Chemical and Biological Engineering Publications Chemical and Biological Engineering
6-25-2014 Characterization of sheared colloidal aggregation using Langevin dynamics simulation Sergiy Markutsya Iowa State University
Rodney O. Fox Iowa State University, [email protected]
Shankar Subramaniam Iowa State University, [email protected]
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This Article is brought to you for free and open access by the Chemical and Biological Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Chemical and Biological Engineering Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Characterization of sheared colloidal aggregation using Langevin dynamics simulation
Abstract Aggregation of colloidal particles under shear is studied in model systems using a Langevin dynamics model with an improved interparticle interaction potential. In the absence of shear, aggregates that form are characterized by compact structure at small scales and ramified structure at larger scales. This confirms the structural crossover mechanism previously suggested by Sorensen and coworkers, that colloidal aggregation occurs due to monomer addition at small scales and due to cluster-cluster aggregation at large scales. The fractal dimension of nonsheared aggregates is scale-dependent. Smaller aggregates have a higher fractal dimension than larger ones, but the radius of gyration where this crossover occurs is independent of potential well depth for sufficiently deep wells. When these aggregates are subjected to shear they become anisotropic and form extended cigar-like structures. The size of sheared anisotropic aggregates in the direction perpendicular to the shear flow is limited by shear-induced breakage because the shear force dominates interparticle attraction for sufficiently large aggregates. Anisotropic aggregates are not completely characterized by a single radius of gyration, but rather by an inertia ellipsoid. Consequently the fractal dimension is no longer an adequate metric to properly characterize them, and to identify changes in their structure from their nonsheared isotropic counterparts. We introduce a new compactness-anisotropy analysis that characterizes the structure of anisotropic aggregates and allows us to distinguish between aggregates from sheared and nonsheared systems. Finally, using the ratio of interparticle force to the shear force fpot,sh we are able to characterize different outcomes of sheared aggregation as a function of dimensionless well depth and Péclet number.
Disciplines Biological and Chemical Physics | Biological Engineering | Chemical Engineering
Comments This article is published as Markutsya, Sergiy, Rodney O. Fox, and Shankar Subramaniam. "Characterization of sheared colloidal aggregation using Langevin dynamics simulation." Physical Review E 89, no. 6 (2014): 062312. DOI: 10.1103/PhysRevE.89.062312. Posted with permission.
This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/cbe_pubs/300 PHYSICAL REVIEW E 89, 062312 (2014)
Characterization of sheared colloidal aggregation using Langevin dynamics simulation
Sergiy Markutsya,1 Rodney O. Fox,2 and Shankar Subramaniam1,* 1Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50011, USA 2Department of Chemical and Biological Engineering, Iowa State University, Ames, Iowa 50011, USA (Received 25 February 2014; published 25 June 2014) Aggregation of colloidal particles under shear is studied in model systems using a Langevin dynamics model with an improved interparticle interaction potential. In the absence of shear, aggregates that form are characterized by compact structure at small scales and ramified structure at larger scales. This confirms the structural crossover mechanism previously suggested by Sorensen and coworkers, that colloidal aggregation occurs due to monomer addition at small scales and due to cluster-cluster aggregation at large scales. The fractal dimension of nonsheared aggregates is scale-dependent. Smaller aggregates have a higher fractal dimension than larger ones, but the radius of gyration where this crossover occurs is independent of potential well depth for sufficiently deep wells. When these aggregates are subjected to shear they become anisotropic and form extended cigar-like structures. The size of sheared anisotropic aggregates in the direction perpendicular to the shear flow is limited by shear-induced breakage because the shear force dominates interparticle attraction for sufficiently large aggregates. Anisotropic aggregates are not completely characterized by a single radius of gyration, but rather by an inertia ellipsoid. Consequently the fractal dimension is no longer an adequate metric to properly characterize them, and to identify changes in their structure from their nonsheared isotropic counterparts. We introduce a new compactness-anisotropy analysis that characterizes the structure of anisotropic aggregates and allows us to distinguish between aggregates from sheared and nonsheared systems. Finally, using the ratio of interparticle
force to the shear force fpot,sh we are able to characterize different outcomes of sheared aggregation as a function of dimensionless well depth and Peclet´ number.
DOI: 10.1103/PhysRevE.89.062312 PACS number(s): 82.70.Dd, 07.05.Tp, 05.40.−a
I. INTRODUCTION In these experiments, changes in aggregate structure are inferred from d , the fractal dimension of aggregates that Aggregation of colloidal nanoparticles is a nonequilibrium f is obtained from light scattering analysis. However, two- multiscale problem that is characterized by a wide range of dimensional (2D) Langevin dynamics (LD) simulations of length and time scales ranging from those associated with Cerda et al. [12] report that shear changes the effective a monomer to superaggregates comprising tens of thousands interaction between monomers by increasing the distance of monomers. Usually colloidal aggregation in real physical between neighboring monomers in aggregates at initial stages systems occurs in the presence of external forces due to of the cluster nucleation (induction period). These changes in gravity [1,2] or shear flow [3–11]. This work is focused on structure are characterized by computing a mean interaction the effects of shear. Shear flow affects the size and structure of energy attributed to the physical bonds that hold the cluster aggregates and the rate at which they are formed. A rich variety together [12]. After the induction period ends, shear is of phenomena are observed depending on the magnitude of the observed to compact the initial fractal aggregates. From these shear rate, the initiation time [11] (i.e., whether shear is applied studies it is not clear why in some cases shear results in after some aggregation has taken place, or right from the onset the formation of more compact aggregates, while in others of aggregation), and the duration of time over which the system it produces structures where the distance between neighbor- is subjected to shear. ing monomers is stretched when compared to nonsheared These phenomena have been experimentally investigated structures. by studying the influence of externally applied shear on the For clarity of exposition we define the terms we use aggregation of latex nanoparticles [8,11]. If shear is applied to characterize aggregates in this work. We always use after aggregates have already formed, it can change aggregate compactness to refer to the characterization of aggregates structure. The magnitude of shear relative to diffusion is in terms of interparticle distance and the magnitude of the characterized by the Peclet´ number, which can be defined potential energy of particles in the aggregate. We denote based on the monomer diameter σ as ramified aggregates as those consisting of smaller compact 1 Gσ 2 clusters formed by cluster-cluster aggregation. For isotropic Pe = , (1) 4 D∞ aggregates in nonsheared systems, compact aggregates have high fractal dimension, and ramified aggregates have a lower where G is the shear rate and D∞ is the diffusion coefficient of the nanoparticle at infinite dilution. In experiments it has been fractal dimension. Compactness as quantified in this work found that moderate shear flow (characterized by Pe = 1–5) by the local volumetric potential energy density (LPED) is results in aggregates with a higher fractal dimension than those not scale-dependent and is therefore useful to characterize found in nonsheared systems [3,8,11]. aggregates in both nonsheared and sheared cases. On the other hand, the fractal dimension is scale-dependent because it depends on the spatial geometry of the aggregates. Therefore, *Corresponding author: [email protected] in sheared systems where aggregates are anisotropic, a higher
1539-3755/2014/89(6)/062312(12) 062312-1 ©2014 American Physical Society MARKUTSYA, FOX, AND SUBRAMANIAM PHYSICAL REVIEW E 89, 062312 (2014) value of compactness as quantified by LPED does not always leading to a classification of sheared aggregating systems on correspond to a higher fractal dimension. the basis on their initial physical parameters. Computational approaches are well suited to answering the question of how shear affects aggregate structure because one can easily vary parameters such as the dimensionless well II. IMPROVED LANGEVIN DYNAMICS SIMULATION depth of the interaction potential and the Peclet` number to investigate their effect on aggregation outcomes. In earlier Langevin dynamics (LD) is a mesoscale simulation method work [13] it was shown that although molecular dynamics that can capture the effect of shear on aggregate restructur- (MD) simulations of aggregation in dilute systems with full ing [14]. The LD model is used to simulate aggregation of solvent interactions are still too computationally expensive, solute particles immersed in a liquid solvent that is subjected mesoscale methods such as LD with modeled solvent inter- to steady, uniform (spatially homogeneous) shear flow. LD actions scale favorably to larger systems while retaining the permits the simulation of larger systems of solute particles than capability of representing structure in aggregating systems. MD [13] because in LD the solvent molecules are not explicitly A coarse-graining procedure recently developed in our group represented, but the solute-solvent interactions are modeled to specify the potential of mean force in LD for aggregating through frictional and random terms, and through modification systems yields time-evolving structure in nonequilibrium ag- of the solute pair interaction potential in the presence of gregating systems that matches very well with MD simulations solvent molecules [14]. For the model system considered in in both diffusion-limited and reaction-limited regimes [14]. this work, the relative magnitude of time scales corresponding With this improved potential of mean force, the authors have to the frictional and pairwise interaction force terms requires confidence that LD simulations of aggregation can reliably evolution of both position and velocity Langevin equations predict important aggregation statistics such as the extent of for accurate LD simulations [13,19,20]. The LD equations for (i) (i) aggregation, time-evolving solute pair correlation function, evolution of the position rα and velocity vα of the ith solute and dynamically scaled cluster size distribution that have been particle in a sheared solvent flow are [21] compared with MD simulations of smaller model systems [14]. dr(i) = v(i)dt, α = 1,2,3, (2) The essential features of this improved LD model that is used to α α study sheared aggregating systems in this work are described 1 in Sec. II. dv(i) =−γv(i)dt + γu dt + F (r(i))dt α α α (i) α The effects of shear on aggregate structure are not easy to m characterize because shear affects aggregates differently on + (i) = 2γσv∞ dWα ,i 1,...,N, (3) smaller length scales (on the order of ten monomer diameters) as compared to larger length scales (on the order of hundred where subscript α denotes the Cartesian coordinate direction, (i) = (i) = monomer diameters). The reason for this probably lies in the m is the mass of ith particle, uα Grβ δ1αδ2β ,(α,β mechanism underlying the formation of aggregates at different 1,2,3) is the mean fluid velocity (of solvent molecules) due scales. As shown earlier [11,15,16], the small-scale structure to imposed shear flow evaluated at the solute particle location (i) = (i) of aggregates arises from monomer addition, while the large- rβ , G is the uniform shear rate, γ kB Tref/m D∞ is the scale structure corresponds to cluster-cluster aggregation. This friction coefficient, D∞ is the diffusion coefficient of solute in scale-dependent modification of aggregate structure by shear 2 = (i) solvent at infinite dilution, σv∞ kB Tref/m is the stationary also means that a single fractal dimension is not sufficient to (i) velocity variance, dWα is a Wiener process increment, N LD characterize the compactness of the aggregate. Expanding on is the total number of solute particles, and F(r) =−∇rU , the ideas of Cerda et al. [12], an LPED metric is proposed where U LD is the effective LD potential of mean force between in Sec. III C to characterize the compactness of anisotropic solute particles in the presence of solvent. This effective LD aggregates that are formed under shear. Aggregate breakage is potential is calculated according to a coarse-graining method observed [6,9] when a sufficiently high shear rate is imposed. developed by Markutsya et al. [14]as Thus shear limits the maximum size of aggregates [3,6–9], LD LJ with the maximum attainable size decreasing with increasing U (rij ) = U (rij ) + U˜2(rij ) shear rate. σ 12 σ 6 4ε − + U˜2(rij ),rij rcut Lekkerkerker et al. [17] have proposed a phase diagram = rij rij for aggregating systems, and Chakrabarti et al. [18] classified 0 rij >rcut, aggregation regimes into (a) equilibrium aggregation, (b) (4) nonequilibrium aggregation, and (c) gelation. Inspired by these works, we identify the most important dimensionless param- where σ is the particle diameter, rij is the scalar separation or eters that enable the classification of outcomes of sheared distance between centers of particles i and j, rcut is the cutoff aggregation as the nondimensional potential well depth and distance chosen to be 2.5 σ , U˜2 is the correction term that takes the Peclet´ number (details of scale analysis and dimensionless into account the effect of solvent molecules on interparticle parameters are given in Appendix B). In Sec. III C it is shown interaction potential [14], and ε is the well depth of Lennard- that the outcomes of sheared aggregation can be characterized Jones interaction potential between solutes. by a set of metrics M that includes the LPED and anisotropy of In this coarse-graining (CG) method for nonequilibrium the aggregates. In Sec. IV these metrics are used to distinguish time-evolving aggregating systems, the effect of solvent can different aggregation outcomes in the parameter space defined be represented as a correction term U˜2 to the solute-solute by the nondimensional well depth and the Peclet´ number, interparticle potential (see Ref. [14] for details). The effective
062312-2 CHARACTERIZATION OF SHEARED COLLOIDAL . . . PHYSICAL REVIEW E 89, 062312 (2014)
6 40 LJ LJ U AA U AA 4 LD LD U AA,eff 20 U AA,eff 2 ∼ ∼
T U T U
B 2 B 2 0 0 -2 -4 ∧ ∧ -20 ∧ ∧ U(r) = U(r)/k U(r) = -6 U(r)/k U(r) = -40 -8 -10 12345 12345 ∧ ∧ r=r/σ r=r/σ (a) (b)
LD LJ FIG. 1. (Color online) Comparison of the effective LD potential U and potential correction U˜2 with the Lennard-Jones potential U at (a)ε ˆ = 8, (b)ε ˆ = 50.
LD potential U LD, the Lennard-Jones potential U LJ, and the influence on aggregation statistics and the system consists of correction term U˜2 are shown in Fig. 1 for systems with mostly large, well-separated aggregates with a small number εˆ = 8 andε ˆ = 50. For the values of potential well depth of nonaggregated monomers. Then a uniform shear flow is 2 used in this work, the potential correction is close to zero applied for tˆ = tD∞/σ = 113. Such a simulation strategy (U˜2 ≈ 0), and the effective LD potential is practically the same allows us to observe changes in the aggregate structure that as the Lennard-Jones potential U LD ≈ U LJ. Nevertheless, occur once uniform shear is applied. In order to characterize the for the sake of consistency and for direct comparison with statistical variability in fractal dimension df measurements, previous work that reported aggregation without shear using three independent simulations were performed with different the effective potential [14], we have retained the slightly initial particle configurations (positions and velocities), all modified effective potential. corresponding to the same initial distribution (uniform for The diffusion coefficient at infinite dilution D∞ and the position and Maxwellian for velocity). correction to the interparticle potential U˜2(rij ) are required The model system chosen to study aggregation consists of input parameters for the improved LD model. The diffu- a system of solute particles interacting in a solvent through an sion coefficient at infinite dilution D∞ is extracted directly effective potential calculated according to Eq. (4), while the from MD simulation of the corresponding system, and the friction and random terms in Eq. (3) account for additional correction to the interparticle potential U˜2(rij ) is obtained solvent effects. This system has a dimensionless volume frac- using the semianalytical coarse-graining procedure described tion nˆ = nσ 3 = 0.032 that corresponds to a volume fraction previously [14]. fv = 0.017. This volume fraction represents a dilute system The LD simulations are performed using the LAMMPS [22] of colloidal nanoparticles in a water bath, but it is not as low software package. The initial spatial configuration of the as in some experiments [3,4,6–8,11] because of computational particles is specified to ensure nonoverlapping particles. This limitations. System parameters used in the present study are is accomplished by spatially distributing the solute particles reported in Table I. according to a hard-core Matern´ point process [23]. The In order to investigate the effect of system size, systems solute particles are assigned a Maxwellian velocity distribution with N = 10 000 and N = 300 000 solute particles were corresponding to their reference temperature Tref.Fromthis considered. No significant difference in aggregation statistics in configuration the system is allowed to equilibrate to the initial (fractal dimension df , and inflection point Rg ) was observed condition for the aggregation simulations by allowing particles between these two systems for both sheared and nonsheared to interact with a Lennard-Jones potential with dimensionless cases. The results in this paper are reported from the system = well depthε ˆ = ε/kB Tref = 1, where kB is the Boltzmann with N 300 000 solute particles in a cube with sides of ∼ constant, and Tref is the reference temperature. length 210σ because that yields better statistics of aggregate properties.
III. EFFECT OF SHEAR ON AGGREGATION STRUCTURE TABLE I. Parameters used to define model aggregating systems In this section the structure of aggregates obtained using LD for LD simulations. simulations are analyzed for systems with and without shear, −9 2 as described in Table I. Initially all systems are simulated εˆ Pe ND∞,10 m /s = 2 = without shear to a dimensionless time tˆ tD∞/σ 3244 8.0 0.0, 2.1, 8.0 300 000 1.95 in order to obtain good statistics of aggregate structure. 50.0 0.0, 2.1, 8.0 300 000 1.54 It is verified that further time evolution has no significant
062312-3 MARKUTSYA, FOX, AND SUBRAMANIAM PHYSICAL REVIEW E 89, 062312 (2014)
The effect of shear flow on aggregation was studied by to the radius of gyration of the cluster Rg by the fundamental imposing different shear rates corresponding to two Peclet´ relation = number values Pe (2.1; 8.0), representative of moderate df Rg shear flow [11,12]. The strength of the interparticle interaction k = k0 , (5) potential in Eq. (4) is controlled by varying the dimensionless a well depthε ˆ of the Lennard-Jones interaction potential. Its where a = σ/2 is the particle radius, σ is the particle diameter, value was chosen based on experience with aggregating and k0 is a constant. For a statistically isotropic system, the systems in previous work [14]. radius of gyration for each cluster can be calculated as
1/2 A. Aggregation without shear 1 k RI = (rcm − r(i))2 , (6) Aggregates were determined based on a distance criterion, g k = according to which two particles are assigned to the same i 1 cluster if the distance between their centers is less than a where rcm denotes the position of the center of mass of the (i) specified distance rcl, which is typically in the range 1.1σ< cluster, and r denotes the position of the center of the ith rcl < 1.5σ . In the present study rcl = 1.4σ, which corresponds particle in the cluster. The radius of gyration given by Eq. (6) to the location of the first minimum in the radial distribution is a good measure of aggregate size for statistically isotropic function g(r), which is approximately the same for bothε ˆ = 8 systems, and so for the nonsheared cases we use Eq. (5) with = = I andε ˆ 50. Aggregation in the absence of shear was analyzed Rg Rg to calculate df . by plotting the size of the aggregate (characterized by the The fractal dimension of the aggregates is obtained accord- number of solute particles in the cluster Ncl)asafunctionof ing to Eq. (5) by fitting the data from Figs. 2(a) and 2(b) with the nondimensional radius of gyration Rg/σ on a logarithmic the power function and extracting the power term. The two scale as shown in Figs. 2(a) and 2(b). The fractal dimension values of fractal dimension shown in the figures are obtained in df is related to the number of monomers (k) in a cluster, and by splitting the data at the inflection point Rg . The statistical
104 104
d = 1.50±0.02 ± f df = 1.48 0.03
103 103 ± ± df = 2.37 0.01 df = 2.40 0.03 cl cl N N
102 102
N=300,000 N=300,000
101 101 51015 51015 σ σ Rg/ Rg/ (a) (b)
(c) (d)
LD = = FIG. 2. (Color online) The fractal dimension df from LD simulations with effective potential Ueff at time tˆ 3244: (a)ε ˆ 8.0, (b)ε ˆ = 50.0. Snapshots for typical aggregate forε ˆ = 50.0 at time tˆ = 3244 for (c) aggregate containing 150 monomers with the radius of gyration Rg = 2.8σ ; (d) aggregate containing 966 monomers with the radius of gyration Rg = 7.1σ .
062312-4 CHARACTERIZATION OF SHEARED COLLOIDAL . . . PHYSICAL REVIEW E 89, 062312 (2014) variability in the fractal dimension df measurements (arising B. Shear-induced aggregation from variability in the initial configuration of the particle Starting from a nonsheared aggregated system described at ensemble) is estimated from three independent simulations the beginning of this section, aggregation in moderate shear (data points from one simulation are shown for clarity). flow (Pe = 2.1) is simulated to a dimensionless time tˆ = 113 2 These plots indicate a multifractal structure corresponding to a in σ /D∞ units. The fractal dimension, scale-dependent mod- phenomenon called structural crossover [24], and suggests the ification of aggregate structure, and behavior of the inflection presence of aggregates with two different fractal dimensions point are analyzed and compared to the nonsheared case. Since in with the transition marked by an inflection point Rg that shear flow results in anisotropic aggregates, Eq. (6) is not corresponds to the number of monomers in an aggregate used for the radius of gyration calculation. For anisotropic in Ncl at the inflection point. Aggregates with Rg smaller than aggregates, the radius of gyration RA is calculated using the in in g Rg and with Ncl
062312-5 MARKUTSYA, FOX, AND SUBRAMANIAM PHYSICAL REVIEW E 89, 062312 (2014)
4 10 ± df = 1.09 0.02
3 ± 10 df = 2.87 0.11 cl N ∧ 2 ε =50 10 ∧ ε =8
101 10 20 30 40 σ Rg/ (a) (b)
LD = FIG. 3. (Color online) (a) The fractal dimension df from LD simulations with effective potential Ueff under shear flow with Pe 2.1at time tˆ = 113; (b) snapshots for typical aggregate forε ˆ = 50.0 at time tˆ = 113 for an aggregate containing 7144 monomers with radius of gyration Rg = 15σ .
in = LJ aggregates with Rg >Rg further decreased to df 1.08 on U (rij ) between particles i and j as defined in Eq. (4)], rij is shearing, and the formation of aggregates with cigar-like the separation distance between solute particles i and j, k is shapes is observed [see Fig. 3(b)]. This suggests that when the number of interior particles in the cluster, and Vcl = kVm shear is applied, all the aggregates in the system are compacted. is the volume of all the bulk particles in the cluster. Particles Moreover, in addition to compaction those aggregates with that lie on the surface of the cluster are not included in the ramified structure in the nonsheared case [see Fig. 2(d)]are definition of LPED, thus excluding the effect of cluster size on rearranged due to mechanical stress created by the shear to LPED. Particles that lie on the surface of the cluster are form cigar-like aggregates. identified using the following procedure. The average absolute From these results, it is possible to conclude that shear potential energy per particle is computed for the cluster. Then flow significantly changes aggregate structure on both small each particle in the cluster with absolute value of potential and large length scales. These changes are observed through energy less than 90% of this average is identified as a surface changes in metrics such as the fractal dimension df , and particle. We then recalculate the average absolute potential through visual observation when large ramified aggregates energy of the cluster excluding the identified surface particles change to cigar-like structures under shear. It is worth and re-examine the remaining particles in the cluster to see noting here that shear-induced anisotropic aggregates are not which of them satisfy the criterion for being identified as a completely characterized by a single radius of gyration, but surface particle. The identification and recalculation steps are rather by an inertia ellipsoid, and consequently the fractal performed recursively until no new surface particles are found. dimension is no longer an adequate metric to properly char- The dimensionless LPED determines the potential energy acterize anisotropic aggregates and distinguish their change of interior particles in the system per unit volume, and it in structure from their nonsheared isotropic counterparts. describes compactness of the aggregate structure. Compact- Therefore, although the fractal dimension df is useful in ness of aggregate structures can also be characterized by signaling changes in aggregate structure, in order to address the average number of nearest neighbors in the aggregate. these issues we define more general metrics to characterize Thus, for a highly packed structure, where the number of aggregate structure and distinguish the effect of shear on them. nearest neighbors is high, the LPED is higher than for a porous structure. The LPED is a useful metric to characterize C. Compactness-anisotropy analysis anisotropic aggregates because it is sensitive to the presence of structural anisotropy in aggregate structures that arise from In this section we describe how the characterization of fewer neighbors in the shear–normal direction. Therefore, aggregates on a compactness-anisotropy map allows us to dis- in this work LPED U/ˆ Vˆ is used in conjunction with an tinguish between aggregate structures obtained from sheared cl anisotropy factor A to classify aggregates produced by shear. and nonsheared systems. We introduce the dimensionless local αβ The anisotropy parameter was calculated after determining volumetric potential energy density (LPED) |U/ˆ Vˆ | as a cl the equivalent inertia ellipsoid with principal axes for each measure of compactness of an aggregate, where cluster calculated according to Eq. (A1)–(A4) and is defined Uˆ U σ 3 1 k k σ 3 as = = U LJ(r ), (7) ˆ ij Vcl Vcl ε kVm = ε Rα i 1 j>i Aαβ ≡ , (8) Rβ where σ is the solute particle diameter,ε ˆ is the potential well– depth, Vm is the volume of a single particle, U is the interaction where the principal axes are rearranged as R1 >R2 >R3. potential [in this case taken to be Lennard–Jones potential With this specification the anisotropy factor value Aαβ is
062312-6 CHARACTERIZATION OF SHEARED COLLOIDAL . . . PHYSICAL REVIEW E 89, 062312 (2014)
14 14 clusters with 100 ∧ surface is relatively large compared to the total number of ∧ Νcl Νcl 11 1400 11 1400 particles in aggregate, which results in fewer samples when 1200 1200 ∧ 1000 ∧ U(r)/V U(r)/V 1000 calculating the potential energy from interior particles in the 10 800 10 800 600 600 400 LPED. If smaller aggregates are excluded, the variation in the 9 9 400 200 200 local volumetric potential energy density is within 10%. 8 8 Based on these results it can be concluded that the formation 0510 0510 A12 A12 of clusters with branchlike structure through the coalescence of smaller aggregates explains the decrease of the fractal dimension df . Moreover, the local structure (at small length 14 14 in scale Ν ∧ cl Νcl be identical with no preferential direction, which supports the 11 1400 11 1400 1200 1200 assumption of statistical isotropy. For nonsheared systems the ∧ ∧ 1000 U(r)/V 1000 U(r)/V 10 800 10 800 average value for the local volumetric potential energy density 600 600 400 ˆ ˆ = 9 9 400 U/Vcl 10.8 does not change significantly with well depth 200 200 of the interaction potential, for systems withε ˆ = 8 andε ˆ = 50. 8 8 0510 0510 The effect of shear on aggregate characterization using A A 13 13 LPED and the anisotropy factors Aαβ is studied by applying shear flow with Pe = 2.1 to aggregating systems with different potential well depthsε ˆ. Figure 5 shows dimensionless LPED 14 14 as a function of the anisotropy factors Aαβ (only clusters with 13 13 k>100 are shown). By comparing these data for systems 12 12 with a uniform shear flow with the nonsheared case (Fig. 4), cl cl ∧ ∧ Ν Ν cl cl several important features can be distinguished. A significant 11 1400 11 1400 1200 1200 ∧ ∧ 1000 asymmetry in anisotropy factors A12 and A13 is observed for all U(r)/V U(r)/V 1000 10 800 10 800 600 600 systems with different potential well depths [Figs. 5(a)–5(d)], 400 400 9 200 9 which is explained by shearing along the x direction (which 200 = 8 8 corresponds to α 1inAαβ ). Moderate shear flow signifi- 0510 0510cantly changes the magnitude of anisotropy factors along the A23 A23 shear flow direction (A12 and A13) from 5 for the nonsheared systemstoupto40forˆε = 8, and up to 10 forε ˆ = 50, in FIG. 4. (Color online) Dimensionless local volumetric potential the sheared cases. At the same time, the magnitude for the 3 energy density Uˆ (r)/Vˆcl in ε/σ units for aggregation without shear, anisotropy factor A23 is in the same range as for the nonsheared as a function of the anisotropy Aij ,wherei,j = 1,...,3forˆε = 8.0 systems [Figs. 4(e) and 4(f) and Figs. 5(e) and 5(f)]. It is andε ˆ = 50.0. Color legend represents the number of monomers in possible to conclude that shear flow introduces anisotropy into each cluster. the aggregating system by changing the statistically isotropic aggregate structure to cigar-like structures. Moreover, this “stretching” of aggregates due to shear flow always greater than unity for α<β. This specification of changes their local structure, which is nicely captured by the anisotropy is different from those proposed [16,28] where the dimensionless LPED. For systems with weaker interparticle mean shape anisotropy for the ensemble of clusters Aαβ was interaction (ˆε = 8), shear at Pe = 2.1 is strong enough to calculated. The mean shape anisotropy does not fit the needs significantly separate particle pairs inside aggregates, and as of this work where anisotropy for every cluster in the system a result the average distance between neighbor monomers needs to be defined. increases (aggregates become less compact). Consequently The reference value for isotropic aggregating systems is the dimensionless LPED decreases to 9.2from10.8(for first established by calculating the LPED as a function of the nonsheared systems) as seen in Figs. 5(a), 5(c), and 5(e). When anisotropy factor Aαβ as shown in Fig. 4. In these plots each interparticle interaction is stronger (ˆε = 50), the same shear point corresponds to a single cluster, and only clusters with flow with Pe = 2.1 is not able to stretch monomer pairs inside k>100 are shown. For both isotropic aggregating systems aggregates to the same extent as in the case ofε ˆ = 8. However, withε ˆ = 8 andε ˆ = 50, a similar pattern is observed for some displacements of particles due to shear force occur that all anisotropy factors Aαβ . This result is consistent with the initiates their local rearrangement. This local rearrangement observation that the aggregates in nonsheared aggregating allows particles to occupy more energy stable locations than systems are statistically isotropic, although the anisotropy they could have due to aggregation without shear. As result factor for individual clusters may differ from unity. However, of these local rearrangements more compact structures were even for isotropic systems, the maximum magnitude of the formed that are supported by increasing value of the average anisotropy factor is as large as 5. Also it is found that smaller dimensional LPED to 12.1. 062312-7 MARKUTSYA, FOX, AND SUBRAMANIAM PHYSICAL REVIEW E 89, 062312 (2014) 14 14 IV. CLASSIFICATION OF AGGREGATION OUTCOMES UNDER SHEAR 13 13 12 12 As discussed in Appendix B, outcomes of sheared aggre- cl cl ∧ Ν ∧ Ν cl cl gation can be classified in a parameter space defined by the 11 7000 11 7000 6000 6000 ∧ dimensionless potential well depthε ˆ and the Peclet´ number Pe 5000 ∧ U(r)/V 5000 U(r)/V 10 4000 10 4000 based on the values of LPED and the anisotropy parameter 3000 3000 2000 2000 9 1000 9 1000 Aαβ for the aggregates. Competition between interparticle attractive forces (characterized byε ˆ) and shear (characterized 8 8 010203040 0510by Peclet´ number Pe) determines aggregation outcomes in A12 A12 sheared systems. The relative magnitude of these forces fpot,sh [defined by Eq. (9) later in this section] is used to distinguish between different aggregation outcomes that are classified 14 14 according to their characteristics: fractal dimension, maximum 13 13 size of aggregates in the shear normal direction, and LPED. 12 12 Shear introduces a new time scale 1/G in aggregating cl cl ∧ Ν ∧ cl Ν cl systems, and an additional force arises between a pair of 11 7000 11 7000 6000 6000 ∧ 5000 ∧ particles due to the shear flow. When shear is applied to an U(r)/V U(r)/V 5000 10 4000 10 4000 3000 3000 aggregating system, particles attain the local flow velocity on 2000 9 2000 ∼ 1000 9 1000 atimescale( 1/γ ). Because of the presence of a velocity 8 8 gradient in a shear flow, a pair of particles at different 010203040 0510locations along the velocity gradient will relax to the local fluid A13 A13 velocity and thus attain different velocities. This relaxation to a different local mean fluid velocity for a pair of particles 14 14 is the source of an additional force due to shear that tends to move particles away from each other. In sheared systems 13 13 the aggregation outcome does not solely depend on the 12 12 cl cl magnitude of pair-interaction force that arises from interaction ∧ Ν ∧ cl Ν 7000 cl 11 7000 potential, but instead it depends on the relative magnitudes 6000 11 6000 ∧ 5000 ∧ U(r)/V U(r)/V 5000 of the pair-interaction force and the shear force. Based on 4000 10 10 4000 3000 3000 this reasoning it is useful to introduce the interparticle force 2000 2000 9 1000 9 1000 normalized relative to shear fpot,sh that is defined as 8 8 010203040 0510 |F (r )|max A A pot min 23 23 fpot,sh = , (9) Fsh(rmin) where |F (r )|max is the maximum absolute interparticle FIG. 5. (Color online) Local volumetric potential energy density pot min 3 force between a pair of particles separated by rmin, and which Uˆ (r)/Vˆcl in ε/σ units as a function of the anisotropy Aij ,where = = is calculated as i,j 1,...,3 for sheared aggregating systems with Pe 2.1. The color legend represents the number of monomers in each cluster. 13 7 max ε σ σ Dashed line represents average LPED value for nonsheared case. |Fpot(rmin)| = 24 2 − , (10) σ rmin rmin where rmin is the pair separation that satisfies the maximum attraction force between pair of particles. The force due to shear between a pair of particles separated by rmin, Fsh(rmin)is From these results it is possible to conclude that LPED calculated as captures the scale-dependent changes in aggregate structure rmin due to shear. When the force due to shear between monomer Fsh(rmin) = mγ Grmin = mγ 4PeD∞ . (11) pairs is significant compared to the force due to interparticle σ 2 potential interaction, aggregates with cigar-like structures By substituting Eq. (10) and Eq. (11) into Eq. (9) and were formed. These cigars are very long along the shear representing values in dimensionless form, a final expression direction with less compact local structure (when compared to for the interparticle force normalized relative to shear is nonsheared aggregates). In the cases where force due to shear 14 8 is relatively small compared to the force due to interparticle 6ˆε σ σ fpot,sh = 2 − , (12) potential interaction, aggregates with cigar-like structures Pe rmin rmin are still observed. However, in this case the “cigars” were = = shorter. Moreover, shear flow allows the local rearrangement where dimensionless units are computed asε ˆ ε/kB Tref,ˆγ 2 ˆ ∞ = ∞ = of particles, and as result more compact local structures γσ/σv∞ , D D /σ σv∞ , and σv∞ kB T/m, and by taking of aggregates are formed. LPED and anisotropy factor Aαβ into account the fact thatγ ˆDˆ ∞ = 1. enable us to discriminate between these different aggregation By analyzing Eqs. (9) and (12) it is possible to conclude structures. that when fpot,sh < 1 the shear force is stronger than the 062312-8 CHARACTERIZATION OF SHEARED COLLOIDAL . . . PHYSICAL REVIEW E 89, 062312 (2014) mp TABLE II. Aggregation metrics (df , R⊥ , LPED) as a function This classification uses the normalized interparticle force of Peclet` number Pe and potential well depthε ˆ. relative to shear fpot,sh as a criterion to identify aggregation outcomes in the parameter space defined by the dimensionless Pe interparticle potential well depthε ˆ and the Peclet` number 2.1 8.0 Pe. In Fig. 6 the color legend represents values of fpot,sh. Based on the correlation between aggregation metrics and the εdˆ Rmp LPED d Rmp LPED f ⊥ f ⊥ value of the normalized interparticle force fpot,sh, three dif- 8 1.08 1.5 9.2 0 0.5 NA ferent regions are identified: (a) no aggregation (fpot,sh < 1); 50 1.08 2.9 12.1 1.13 2.0 10.0 (b) aggregation with less dense local structure (1 fpot,sh < 4) corresponding to LPED values less than that of nonsheared aggregates (10.8fromFig.4); and (c) aggregation with interparticle interaction force, and in such a system the compact local structure (fpot,sh > 4) corresponding to LPED breakage of aggregates is expected. When fpot,sh > 1the values greater than that for nonsheared aggregates. The dashed formation of aggregates in a system is expected [this can be line represents the border between regions where aggregates achieved by increasing dimensionless potential well depthε ˆ do not form (fpot,sh < 1) and the region where formation of or by decreasing shear flow intensity Pe as can be seen from aggregates is observed (fpot,sh 1). The dotted line separates Eq. (12)]. It should be noted that the nondimensional interparti- regions when aggregates with compact local structure (when cle force is defined based on particle-particle interaction force, compared to nonsheared aggregates) are formed. Consider two which may not be an appropriate one when cluster-cluster systems that fall in the region characterized by less dense local aggregation occurs. However, this criterion for classification aggregates (indicated by arrows with aggregation metrics in of aggregation outcomes should still be useful, but it may Fig. 6) that start from significantly different initial conditions: change the magnitude of the normalized interparticle force (ˆε;Pe)= (8; 2.1) and (50; 8.0). Since these two systems are in fpot,sh at which this transition occurs. the same region in parameter space as defined by their fpot,sh LD simulations of aggregation in the presence of shear at values, their aggregation metrics (fractal dimension, size of different Peclet` numbers Pe and with different potential well largest aggregates and LPED) have very similar values even depthsε ˆ are performed and aggregates are characterized in though their initial conditions are significantly different (see terms of the fractal dimension df , the most probable aggregate Table II and Fig. 6). The LPED value for both systems is less mp size in the direction perpendicular to the shear flow R⊥ , than that for corresponding nonsheared aggregating systems and the local volumetric potential energy density LPED, as (LPED = 10.8), indicating less dense local aggregates. On the shown in Table II. The most probable aggregate size in the other hand, a system with initial conditions (ˆε;Pe)= (50 ; 2.1) mp direction perpendicular to the shear flow R⊥ is computed by has fpot,sh > 4 and according to this classification will form first determining the equivalent inertia ellipsoid with principal compact aggregates. As seen in Fig. 6 (symbol in locally mp axes R1,R2,R3 for each aggregate. Then, R⊥ is the mode of compact aggregate region with arrow pointing to it and values the R2 distribution (if more than one mode exists, the highest of aggregation metrics), this system has LPED = 12.1, which value of R2 is chosen). The aggregates are then classified is consistent with the classification. Systems with fpot,sh < 1 according to the value of the normalized interparticle force are also found to exhibit aggregation metrics consistent with fpot,sh as shown in Fig. 6. their classification as nonaggregating. Based on this analysis of metrics for sheared aggregating systems we conclude that the normalized interparticle force fpot,sh is useful for classifying 10 No aggregation Aggregation aggregation outcomes under shear. (0,0.5,-) fpo t,sh 8 5 (1.13,2.0,10.0) 4 V. SUMMARY 3 2 Here we summarize our observations on the effect of 6 1 0 shear on aggregation as revealed by the Langevin dynamics Pe simulations. For the value of friction coefficient used in these 4 Locally (1.08,1.5,9.2) Compact simulations, once shear is applied to an aggregating system Aggregates all the particles attain the local flow velocity in a very short 2 (1.08,2.9,12.1) (d ,R ,LPED) time that is on the order of the dissipation time scale ∼8/γ .In the sheared cases the aggregate structure depends not only 0 10 20 30 40 50 on the magnitude of the interparticle force represented by ε∧ the potential well depthε ˆ but also on the shear intensity characterized by Pe. Therefore, aggregation outcomes under FIG. 6. (Color online) Normalized interparticle force relative to shear are classified on the basis of the ratio of the interparticle shear fpot,sh (color legend) as a function of the dimensionless force to the shear force, represented by fpot,sh. When shear interparticle potential well depthε ˆ,andPeclet` number Pe. The dashed is dominant (fpot,sh < 1), then energy that is transferred from line represents the boundary between nonaggregating and aggregating kinetic energy of mean velocity into fluctuating kinetic energy systems, and the dotted line identifies the region where compactness overcomes the interparticle interaction energy and aggregate of local structure is observed. Values in brackets for selected systems breakage is observed. When the shear force is of the same order mp represent df , R⊥ , and LPED, respectively. of magnitude as the interparticle force and 1 fpot,sh < 4 062312-9 MARKUTSYA, FOX, AND SUBRAMANIAM PHYSICAL REVIEW E 89, 062312 (2014) then aggregates that are larger than the most probable size of ACKNOWLEDGMENT aggregate Rmp break down until their size satisfies R 062312-10 CHARACTERIZATION OF SHEARED COLLOIDAL . . . PHYSICAL REVIEW E 89, 062312 (2014) TABLE III. Classification of scales encountered in LD simulation of aggregation under shear where τF is the time scale associated with the interparticle force, τv is the characteristic velocity autocorrelation (1) (k) A time of a monomer (τv )oracluster(τv ), Rg is the aggregate radius mp of gyration, R⊥ is the macroscopic length, and G is the shear rate. Microscale Mesoscale Macroscale A mp Pe Length σRg R⊥ −1 −1 Time γ τF τv G A × mp Velocity σv∞ Rg /τv G R⊥ the solute particles arising from the LD dynamical equations [Eqs. (2) and (3)] [13]. However, it is not known apriori which set of dimensionless parameters is most useful for characterizing aggregation outcomes. While metrics such as I the radius of gyration Rg , fractal dimension df , extent of Pe aggregation ξ are used for nonsheared systems, the appropriate metrics for sheared aggregating systems are not yet estab- lished. Aggregation introduces clusters characterized by their A radius of gyration Rg , which introduces a mesoscale-length scale, and shear flow introduces a macroscale time scale as shown in Table III. A dimensional analysis based on the Buckingham theorem is used to reduce these physical FIG. 7. (Color online) (a) ξ = 1 − M0(t) is the extent of aggrega- parameters to a nonunique set of dimensionless parameters . M0(0) The dimensionless parameters that represent length and time tion where M0 is zeroth moment of total concentration of clusters. scale ratios of macro- (or meso-) to meso- (or micro-) scales (b) Candidate paramater space defined by the dimensionless potential ˆ are useful in characterizing scale-separated or scale-overlap well depthε ˆ, dimensionless diffusion coefficient D∞,andPeclet´ regimes. number Pe for characterizing aggregation outcomes. The magnitude of the Deborah number Gτ (1) delineates the scale-separated and For sheared aggregating systems, the dimensional physical v scale-overlap regimes. (c) Deborah number De = Gτ (1) as a function parameters were scaled to corresponding nondimensional v of the dimensionless potential well depthε ˆ and Peclet´ number Pe for counterparts in a similar manner as for nonsheared aggregating LD simulations. systems in terms of (Dˆ ∞,εˆ) as previously described [13], where Dˆ ∞ is the dimensionless diffusion coefficient andε ˆ is the dimensionless Lennard-Jones potential well- depth [see separation. By analogy, the aggregation outcome can be Fig. 7(a)]. The productε ˆDˆ ∞ can be interpreted as 1/(γτF ), − monitored by the ratio of microscale length scale to macroscale the ratio of the frictional time scale γ 1 to the interparticle length scale as force time scale τF [13]. When an aggregating system is subjected to shear, an (1) additional dimensionless parameter (Peclet` number Pe) related D∞τv mp , (B1) to the shear rate G arises [see Eq. (1)]. Therefore, outcomes R⊥ of sheared aggregation could be classified in a candidate (1) parameter space that is shown in Fig. 7(b). Also, shear limits where D∞τv represents the microscale length scale and the size of aggregates that introduces an additional length scale corresponds to the distance a single particle will travel during mp (1) R⊥ , the most probable size of aggregates in the shear-normal time τv . direction. The presence of clusters with a range of sizes in From these time and length scale ratios, the scale-separated an aggregating system results in as many time scales. On the regime can be determined as microscale level, the system is characterized by the velocity (1) (1) autocorrelation time scale for a single particle τv , and particle D∞τv size σ .Onthemesoscale level, when aggregates are formed Gτ (1) 1; 1, (B2) v Rmp from a single particles, systems are characterized by the ⊥ velocity autocorrelation time scale for a cluster containing and the scale-overlap regime is defined as (k) A k monomers τv , and a radius of gyration of aggregate Rg . (1) At the macroscale, the system is characterized by the shear D∞τv −1 (1) time scale (represented by G ) and the most probable size Gτv 1; mp 1. (B3) mp R⊥ of aggregates R⊥ that is limited by shear (see Table III). A sheared aggregating system can be characterized by the ratio of Based on this analysis, different values for the time scale (1) (1) macroscale to microscale timescales Gτv that corresponds to ratio Gτv should correspond to scale-separated or scale- the Deborah number De and defines the extent of timescale overlap regimes in the candidate parameter space shown 062312-11 MARKUTSYA, FOX, AND SUBRAMANIAM PHYSICAL REVIEW E 89, 062312 (2014) in Fig. 7(b). A proposed aggregation map is constructed particles that leave the aggregate lose their “memory” faster by analogy to the aggregation map reported for nonsheared than those left in the aggregate, and overall, the average aggregating systems [Fig. 7(c)][13] with an additional axis velocity autocorrelation time calculated for a single particle (1) for the Peclet` number to represent shear. τv decreases. For all LD simulations of systems described in Table I, Earlier analysis of nonsheared aggregation by the authors the velocity autocorrelation time scale for a single particle led to the following conclusions: (a) aggregation in a system (1) (1) τv is computed, and the values of Gτv as a function of the occurs only if the dimensionless potential well depthε ˆ is dimensionless potential well depthε ˆ and Peclet´ number Pe are greater than some critical valueε ˆcr and (b) the dimensionless shown in Fig. 7(c). From the delineation of the parameter diffusion coefficient does not control the aggregation outcome, space in Fig. 7(b) it is possible to conclude that for all but controls only the rate of aggregation [13]. Based on simulated systems reported in Table I, only the scale–separated these findings for nonsheared systems, we conclude that regime is observed and shear does not change the Deborah the dimensionless diffusion coefficient is not a relevant = (1) number De Gτv significantly, since decrease in the shear parameter for classifying sheared aggregation outcomes, time scale (by increase in the shear rate G) decreases the and systems withε> ˆ εˆcr are chosen to ensure aggregation (1) velocity autocorrelation time scale for a single particle τv . in the shear simulations. Therefore, outcomes of sheared An explanation for this phenomena is based on the fact that as aggregation can be classified in a parameter space defined intensity of the shear flow (Pe) increases, the most probable by the dimensionless potential well depthε ˆ and the Peclet´ mp size of aggregates R⊥ decreases [3,6–9]. 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