Experimental Characterization of Breakage Rate of Colloidal

Nov 6, 2014 - Note that the range of strain rates is broader than what is accessible by the state of the art optical systems like 3D-PTV, despite its ...
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Experimental Characterization of Breakage Rate of Colloidal Aggregates in Axisymmetric Extensional Flow Debashish Saha,† Miroslav Soos,*,‡ Beat Lüthi,† Markus Holzner,† Alex Liberzon,§ Matthaus U. Babler,∥ and Wolfgang Kinzelbach† †

Department of Civil, Environmental and Geomatic Engineering, Institute of Environmental Engineering, ETH Zurich, 8093 Zurich, Switzerland ‡ Department of Chemistry and Applied Biosciences, Institute for Chemical and Bioengineering, ETH Zurich, 8093 Zurich, Switzerland § School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel ∥ Department of Chemical Engineering and Technology, KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden S Supporting Information *

ABSTRACT: Aggregates prepared under fully destabilized conditions by the action of Brownian motion were exposed to an extensional flow generated at the entrance of a sudden contraction. Two noninvasive techniques were used to monitor their breakup process [i.e. light scattering and three-dimensional (3D) particle tracking velocimetry (3D-PTV)]. While the first one can be used to measure the size and the morphology of formed fragments after the breakage event, the latter is capable of resolving trajectories of individual aggregates up to the breakage point as well as the trajectories of formed fragments. Furthermore, measured velocity gradients were used to determine the local hydrodynamic conditions at the breakage point. All this information was combined to experimentally determine for the first time the breakage rate of individual aggregates, given in the form of a size reduction rate KR, as a function of the applied strain rate, as well as the properties of the formed fragments (i.e., the number of formed fragments and the size ratio between the largest fragment and the original aggregate). It was found that KR scales with the applied strain rate according to a power law with the slope being dependent on the initial fractal dimension only, while the obtained data indicates a linear dependency of KR with the initial aggregate size. Furthermore, the probability distribution function (PDF) of the number of formed fragments and the PDF of the size ratio between the largest fragment and the original aggregate indicate that breakage will result with high probability (75%) in the formation of two to three fragments with a rather asymmetric ratio of sizes of about 0.8. The obtained results are well in agreement with the results from the numerical simulations published in the literature.

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INTRODUCTION The formation of aggregates composed of primary particles is of central importance in many fields of industrial practice such as polymer processing, crystallization, coagulation processes, wastewater treatment, formation of marine snow, etc.1−5 Aggregates are commonly formed under reduced electrostatic stabilization of primary particles (e.g., by addition of a suitable electrolyte such as salt or acid). Consequently, these particles undergo aggregation driven by Brownian motion or the action of hydrodynamic shear resulting in a formation of fractal-like objects.6−16 As these aggregates grow in size, their interaction with the surrounding flow becomes stronger, which results in cluster restructuring or even their breakup.17−26 The strong dependency of these mechanisms on the primary particle size and material properties, initial aggregate size and structure, the applied flow field, interparticle forces, the composition of the surrounding environment, and the fact that the breakup process © 2014 American Chemical Society

is very fast, complicate proper experimental characterization of the underlying mechanisms. A wide variety of flow devices encompassing laminar as well as turbulent conditions have been used in the literature to experimentally investigate the aggregate breakup. In particular, turbulent jets27 and stirring devices28−31 are typical configurations used for studying the breakage of colloidal aggregates under turbulent flow conditions. Although practically relevant, the highly heterogeneous distribution of the turbulent velocity field limits their application for fundamental studies. Laminar flow, on the contrary, for which the velocity field can be easily characterized, is better applicable for fundamental studies. In the literature, different types of laminar flows such as simple shear flow occurring in Couette flow devices,22,32−35 contractReceived: July 8, 2014 Revised: October 13, 2014 Published: November 6, 2014 14385

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ing nozzles generating an extensional flow,19,24,36,37 or a twodimensional straining flow formed in four-roll mills,32 have been used to study aggregate breakup. Due to the fact that the breakup process is very fast, it is typically not monitored directly, but the rather common outcome from these studies is restricted to the scaling of fragment size as a function of applied shear rate or hydrodynamic stress. Numerical techniques such as the discrete element method (DEM) and Stokesian dynamics have been used to model the breakup process, assuming different levels of approximation of interparticle and hydrodynamic interactions between individual particles composing a cluster.18,20,21,36−39 The obtained results are commonly reported in the form of the scaling of the maximum stable cluster size as a function of the applied shear rate or hydrodynamic stress but are rarely compared with experimentally measured values.20 As an example, Harshe and Lattuada21 evaluated the breakage rate as a function of the number of primary particles composing an aggregate, its initial fractal dimension, stress magnitude, and flow type, providing information needed for the modeling of the breakage process using the population balance equation. By modeling the first breakage event (i.e., the separation of any two primary particles composing an aggregate over a distance of several nanometers), and taking the inverse of this time, the breakage rate, KB, was calculated. It was found that KB follows a power law dependency of the product of aggregate size and applied stress, with values of the corresponding exponents, depending only on the aggregate fractal dimension and the type of the flow field. The results of Harshe and Lattuada21 are supported by theoretical works of Bäbler et al.14,38 and O’Conchuir and Zaccone.39 In these works, the authors approximated the aggregates as fractal objects and modeled different levels of aggregate strength in laminar flow and concluded that the breakage rate of the fractal aggregates in the presence of shear will follow a certain power law, which will be a function of the internal cluster morphology (i.e., df). The goal of this study is to provide direct quantitative measurements of individual breakage events of aggregates exposed to the flow in order to advance our understanding of colloidal breakup and to validate the results of the abovediscussed numerical studies, based on our previous work.19,20,40,41 Due to a high probability of a breakage event occurring at the entrance of a sudden contraction,36 aggregates were exposed to elongational flow in a contracting rectangular nozzle. Three-dimensional particle tracking velocimetry (3DPTV), a nonintrusive Lagrangian flow measurement technique based on imaging,40,41 was used to record the trajectories of individual aggregates up to the breakage point. In parallel to the detection of the breakage point, a detailed flow field characterization of the rectangular nozzle was performed to understand the underlying breakup mechanism. To extend the results obtained from 3D-PTV, the same type of aggregates were broken by an elongational flow generated in a smaller circular contracting nozzle, and the formed aggregates were characterized using the small angle light scattering (SLS) technique.19,20 The two experimental techniques complement each other and represent a robust tool to experimentally evaluate all parts required to construct a breakage rate expression and a fragment distribution function similar to that obtained by Harshe and Lattuada21 using Stokesian dynamics.

Article

MATERIALS AND METHODS

Preparation of Aggregates. The aggregates used in our experiment were prepared from surfactant-free monodisperse white sulfate polystyrene particles with the diameters 420 nm manufactured by Interfacial Dynamics Corp. (IDC), Portland, OR (product no. 1800, cumulative variation = 2%, batch no. 642, 4, solid weight fraction = 8.0%, surface charge density = 4.8 μC/cm2). To prepare the aggregates large enough for an optical detection, 0.3 mL of original latex solution was diluted in 15 mL deionized water and then mixed with 15 mL of 2.5 molar NaCl, resulting in a final solid volume fraction of 10−4. A high amount of salt was used to completely screen any electrostatic repulsive forces between the primary particles and to achieve density matching of the liquid with polystyrene nanoparticles (1050 kg/m3) to prevent sedimentation of the formed aggregates. Approximately 12 h were required to grow the aggregates under stagnant conditions into large flocks, with sizes ranging from several hundreds of microns up to several millimeters (see Figure 1). Since these aggregates have to be introduced into the breakup chamber, they were prepared directly in a syringe, which was subsequently used for their injection.

Figure 1. (a) Image of statically grown aggregates within a syringe (b) together with their highly magnified view. Breakup Devices. Two experimental setups were used in this work to study the breakup of the aggregates. The first one was a circular contracting nozzle mounted between two syringes used previously for breakup studies (see Figure 2, panels a and b).19,21,42 The evolution of the aggregate size during a breakup experiment was measured by light scattering. This technique cannot provide information about individual breakup events and is limited to ensemble-averaged properties of an aggregate population. To overcome this drawback, the second setup was constructed allowing full access for optical detection of the individual aggregate breakup events. It consists of two identical rectangular chambers (75 × 25 × 20 mm) connected through a rectangular orifice (20 × 3 × 3 mm) covered from the top by a transparent glass (see Figure 2c). As can be seen, there are three inlets in the bottom part. Two transverse inlets are used for the carrier fluid with a flow rate of 88 mL/min, while the inlet at the bottom is used for the injection of aggregates. Light Scattering Measurement. To characterize the size and internal structure of the initial population of aggregates after their preparation, they were very gently introduced into the light scattering device (Malvern Mastersizer 2000, Malvern, U.K.), applying a flow rate of 10 mL/min. The measured scattered light intensity, I(q), was recorded and used for aggregate characterization. It can be expressed as43,44 I(q) = I(0)P(q)⟨SF(q)⟩

(1)

where I(0) is the zero-angle intensity, P(q) is the form factor (due to primary particles), ⟨SF(q)⟩ is the average structure factor (due to the arrangement of primary particles within the aggregates), and q is the scattering vector amplitude. The Guinier approximation of the structure factor ⟨SF(q)⟩43,44 was used to evaluate the root-mean 14386

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Figure 2. Schematic showing the devices used for the breakage experiments. (a and b) Circular contracting nozzle located between two syringes, (c) 3D-PTV setup with two traverse inlets at the bottom used for the flow, carrying the aggregate and the outlet located at the top. Aggregates are introduced into the flow from the bottom indicated by an arrow. (d) The schematic illustrates how a pyramid shaped image splitter is aligned with the center of the camera chip allowing a stereoscopic view using a single camera.

Table 1. Operating Conditions Used during the Breakup Experiments in the Circular Contracting Nozzle

*

dnozzle (mm)

Q (mL/min)

U (m/s)

Renozzle (−)

Save* (1/s)

τmax* (Pa)

E** (1/s)

0.5 1.0 3.0

90 50 90

7.64 1.06 0.21

3820 1061 637

1.55×105 8.83×102 4.83×102

684.0 37.7 1.9

3.97×104 2.76×103 1.84×102

Values are calculated using CFD software, FLUENT v6.2.19,20

**

According to Metzner,55 E = {Q(sin α)3}/{πr3nozzle(1 − cos α)}

square radius of gyration, ⟨Rg⟩, of a population of aggregates which provides an estimate of the aggregate size with an accuracy of 5 to 10%. Furthermore, assuming that the produced aggregates behave as fractal-like objects, further information which can be extracted from the light scattering data is the aggregates’ internal structure characterized by fractal dimension, df.43−45 Therefore, by plotting ⟨SF(q)⟩ versus q in the range of q values from 1/⟨Rg⟩ to 1/Rp, where Rp is the radius of the primary particle, using a double logarithmic plot results in a straight line with the slope equal to −df. Typical values of df are ranging from approximately 1.8 measured for the aggregates grown under fully destabilized stagnant conditions46,47 to approximately 2.7, which is a characteristic value of df measured for the aggregates grown under flow conditions.19,20,30,31 Optical Arrangement and 3D Particle Tracking Velocimetry (3D-PTV). 3D-PTV is a nonintrusive and optical flow measurement technique tracking the motion of the particles carried by a fluid. A stereoscopic observation system shown in Figure 2d was used to track aggregates as they move in the flow and record the breakage events. On the basis of the previous work of Higashitani et al.,36 we chose the observation domain starting from approximately 4 mm upstream of the orifice entrance and expanding toward the end of the channel. A cost-effective optical design introduced by Hoyer et al.41 that requires only a single camera for the stereo vision was used without compromising the robustness of the measurement provided by the generic setup of a synchronized four camera system. Figure 2d and Figure SM 1 of the Supporting Information show the laboratory view of the system together with the flow chamber. The high speed camera used for this experiment is a FASTCAM SA5 from Photron USA, Inc., which can record up to 7000 frames per second at a maximum spatial resolution of 1024 × 1024 pixel. With 16 GB built-in memory, it is capable of storing 10918 images at full resolution. In this study, a

Nikkor Micro 60 mm lens was used with an f-stop 11 and the imaging frequency was 4000 frames per second. Prior to the experiments, the 3D-PTV system was calibrated using a calibration block with uniformly bright target points of diameter 300 μm, which are set apart by 1 mm from each other and distributed in three equidistant layers (see Figure SM 2 of the Supporting Information). In all experiments, a 20 W ArIon continuous wave laser (λ = 513 nm) illuminated the observation domain.

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RESULTS AND DISCUSSION Breakup of Aggregates in Extensional Flow Measured by Light Scattering Using Circular Nozzle. The aggregates from the same population as tested for the individual breakage events using 3D-PTV were diluted by 1.25 M NaCl solution (density matched to the primary particle material) down to a solid weight fraction of 2 × 10−6 and broken in a circular contracting nozzle located between two syringes.19,20 This low concentration was used to prevent reaggregation of the formed fragments, and no visible aggregation was observed over the period of several hours after completing the breakage experiment. Furthermore, when comparing the characteristic time of breakage (approximated as an inverse of the applied strain rate) with that of aggregation48 (considering measured fragment sizes being above one micron, and therefore, dominated by shear aggregation mechanism), it was found that characteristic breakage time is at least three orders of magnitude shorter than that of aggregation, indicating that under investigated conditions breakage was the dominant mechanism. The evolution of the size of the fragment 14387

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Figure 3. Evolution of the ⟨Rg⟩ measured by light scattering along the breakage experiment (a) using combinations of nozzle diameter and flow rate. (■) dnozzle = 0.5 mm, Q = 90 mL/min (Renozzle = 3820), (●) dnozzle = 1.0 mm, Q = 50 mL/min (Renozzle = 1061), and (▲) dnozzle = 3.0 mm, Q = 90 mL/min (Renozzle = 637). (b) Plot of ⟨SF(q)⟩ versus q⟨Rg⟩ measured for the initial aggregates (□) and after one pass (○) through the contracting nozzle using dnozzle = 3.0 mm, Q = 90 mL/min.

An example of ⟨SF(q)⟩ plotted as a function of q⟨Rg⟩ measured for the initial aggregates prepared in this work is shown in Figure 3b (□). It can be seen that the obtained value of df equal to 1.8 is in agreement with the values measured for the statically grown aggregates published in the literature.20,46,47 Similarly df was measured also for the population of fragments generated after one pass through the circular contraction (○ in Figure 3b). It was found that, the df increases from initial value of about 1.8 to 2.2. Also due to the long-range of a power law region (more than 2 orders of magnitude), the error related to an estimation of df is very low, on the order of 0.5% of the determined values, which is smaller than the observed difference of the data shown in Figure 3b. Such an increase is in agreement with our previous work,20 clearly indicating the presence of significant restructuring during aggregate breakup. Due to the large size of the primary particles and rather small relative size of the formed clusters with respect to the primary particles, further evaluation of df along the breakage process was not performed.30,47,50 Flow Field Analysis in the Rectangular Nozzle. The flow field in the contracting orifice setup is laminar (Re = 622) and stationary, allowing for studying the flow velocity field separately from the aggregate breakage recording. For this, the flow was seeded with neutrally buoyant flow tracers (Evonik Industries, Germany). Previous studies in contracting nozzles36,51 indicated that the nozzle entrance is the most probable zone for the aggregate breakup. Hence, detailed flow field characterization covering a region approximately 4 mm upstream of the orifice to approximately 15 mm downstream of the orifice entrance was performed. An ensemble of the flow tracer trajectories together with the channel geometry from different viewing angles is shown in Figure SM 3 of the Supporting Information. The Lagrangian measurements of the velocity were interpolated onto a uniform Eulerian grid with a spacing ΔL equal to 0.5 mm and used for the calculation of spatial velocity gradients. In an axisymmetric converging flow generated by a sudden contraction, a sharp increase in velocity is observed close to the constricted regime. In fact, this is also the case for the studied geometry. With a volumetric flow rate of 88 mL/ min, the velocity increases from about 4 mm/s in the upstream of the orifice entrance to approximately 25 cm/s in the orifice and in the channel. Figure SM 4 of the Supporting Information illustrates the velocity magnitude distribution in the region of interest. In the channel, an equal magnitude in velocity of about 10 cm/s close to the both side walls was found, proving that the flow pattern is symmetric with respect to the midplane.

population was monitored by light scattering. The nozzle radius and flow rate were varied in a way to cover a broad range of strain rates covering an interval from about 400 up to approximately 1.4 × 105 s−1 (see Table 1). Note that the range of strain rates is broader than what is accessible by the state of the art optical systems like 3D-PTV, despite its highspeed recording and spatial resolution capabilities. An example of the evolution of the mean radius of gyration ⟨Rg⟩, evaluated from light scattering data using the procedure described in the Supporting Information, measured along the breakage experiment is presented in Figure 3a. At first, we notice that the initial aggregate size with a diameter of 200 microns (see the inset of Figure 3a) is substantially smaller compared with the original aggregate size shown in Figure 1. The reason for this is that in order to perform a breakage experiment it was necessary to transfer the aggregates prepared under static conditions inside the syringe to the measuring chamber. This manipulation results in the disintegration of the large aggregates to the initial size shown in Figure 3a. However, it was found that this early disintegration was well-reproducible and little variation of the initial aggregate size was seen among the repetitions (i.e., the variation in the measured ⟨Rg⟩ is approximately 10%). This allowed us to take the aggregate size obtained from the first light scattering measurement as the starting point of the breakup experiment. From the measured values of ⟨Rg⟩, it can be seen that the decrease of the average aggregate size is more pronounced at the beginning of the process (see the inset of Figure 3a) with faster decay measured for higher strain rates as compared to the experiments with lower strain rate. In contrast, to reach the steady state, independent of the applied strain rate, approximately 200 passages through the nozzle are required. As their size decreases with the increasing number of passages, a very high hydrodynamic stress, which is present only in the very thin near-wall region, is required to break the aggregates further down to reach smaller sizes.19,20 As previously shown by Soos et al.,19 the volume fraction of liquid passing through this highshear region is very small, and many passages are necessary until each aggregate passes through this zone and is subsequently broken to its steady state size. In agreement with our previous works19,20 also in this study the obtained steady state aggregate size decreases with the increase of the applied Reynolds number in the nozzle, Renozzle. This is in accordance with the previous works in the literature,18,21,24,49 showing that for each applied hydrodynamic stress there is a critical stable aggregate size below which the aggregate would not break further. 14388

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Figure 4. (a) Comparison of the eigenvectors of the rate of strain tensor Sij multiplied with their corresponding eigenvalues Λi in a stream-wise view. (b) Orientation of λ1 and λ3 along the stream-wise direction and (c) a PDF of their magnitude. (d) Contour plot of the strain rate together with an iso-surface of the maximum strain spans from the front of the orifice and exists inside the channel close to the wall. The iso-surface renders the maximum strain rate which appears close to the orifice entrance and extends inside the orifice, embracing the channel boundary.

straining exists near the central region of the converging flow, away from the surrounding walls of the chamber and channel. Thereafter, deep inside the channel, Λ2 becomes vanishingly small which leaves Λ1 and Λ3 in balance: close to the wall inside the channel and near the chamber boundary, extensional and compressive stresses align themselves at 45° with respect to the wall. This is characteristic of the simple shear similar to that present in a Couette flow. In the following, the strain rate magnitude S = (SijSij)1/2 = (ΣΛ2i )1/2 will be used as a characteristic measure of the velocity gradient. A summary of the obtained values is presented in Figure 4d. The contraction is characterized by an increase of S over several orders of magnitude. More precisely, in the chamber, S is on the order of 0.3−1 s−1 and reaches values on the order of 102 s−1 in the center of the orifice (Figure 4d). Therefore, at the orifice, both the contraction and the wall shear give similar contributions to the total stress. Beyond the orifice and deep inside the channel, as the velocity profile quickly approaches the laminar profile of a rectangular duct, the contribution of the boundaries is dominating. Breakup of Aggregates in Extensional Flow Measured by 3D-PTV Using Rectangular Nozzle. Even though the whole population of initial aggregates is characterized by df around 1.8, indicating a very open fractal-like structure (see Figure SM 1 of the Supporting Information), the formed individual aggregates differ from each other in size, shape, and most probably also in morphology. Not all aggregates passing the orifice broke up. Therefore, in the following analysis, we discarded those aggregates whose size is comparable to the width of the orifice as they mostly undergo rigid body elastic deformation and resist breakup. We also disregarded the breakage events of smaller aggregates for which we could not

Furthermore, the spatial extent of the high velocity (i.e. > 10 cm/s) zone starts from approximately 1 mm upstream of the orifice position and extends to the limit of the observation domain (i.e., 15 mm downstream from the orifice entrance). To further study the structure of the velocity field, we decompose the measured velocity gradient tensor, Aij = ∂ui/∂xj into the symmetric part that represents the rate of strain tensor, Sij, and the antisymmetric part which is defined as the rate of rotation tensor, Rij. We study the evolution of Sij based on the eigenvalues Λi of Sij and the corresponding mutually perpendicular eigenvectors λi that determine the principal axis of strain. In accordance with the definition for incompressible flow,52 the trace of Sij sums up to zero, i.e., ∑Λi = 0. By convention, Λ1 represents the most stretching eigenvalue and is always positive and Λ3 is the most compressive one and is always negative. The intermediate eigenvalue, Λ2, can be either compressive or extensional. The rotation tensor Rij is studied using the vorticity vector ω = ▽ × u that describes the rate of rotation of a fluid element. Inspection of the flow field using these quantities shows that in the midplane near the orifice entrance, we observed nearly irrotational flow. As summarized in Figure 4, a fluid element approaching the orifice will be deformed due to the combined stress exerted by all three principal components of Sij. From approximately 3 mm upstream of the orifice, the intermediate eigenvector, λ2, takes part in compression together with the conventionally compressive λ3. Before reaching the orifice, the flow is almost two-dimensional (i.e., there is an approximate radial symmetry and Λ2 ∼ Λ3) (i.e., the intermediate and the compressive eigenvectors are nearly indistinguishable). When looking along the stream-wise direction of the flow (see Figure 4, panels a and b), it becomes clear that uniaxial 14389

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Figure 5. Example of the snapshots showing breakup of a single aggregate resulting into (a) two and (b) five fragments. The flow direction is from the bottom to top and aggregates are indicated by arrows. Corresponding values of the strain rates along the individual trajectories are presented in panel (c). Different markers were used to distinguish between different trajectories. The location of the orifice entrance is indicated by the vertical line.

evaluated. We tested the change of the total sum of pixels of each aggregate which was found be too noisy due to inhomogeneous light distribution in the observation volume and angle-dependent scattering properties of the particles (see Figure SM 5 of the Supporting Information). Instead, a moving time integrated change in the total number of pixels along the Lagrangian trajectory of the aggregate was used. Namely, a change in the number of pixels over 30 time steps is integrated along the parent track. If there exists a drop exceeding an empirical tolerable deviation in total pixel number then it provides an indication of a breakage event. This procedure is less sensitive to noise and allows identifying the moment of time at which the change of the aggregate size is significant. Furthermore, due to subpixel accuracy (about 0.2 pixel), the error in the total area of an aggregate as well as the detection of an aggregate boundary were less than 5%. We used this

detect the detached parts as fragments. In particular, this was the case when aggregates are moving along the wall exhibiting two phenomena. Either they are instantly sheared into many fragments or some of them are elongated into a thin threadlike shape and then break apart into many fragments. In both cases, detecting the breakage location and the number as well as the size of the produced fragments is the subject of great uncertainty. To eliminate this effect, breakage events occurring near the wall were omitted from further analysis. In the following, we analyze the breakage events filtered according to the initial aggregate size and a certainty of formation of distinguishable fragments. An example of two such events generating two and five distinguishable fragments is presented in Figure 5. To precisely detect the breakage point, several aggregate image parameters recorded by the 3D-PTV software were 14390

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Figure 6. (a) Breakage locations the isometric view, from the (b) top and (c) stream-wise view and labeled with the strain rate magnitude. The flow direction is in the increasing y direction.

diameter of 90 and 810 nm in extensional flow, indicating similarity between the used breakage setups. Scaling of Fragment Size with Applied Stress. To compare the maximum stable fragment size generated during the experiment from the individual breakage events with the aggregate sizes measured after a single pass through the contracting nozzle by light scattering, the following assumption was made. In the 3D-PTV experimental setup every breakage event results in a formation of several fragments where the size of the largest fragment represents the maximum possible fragment size which survives the local value of the strain rate at the breakage point. Therefore, this value was taken as critical stable size for a given local value of the strain rate. In contrast, aggregates passing through the nozzle and analyzed by light scattering are characterized by a different strain rate history depending on their trajectory. To take this into consideration, the measured ensemble averaged diameter, Dg = 2⟨Rg⟩, was used as a characteristic size of the formed fragments. The variation of the strain rate along the circular nozzle radius is presented in Figure SM 9 of the Supporting Information. To characterize its intensity by a single value, a flow rate weighted value of the strain rate at the nozzle entrance was used for the comparison. It was calculated as a flow rate-weighted axial maximum of the strain rate over the surface defined by the locus through the maxima of the considered strain rate (illustrated by black lines in Figure SM 9 of the Supporting Information) according to53

detection algorithm to analyze the Lagrangian tracks of the parent aggregates as well as formed fragments (see Figure SM 6 of the Supporting Information). It was found that in both cases, the first breakage event takes place approximately 1 mm upstream of the orifice where the magnitude of strain rate (S) approaches its maximum (see Figure 5c). Furthermore, as can be seen from Figure 5c, the trajectories of the formed fragments closely follow the trajectory of the parent aggregate, confirming the precision of the used detection algorithm. In the case of five fragments, other breakage event occurs just after the nozzle entrance. The trajectory of the last fragment is not presented because it occurs deeply inside the nozzle channel where a stereoscopic view of the 3D-PTV system was shielded by the nozzle corners prohibiting the precise detection of the breakage location. It is noteworthy that though the initial size of the aggregates was comparable and they were exposed to an approximately comparable magnitude of strain rate, they broke into a different number of fragments. This is probably due to the differences in the internal structure of individual aggregates. To take into account the variability of individual aggregates, we recorded approximately 450 breakage events. The detected breakage locations are presented in Figure 6. It can be seen that most of the breakage events (approximately 300) take place approximately 1 mm in front of the orifice entrance where the strain rate is maximum with an average strain of S = 102 s−1. Furthermore, this region is characterized by a strong uniaxial stretching where both Λ2 and Λ3 become comparable (see Figure SM 7 of the Supporting Information). This is in agreement with the previous studies where breakup of aggregates was studied using a contracting nozzle or orifices, concluding that elongational flow is more destructive compared to simple shear flow.19,20,36,51 Since several breakage events occurred deep inside the nozzle where Λ2/Λ3 was below 0.2 (see Figure SM 7 of the Supporting Information), indicating that breakage was dominated by the simple shear, these trajectories were excluded from further analysis. This reduced groups of approximately 300 breakup events was used in the following analysis. Similar to the circular nozzle, also in this case the variation of the df after aggregate breakage was monitored by light scattering. It was found that df increases from an initial value of 1.8 to 2.2 (see Figure SM 8 of the Supporting Information) which was as similar to that measured for the circular nozzle geometry, confirming substantial fragment restructuring. This is in agreement with the work of Harshe et al.,20 where a similar increase was observed when breaking statically grown aggregates composed of polystyrene primary particles with

S ave =

∫A (U·n)Sda ∫A (U·n)da

(2)

where A is the surface, da is the surface element, and n is the unit normal to A. A summary of the obtained values for the applied conditions is presented in Table 1. As can be seen from Figure 7, despite the above-mentioned differences, both data sets follow the same scaling indicated by a solid line with the slope equal to −0.6. This is in good agreement with Kobayashi et al.,37 who found a scaling exponent equal to −0.56. Their absolute sizes, however, are larger by approximately one order of magnitude which can be due to the difference in particle size (420 nm used in this work and 1356 nm used by Kobayashi et al.), surface chemistry and method for estimating the strain rate. Furthermore, considering that the aggregates behave as fractal objects, with the average number of primary particles ⟨i⟩ being proportional to the aggregate size according to ⟨i⟩ ∝ ⟨Rg⟩df, and combining it with the measured scaling of ⟨Rg⟩ as a function of the applied strain rate presented in Figure 7, we find that ⟨i⟩ ∝ S−1.32. This scaling of ⟨i⟩ with S is comparable with the results 14391

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collecting large numbers of breakup events. Likewise, in the circular nozzle experiments, we only had access to the mean radius of gyration after the aggregates had passed through the nozzle. Hence, to analyze the breakage data, an alternative breakup rate has to be used that would be applicable to both the observation of individual breakup events obtained when using 3D-PTV and the average quantities obtained when using lights scattering. Such an approximation can be derived by considering the size reduction rate of an ensemble of aggregates, defined as Figure 7. Scaling of the maximum fragment size as a function of applied strain rate evaluated from the light scattering data (▲) and those measured by 3D-PTV technique (black ●, gray ●, and white ○). Circles from black to white correspond to two, four, and six fragments.

KR = −

(5)

where ⟨R⟩ is the mean aggregate size. The relation between the size reduction rate and the number based breakup rate is readily ∞ established. Substituting eq 5 for ⟨R⟩ = (∫ ∞ 0 NξRξ dξ)/(∫ 0 Nξ dξ), where Rξ and Nξ are the size and the number of aggregates of mass ξ (different definition of the mean aggregate size such as the mean radius of gyration, see the Supporting Information, lead to similar results), and taking the time derivative inside the integral, we obtain

of Harada et al.49 and by Harshe and Lattuada.21 In both cases, it was found that for aggregates with df around 2.2 the maximum number of primary particles composing an aggregate scales with applied shear rate according to a power law with the slope equal to −1.1. This finding further support the validity of the presented results. The larger variation of the data obtained from the 3D-PTV method can be explained by the variation of the aspect ratio of individual aggregates. In fact, it was found that its value varies from 1 (sphere) up to approximately 2 (prolate ellipsoid) with an average value around 1.4. These values are in good agreement with our previous measurement of aggregate aspect ratio using microscopy imaging.29 Taking into consideration the difference in number of the aggregates analyzed by both techniques (single aggregate for 3D-PTV versus several millions in the case of light scattering), overall agreement between both techniques is satisfactory, indicating comparability of both techniques. Breakage Rate Measurements. A general definition of the breakup rate of an aggregate of mass ξ can be written as follows:21,38 1 fξ = ⟨τξ⟩ (3)

∞ dNξ

KR = −

∫0

Rξ dξ

dt ∞

∫0 NξRξ dξ

∞ dNξ

+

∫0

dt

dξ

∞

∫0 Nξ dξ

(6)

The time derivative of the number of aggregates in the absence of aggregation is governed by a population balance equation (PBE):54 dNξ dt

= −fξ Nξ +

∫ξ

∞

gξ , ξ ′fξ ′ Nξ ′ dξ′

(7)

where gξ,ξ′ is the number based fragment distribution function. Substitution into eq 6 results in, 1

KR =

∞

∫0 NξRξ dξ −

−

where τξ is the time lag between the release of the aggregate and its breakup, and ⟨·⟩ denotes an ensemble average over many realizations. Breakup of an aggregate occurs when the hydrodynamic stress acting on the aggregate overcomes its strength. In the case where the stress exerted on the aggregate is homogeneous and temporally uncorrelated, breakup follows a Poisson process such that we equivalently can define the breakup rate as38

fξ = −

dln⟨R ⟩ dt

1 ∞

∫0 Nξ dξ

∫0

∫0

∫0

∞

Rξ × (

∫ξ

∞

fξ NξRξ dξ ∞

gξ , ξ ′fξ ′ Nξ ′ dξ′)dξ] ∞

fξ Nξ dξ −

[

∞

[

∫0 ∫ξ

∞

gξ , ξ ′fξ ′ Nξ ′ dξ′dξ] (8)

Assuming a monodisperse aggregate size distribution Nξ ∼ δ(ξ − ξ0), with ξ0 being the initial aggregate mass, the above reduces to ⎡ KR = fξ × ⎢ 0 ⎢⎣

dln Nξ(t )

(4) dt where Nξ(t) is the number of aggregates of mass ξ at time t after their release. Even though eqs 3 and 4 present valid definitions of the breakup rate, and recent numerical studies have used them for measuring breakup rates in simulations,21,38 their application requires the observation of a large number of aggregates over a sufficiently long time, which is beyond our experimental capabilities. The complexity of detecting breakup events in the 3D-PTV experiments forced us to run the experiments with a “one aggregate at a time” strategy, which prevented us from

∫0

∞

⎛ Rξ ⎞ ⎤ ⎟ dξ ⎥ gξ , ξ ⎜⎜1 − 0 R ξ0 ⎟⎠ ⎥⎦ ⎝

(9)

Equation 9 implies that the size reduction rate is proportional to the number based breakup rate, with the proportionality factor depending on the daughter distribution function gξ,ξ′. As shown below, gξ,ξ′ shows only a weak dependency on the shear rate and the size of the mother aggregate, which reduces the proportionality factor in eq 9 to a constant. The important result here is that the size reduction rate is more easily approximated from our experiments, compared to the number based breakup rate typically used in numerical experiments. Writing eq 5 in the form KR ≈ −(1/⟨R⟩)[(Δ⟨R⟩)/(Δt)] and taking Δ⟨R⟩ as the difference between the original aggregate 14392

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Figure 8. (a) Breakup rate KR as defined in eq 10 as a function of strain rate evaluated for the aggregates composed of primary particles with dp = 420 nm and initial df = 1.8 measured by light scattering (▲) and by 3D-PTV (circles). Also shown are data of Soos et al.19 and Harshe et al.20 for two primary particle sizes of 90 and 810 nm, respectively, and two values of df equal to 1.8 and 2.7 [see panel (b) for the explanation of symbols]. Lines indicate scaling of the data with strain rate. (b) Scaling of KR as a function of D0 × τβ obtained from panel (a). Line represents a linear scaling. (c) Probability distribution function of formed fragments evaluated from 280 breakage events measured by 3D-PTV. (d) Fragment distribution function for the largest fragment (symbols) together with the best fit to the data using Schultz-Zimm distribution function56 (line).

and the largest fragment, and Δt as the inverse of the local shear rate, we obtain KR ≈

D0 − DLF S D0

Lattuada,21 values of KR (see Figure 8a) were plotted as a function of the product of D0 × τβ with τ denoting the hydrodynamic stress (product of dynamic viscosity and strain rate) and the exponent β being the log−log slope determined from Figure 8a. As can be seen from Figure 8b, all the results follow the same scaling with respect to D0 × τβ, indicating a linear dependency of KR on the initial aggregate size. A similar finding was obtained also by Harshe and Lattuada21 when considering the breakup of aggregates in a purely elongational flow. Lastly, it is worthwhile to notice that power law expressions for the breakup rate are often applied to model breakup in the framework of PBE. However, except the numerical simulation study of Harshe and Lattuada,21 we are not aware of any model that would allow for deriving such expressions. The use of power-law rate expressions therefore relies on estimating the parameters by fitting the PBE model to experimental data. Here, instead, we determine the breakup rate from the direct measurements of breakup events without assuming any functional form of the breakup rate, and the power-law behavior seen in Figure 8 therefore presents an independent determination of the functional form of the breakup rate. Fragment Distribution Function. The final quantity which can be extracted from 3D-PTV data is the number and size of the formed fragments. The latter is reported in the form of a size-based fragment distribution function (FDF) defined as the ratio of the size of the largest fragment with respect to the initial aggregate size. Additionally, we report the relative size of the largest fragment with respect to all other fragments formed. Figure 8c shows the distribution of the number of fragments for all analyzed breakage events. As can be seen, a typical breakage event will result with probability of 75% in the formation of two to three fragments, while the formation of five and more fragments has a very low probability of slightly above 10%. The corresponding size ratios of the largest fragment to the initial

(10)

where D0 is the aggregate size before breakup, DLF is the size of the largest fragment, and S is the strain rate at the position of breakup measured by 3D-PTV. When using light scattering in the circular nozzle experiments, the size reduction rate is approximated likewise by eq 10 with D0 and DLF replaced by the mean radius of gyration before and after passing the nozzle, respectively. The strain rate in this case is given in Table 1. A comparison of KR values evaluated for the aggregates with df equal to 1.8 composed of primary particles with diameter of 420 nm obtained from both experimental techniques discussed above is presented in Figure 8a. In addition, KR values obtained from the breakage data measured by Soos et al.19 and Harshe and et al.20 who used different primary particles with diameters of 90 and 810 nm, respectively, and two different initial values of df equal to 1.8 and 2.7, are plotted. It can be seen that for all breakage experiments, the evaluated breakage rate follows a power law with respect to the applied strain rate. In addition, independent of the primary particle size, a smaller slope of about 1.25 was found for the aggregates grown under static conditions characterized by an initial df around 1.8. In contrast, aggregates produced under turbulent conditions with df around 2.7 exhibit stronger dependency on the strain rate with a slope equal to 1.65 (see Figure 8a). The obtained scaling is almost identical with that obtained by Harshe and Lattuada21 using Stokesian dynamics with exponents equal to 1.37 for aggregates with an initial df of 1.8 and equal to 1.58 for aggregates with an initial df of 2.7, respectively. Furthermore, based on the results of various initial sizes of the aggregates, we can evaluate the dependency of KR on the initial aggregate size D0. In particular, following a similar approach as that used by Harshe and 14393

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aggregate calculated for two, three, four, and five fragments are presented in Figure 8d. It can be seen that independent of the number of formed fragments, the FDF is asymmetric with a characteristic maximum around 0.78 ± 0.02. When evaluating the ratio of the largest fragment with respect to the smaller ones, it was found that it varies from 1.2 to 2.5, depending on the number of fragments (see Figure SM 10 of the Supporting Information). The rather similar size of the second largest fragment can be explained by the fact that the formed fragments are more prolonged with respect to the original aggregates. This is in agreement with the sequence of the aggregate images presented in Figure 5 as well as with the results presented by Harshe et al.20 when performing Stokesian dynamics of the aggregates breakup in extensional flow. Although a direct comparison of the FDF obtained in this study and those presented by Harshe and Lattuada21 is not possible (they use mass-based distributions while our data are all based on the aggregate sizes), in both cases, the formed fragments follow asymmetric distributions, further supporting the applicability of the presented method to study the breakup of aggregates in extensional flow.

Article

ASSOCIATED CONTENT

* Supporting Information S

Characterization of formed aggregates with light scattering; relation between size reduction rate and breakup rate; experimental setup used to study aggregate breakup; 3D calibration block composed of three layers with dots; example of flow trajectories recorded during flow field characterization in the orifice; velocity magnitude and velocity vector in the orifice shown in three different locations; change in total pixel number and gray value during aggregate breakup; example of parent and fragments trajectories detected for breakage event; breakage locations inside the flow domain; characterization of the aggregates internal structure using light scattering data; contour plot of the strain rate magnitude S; and size ratio of largest gragment and smaller fragments. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +41 44 6334659. Fax: +41 44 6321082.

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CONCLUSIONS In the presented work, a combination of two noninvasive techniques [i.e., light scattering and 3D-particle tracking velocimetry (3D-PTV)] was used to monitor the breakage of statically grown aggregates composed of 420 nm polystyrene particles exposed to an extensional flow through a contracting nozzle. The light-scattering method was applied over a broad range of strain rates and aggregates sizes, allowing for a statistical determination of the formed fragment size and morphology after the breakage event. Important information about the local hydrodynamic condition of the individual breakage event, the number of formed fragments and their distribution function was obtained using a complementary 3DPTV. This optical method provides trajectories of the individual aggregates undergoing breakup and the local strain rates along with the number of formed fragments and their size distribution. The detected breakage locations indicate that most of the aggregates break approximately 1 mm upstream from the entrance into the orifice where the strain rate magnitude is highest and approximately equal to 100 1/s. From the eigenvalues of the rate of strain tensor, we found that at this point, aggregates are exposed mostly to the uniaxial straining. The obtained information was used to experimentally approximate for the first time the breakage rate of aggregates in the form of a size reduction rate KR, the probability distribution function of the number of formed fragments, as well as the size ratio between the largest fragment and the original aggregate. It was found that KR scales with the applied strain rate according to a power law with the slope dependent on the initial aggregates’ fractal dimension only. This scaling is in close agreement with that obtained previously by other authors using numerical simulation. The measured number of the formed fragments indicates that each breakage event will most probably result in two to three fragments (probability of 75%) with highly asymmetric size ratio of the largest fragment to the original aggregate having a maximum around 0.8. The obtained data provide a first experimentally derived breakage kernel, which can be used to solve the population balance equation to model the breakage processes.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the Swiss National Science Foundation (Grants 119815 and 200020-147137/1).

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REFERENCES

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