and Contact of Elastic Solids. Proc. R. Soc. London, Ser. A 1971, 34, (3) Hong, C. W. From long-range interaction to solid-body contact between colloidal surfaces during forming. J. Eur. Ceram. Soc. 1998, 18, (4) Marshall, J. S. Particle aggregation and cature by walls in a articulate aerosol channel flow. J. Aerosol Sci. 007, 38, (5) Zaccone, A.; Wu, H.; Gentili, D.; Morbidelli, M. Theory of activated-rate rocesses under shear with alication to shear-induced aggregation of colloids. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 009, 80, No (6) Xie, D. L.; Wu, H.; Zaccone, A.; Braun, L.; Chen, H. Q.; Morbidelli, M. Criticality for shear-induced gelation of chargestabilized colloids. Soft Matter 010, 6, (7) Zaccone, A.; Gentili, D.; Wu, H.; Morbidelli, M. Shear-induced reaction-limited aggregation kinetics of Brownian articles at arbitrary concentrations. J. Chem. Phys. 010, 13, No (8) Sugimoto, T.; Kobayashi, M.; Adachi, Y. The effect of double layer reulsion on the rate of turbulent and Brownian aggregation: Exerimental consideration. Colloids Surf., A 014, 443, (9) Soos, M.; Kaufmann, R.; Winteler, R.; Kroua, M.; Luthi, B. Determination of Maximum Turbulent Energy Dissiation Rate Generated by a Rushton Imeller through Large Eddy Simulation. AIChE J. 013, 59, (10) Henry, C.; Minier, J. P.; Lefevre, G. Towards a descrition of articulate fouling: From single article deosition to clogging. Adv. Colloid Interface Sci. 01, 185, (11) ODEPACK Library. htt:// (1) Marshall, J. S. Discrete-element modeling of articulate aerosol flows. J. Comut. Phys. 009, 8, (13) Harshe, Y. M.; Lattuada, M. Breakage Rate of Colloidal Aggregates in Shear Flow through Stokesian Dynamics. Langmuir 01, 8, (14) Butt, H. J.; Caella, B.; Kal, M. Force measurements with the atomic force microscoe: Technique, interretation and alications. Surf. Sci. Re. 005, 59, (15) Hodges, C. S.; Cleaver, J. A. S.; Ghadiri, M.; Jones, R.; Pollock, H. M. Forces between olystyrene articles in water using the AFM: Pull-off force vs article size. Langmuir 00, 18, (16) Schmitt, F. J.; Ederth, T.; Weidenhammer, P.; Claesson, P.; Jacobasch, H. J. Direct force measurements on bulk olystyrene using the bimorh surface forces aaratus. J. Adhes. Sci. Technol. 1999, 13, (17) Israelachvili, J. Intermolecular and Surface Forces; Elsevier Science: Amsterdam, 010. (18) Li, S.-Q.; Marshall, J.; Liu, G.; Yao, Q. Adhesive articulate flow: The discrete element method and its alication in energy and environmental engineering. Prog. Energy Combust. Sci. 011, 37, (19) Wu, H.; Tsoutsoura, A.; Lattuada, M.; Zaccone, A.; Morbidelli, M. Effect of Temerature on High Shear-Induced Gelation of Charge- Stabilized Colloids without Adding Electrolytes. Langmuir 010, 6, (0) Smoluchowski, M. U ber Brownsche Molekularbewegung unter Einwirkung aüßerer Kra fte und den Zusammenhang mit der 70 verallgemeinerten Diffusionsgleichung. Ann. Phys. (Berlin, Ger.) 1915, 353, (1) Guery, J.; Bertrand, E.; Rouzeau, C.; Levitz, P.; Weitz, D. A.; Bibette, J. Irreversible shear-activated aggregation in non-brownian susensions. Phys. Rev. Lett. 006, 96, No dx.doi.org/10.101/la500101x Langmuir 014, 30,