Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using mono-dispersed polystyrene latex particles

CIS-01576; No of Pages 14 Advances in Colloid and Interface Science xxx (2015) xxx–xxx

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Historical perspective

Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using mono-dispersed polystyrene latex particles Lili Feng c, Martien Cohen Stuart b, Yasuhisa Adachi a,⁎ a b c

Graduate School of Life and Environmental Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba-shi, Ibaraki 305-8572, Japan Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6307 HB Wageningen, The Netherlands School of Water Conservancy, North China University of Water Resources and Electric Power, No. 36, Beihuan Road, Zhengzhou, China

a r t i c l e

i n f o

Available online xxxx Keywords: Colloidal flocculation Dynamics of polyelectrolyte adsorption Proximate stagnant layer Charge neutralization Bridging flocculation

a b s t r a c t The dynamic behavior of polyelectrolytes just after their encounter with the surface of bare colloidal particles is analyzed, using the flocculation properties of mono-dispersed polystyrene latex (PSL) particles. Applying a Standardized Colloid Mixing (SCM) approach, effects of ionic strength and charge density of polymer chain on the rate of flocculation, the electrophoretic mobility of particle coated with polyelectrolyte, and the thickness of adsorbed polymer layer were analyzed, focusing on distinguishing features of two modes of flocculation, namely bridging formation and charge neutralization. In the case of excess polymer dosage, the bridging flocculation clearly highlights the transient behavior of polymer conformation from random-coil-like in bulk solution to increasingly flatten on the surface. The adsorption of polymer chains leads to a stagnant layer of solvent near the solid wall, which is confirmed by electrokinetic data. In the regime near optimum dosage two cases emerge. For high charge density polymer, charge neutralization is dominant and advantageous for the continuous progress of flocculation by heterogeneous double layer interaction. As a function of elapsed time after the onset of mixing, crossover from bridging to charge neutralization is found. In the case of low charge density polymer, bridging flocculation is the mechanism. Fluid mixing is concluded to have an essential role in the formation of bridges. © 2015 Elsevier B.V. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymer chains in aqueous solution and at interfaces . . . . . . . . . . . . . . . . . . . . . . . 2.1. State of chains in solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. State and dynamic behavior of polymer and polyelectrolyte chains at interfaces . . . . . . . . 3. Experimental section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Flocculation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Experimental tools for flocculation properties . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Standardization of colloid mixing (SCM) in terms of collision process . . . . . . . . 3.2.2. Evaluation of the thickness of an adsorbing polymer layer from the collision frequency 3.2.3. Hydrodynamic layer thickness from diffusion coefficient . . . . . . . . . . . . . . 3.2.4. Electrokinetics and flocculation rate . . . . . . . . . . . . . . . . . . . . . . . 3.3. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Application of SCM to the dynamics of flocculation at excess dosage . . . . . . . . . 3.3.2. The progress of polyelectrolyte adsorption at excess dosage . . . . . . . . . . . . . 3.3.3. Relaxation of polyelectrolyte layers after overshooting . . . . . . . . . . . . . . . 3.3.4. Colloidal stability near the isoelectric point . . . . . . . . . . . . . . . . . . . . 3.3.5. Isoelectric point shift versus ionic strength . . . . . . . . . . . . . . . . . . . . 3.3.6. Analysis of orthokinetic flocculation near the isoelectric point . . . . . . . . . . . . 4. Conclusions, remarks and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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⁎ Corresponding author. E-mail address: [email protected] (Y. Adachi).

http://dx.doi.org/10.1016/j.cis.2015.09.004 0001-8686/© 2015 Elsevier B.V. All rights reserved.

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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1. Introduction The use of water soluble polymers as flocculants to control the stability of colloids has a long history. A very early introduction of the effect of adding polymer on the flocculation can be found in Sanskrit literature (ca. 2000 B.C.) [1]. Nowadays, flocculation is commonly used in, e.g., conventional water and waste water treatment processes [2], mineral processing [3], and pulp and paper making [4] to enhance the precipitation, floatation and filteration of suspended matter. Synthetic polymers usually displace the natural biopolymers extensively used in ancient times. Despite long history and widespread use, the scientific understanding of the flocculation mechanism is still under discussion, at an elementary level. The main difficulty is the lack of knowledge on the transient dynamics of polymer chains from their state in bulk solution to that at interface. Extraction of the elemental processes involved in the flocculation is also impossible in such notoriously chaotic mixing condition. Two mechanisms of colloidal flocculation: bridging and charge neutralization, have been recognized [5–21]. Depletion flocculation, induced by non-adsorbing polymer (negative adsorption) [22,23], will not be discussed here. In Fig. 1, the role of nonionic polymer or polyelectrolyte in the flocculation and the stabilization is briefly illustrated. One can easily imagine that immediately after adding polymers to the colloidal dispersion, the polymers which have strong affinity to the particle surface start to adsorb onto the ‘starved’ particle surfaces. Bridging is formed when one polymer chain adsorbs on two or more particles (Fig. 1(a)). This concept was proposed in 1952 by Ruehrwein and Ward to explain the formation of soil aggregates [5]. In their experiment, the optimum polymer dosage exists and gives maximum flocculation efficiency. The presence of optimum dosage was explained by the concept of successful collision to result in floc formation. That is, the collisions between particles with polymer coated surfaces and uncoated surfaces are successfully making flocs [7]. For the polymer chains with opposite sign of charge to the colloidal particle, the polyelectrolyte adsorption will result in the reduction of the total amount of charge of the colloidal particle, and excess polyelectrolyte adsorption will eventually reverse the particle charge. Usually the most effective flocculation takes place near to complete charge neutralization [8] (Fig. 1(b)).

However, in a concentrated cellar suspension, it is reported that the flexibility, size and charge density of polyelectrolyte chain will influence the optimum flocculation dosage [24]. Repulsion between two surfaces will appear when the amount of adsorption exceeds a certain threshold. In the case of neutral polymer, the overlap of protruding part of adsorbed polymer layer will disturb the approach of two surfaces (Fig. 1(c)). In the case of polyelectrolyte conveying opposite charge, usually charge reversal takes place and colloidal dispersion will be electrostatically stabilized by the interaction between overcharged surfaces (Fig. 1(d)). For both mechanisms of flocculation, the key factor is a change of morphology of the adsorbing polymer chains from their initial conformation in bulk solution to a final state on the surface [9,25,26]. As is well known, sufficient efforts have been devoted to the statistical thermodynamics of nonionic polymer and polyelectrolyte, in solution as well as at interfaces [27–30]. However, there is limited information on the dynamics of polymer adsorption and subsequent flocculation. Probably, one of the biggest remaining issues is that the most important event of colloidal flocculation takes place at the beginning of mixing operation where the system is brought to non-equilibrium state in turbulent flow [31]. In the present review, our attempts to analyze the initial stage of flocculation, in terms of transport phenomena, started about two decades ago [32] are summarized. Our original idea has been very simple, namely, to consider the collision process between mono-dispersion of colloidal latex [33] and to extend the same idea to the collision between colloidal particles and polymer molecules [26,32,34–41]. Accumulation of data for various conditions has made it possible to correlate the obtained results to the elementary description on the nonionic polymer and polyelectrolyte dynamics when chains get attached from bulk solution to the interface [20,25,42,43]. Although our study began as a fundamental study to understand flocculation in water treatment, the results are relevant for a general understanding of the non-equilibrium dynamic properties of polyelectrolytes at interfaces and therefore also relevant for other fundamental aspects of, e.g., biological interaction of DNA and protein [44,45], microbiological column sensing [46], cell glue [47], drug delivery systems [48], thin films and multilayers [49,50], and chemical sensors [51].

Fig. 1. Effects of water soluble polymers and polyelectrolytes with an affinity to colloidal surfaces. (a) Bridging. (b) Charge neutralization. (c) Steric stabilization. (d) Electric double layer formation due to the charge reversal of colloidal surface.

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

L. Feng et al. / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

2. Polymer chains in aqueous solution and at interfaces

3

2.2. State and dynamic behavior of polymer and polyelectrolyte chains at interfaces

2.1. State of chains in solution The ability to stabilize or destabilize colloids is apparently a shared feature of both nonionic polymers and polyelectrolytes. That is, both are able to induce flocculation by means of adsorption. For high molecular weight polymer, both feature a narrow and clear window at optimal dosage [34,52], and a relatively stronger floc is formed [53] as compared to that of coagulation by only salt. Despite these common characteristics, the underlying mechanisms for both types of polymers are rather different. Theoretical treatment of neutral polymers in solution was first developed on the basis of statistical thermodynamics by Flory [27]. Later, De Gennes introduced an elegant approach on the basis of scaling concept to describe polymers in solution as well as in confined space [29]. Both approaches successfully describe the morphology of random coils, or self-avoiding random walks. While, in the case of polyelectrolyte, two extra important length parameters have to be introduced due to the presence of counter ions. One is the Bjerrum length, lB, [54] defined as, lB ¼

e2 4πεkT

ð1Þ

and the other is the Debye length, κ−1, the thickness of the electrical double layer [55–57] determined by κ −1 ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X εkT= e2 ni0 zi :

ð2Þ

i

The Bjerrum length represents the ratio of thermal energy to the Coulombic interaction between monovalent ions in aqueous solution. In water at room temperature, this value is about 0.7 nm. It is an important parameter relating to counter ion condensation and the flexibility of chains at high salt concentration [58]. On the other hand, even in the absence of added salt, there exists a certain concentration of counter ions. At such extremely low salt concentrations, the repulsion between charged segments is important. The Debye length is a characteristic length of the ionic atmosphere, and determines the range of repulsion between charged segments. This length is much longer than the size of segments, except for extremely high ionic strength, like sea water. For the case of very low ionic strength (e.g., pure water), the reduced viscosity increases strongly, implying extensive swelling of the polyelectrolyte [59]. This behavior can be understood from the idea that a long polyelectrolyte chain in solution is a random coil of which the persistent length, lp, will increase with increasing Debye length. In 1976, De Gennes et al. [60] proposed the simple idea that the polyelectrolyte chain would be stiff on the scale of the Debye length. Their idea essentially implied, lp  κ −1 :

Just as it is recognized that a polymer chain in solution takes the conformation of a random coil (or self-avoiding random walk), it is widely accepted that a polymer chain adsorbed on the colloidal surface takes a so-called train-loop-tail conformation [64,65]. The equilibrium conformation is statistically predicted by self-consistent mean field calculations or using the scaling concept [66,67]. In the case of polyelectrolytes with opposite sign of charge to the surface, there exists a long range electrostatic attraction between charged segments of the chain and the surface which tends to dramatically change polyelectrolyte conformation to a flat structure, like a carpet. This tendency will be weakened by the presence of counter ions which will shield the electrostatic interactions, as depicted in Fig. 2 [30,68–70]. The trend is essentially the same as that observed in the exchange behavior of polycation and polyanion in the polyelectrolyte complexes [71]. It should be noted that the concept of random coil or train-loop-tail structure is the result of equilibrium thermodynamic analysis. In practice, however, the transient behavior connecting the two states is of vital importance when one makes an analysis of irreversible process, thereby forming a new domain of study concerning the dynamics of polyelectrolytes at interfaces. Since this domain is important for, e.g., biological systems and tunable smart materials, the literature abounds with results from numerous efforts to characterize transients, such as theoretical treatments [30,68,72–74], numerical approaches [75], direct observation in well-controlled systems [76], measurements of kinetics of adsorption onto well-defined surfaces [77,78], and subsequent exchange and desorption, which often have rather long characteristic time scales [79–89]. When one makes an analysis of the process of colloidal flocculation, the transient behavior of polymer chain in solution to the adsorbed state on the colloidal surface is critically enrolled in the process of flocs formation. A schematic diagram of flocculation induced by addition of polymer flocculants is illustrated in Fig. 3. Upon mixing the polymer solution with the colloidal suspension, the system is brought to a non-equilibrium state. The ensuing flocculation process consists of several elementary events: (1) dilution of the polymer flocculants into a homogeneous solution; (2) collision between colloidal particles; (3) transportation of polymer chains toward the surface of colloidal particles; (4) re-conformation of adsorbed polymers on the surface of colloidal particles; (5) formation of a bond (or bridge) between colloidal particles; and (6) structural rearrangement and breakup of flocs.

ð3Þ

Shortly thereafter, Odijk [61] and later, Skolnick and Fixman [62] derived rather rigorously, lp  κ −2 :

ð4Þ

After that period, the debate how to choose between Eqs. (3) and (4) has been continued until now [63]. It should be noted that at the dilute limit of ionic strength both two equations predict remarkably long persistent lengths for charged chains. This factor is important when polyelectrolyte chains act as bridging agents between two or more colloidal particles.

Fig. 2. Illustration of the effect of ionic strength on polyelectrolyte conformation at an interface. At low ionic strength, polyelectrolyte chain will adsorb to oppositely charged surface in a flat conformation. At high ionic strength, the presence of indifferent ions weakens the electrostatic interaction between the charged segments and oppositely charged surfaces and loosens the conformation of polyelectrolytes.

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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L. Feng et al. / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

place during the early stage of mixing [6,91], and that is the reason why we take a closer look at the mixing process. 3. Experimental section 3.1. Flocculation properties

Fig. 3. Schematic diagram and elementary processes of flocculation with adsorbing polymers [31].

All these elementary processes occur simultaneously under the given mixing conditions which are usually turbulent flow. Trap of colloidal particles to stick together induced by the adsorbing polymer chains which will take places at a certain stage of adsorption is a process of flocculation. With this picture, it is not so difficult to understand the importance of any large conformational changes of the polymer chain, starting at the very beginning moment of collision of a random coil with the bare surface of colloidal particles. What will happen depends on the collision frequency and, thereby, on hydrodynamic conditions. One idea is to suppose that during this early stage of the process one has nonequilibrium trapping of polymer chains, which enters a glassy state when on the surface of colloidal particles, as depicted in Fig. 4. In the case of more concentrated polymer layers, pinning and entanglement of adsorbing chains may take place, which will lead to the increase in adsorbed amount and in layer thickness [90]. Experimentally, results concerning the flocculated state varies strongly, depending on the way of colloid mixing. This implies that the most important event takes

Although it is underlined that transient conformation of adsorbing polymer at the initial stage of flocculation is significant, few studies are available to give direct evidence [25]. Since colloidal flocculation is largely influenced by the state and dynamics of adsorbing polymer chains, the process of flocculation, in a well-defined system, can inform us about the dynamics of polymer chains on the surface of colloidal particles. In this review, we focus on the formation of bridges by protruding chains, taking into account the dynamics of re-conformation on the surface of colloidal particles, and on the effect of patch-wise attractive interaction between colloid particles coated with oppositely charged polyelectrolytes. In order to get a measure of the protruding part of polymer chains on the colloidal surface, we introduced a method of standardization of colloid mixing (SCM) which is based on the concept of collision radius in standardized mixing flow. This method will provide us the effective radius of the colloidal particles in terms of collision frequency [33–40]. The obtained results were enforced by the hydrodynamic Stokes diameter of colloidal particles with adsorbed polymer on the basis of single particle tracing. Additional data on the colloidal stability estimated from the rate of flocculation by particle counting, and the charging property estimated by the measurement of electrophoretic mobility were also employed to confirm the picture of transient behavior of polyelectrolyte adsorption. In Table 1, properties of colloidal particles and polymer flocculations used in our experiments are summarized. As colloid materials, a model dispersion consisted of mono-dispersed polystyrene latex (PSL) particles is used. As for polymer flocculants, homo-polymer of polytrimethylamino-ethylmethacrylate (hereinafter referred as PTMA) with different molecular weights and co-polymer of poly-acryl amide and trimethylamino ethylmethacrylate (herein after referred as PTMC) are employed. 3.2. Experimental tools for flocculation properties 3.2.1. Standardization of colloid mixing (SCM) in terms of collision process The extension of polymers adsorbed on the surface of colloidal particles can be detected as an increment of the collision radius of colloidal

Fig. 4. Dynamics of polymers in the course of adsorption. (a) The random coil conformation will change into more or less flatten loop-tail-train conformation. (b) In the case of high polymer concentration, each polymer chain will have more possibilities to become trapped by neighboring chains: pinning and entanglement.

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

L. Feng et al. / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

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Table 1 Material applied in the model experiment.

The validity of this expression was confirmed by our measurements of the coagulation rate as a function of a0. The special mixing device, namely an end-over-end rotated forked flask (Fig. 5), is employed to produce a turbulent flow [37]. Mixing is induced by the rocking motion of the flask end-over-end at a fixed frequency of 1 [Hz]. Our analysis, using salt-induced rapid coagulation of PSL particles with 2 a0 = 1356[nm], yields the following values: ε = 706 [cm 2/s 2] and α T = 0.38, by assuming the value of Hamaker constant as A = 6.0 × 10− 21[J] [34]. The knowledge of the collision frequency can now be applied to the analysis of the adsorption process of a polymer to a colloidal particle, assuming that the rate of adsorption is limited by the transport of polymer molecules from bulk solution to the surface of a colloidal particle [26]. Since polymer molecules are too small to neglect the contribution of Brownian motion, this transport has contributions from both diffusion and convection (shear). Hence, the rate of polymer supply to the reference colloidal particle, Jp, must be expressed by the sum of the diffusion flux induced by Brownian motion and the convective flux induced by turbulent mixing. That is,

1) Estimated from the method proposed by Kobayashi [92]. 2) Used for the analysis of Brownian Motion (3.2.3).

J p ¼ 4πDop Rop Nop þ

particles, via an increment in the rate of flocculation. In general, the collision frequency depends on the effective shear rate, i.e., on the details of the flow pattern, and it is difficult to assess a priori in an agitated fluid. However, if one assumes that all collisions between colloidal particles in turbulent mixing flow successfully lead to coagulation, a particular turbulent flow can be calibrated in terms of its generated collision frequency based on the measurement of the coagulation rate. By using exactly the same standardized flow, one can then evaluate the effect of polymer chains by monitoring the rate of flocculation. This is what we call the Standardized Colloid Mixing (SCM) method. We first consider rapid coagulation of mono-dispersed colloidal latex induced by a sufficient amount of salt in a turbulent flow. An equation for the rate of coagulation can be derived by invoking the hypothesis of local isotropy of turbulence proposed by Kolmogorov [93]. The rate of coagulation, in terms of the change in particle number concentration N(t) [34] as a function of time, is derived as

Here, Dop, Rop, ap, Np denote the relative diffusion constant between polymer molecules and colloidal particles, the collision radius between polymer molecules and colloidal particles, the radius of polymer molecules, and the number concentration of polymer molecules, respectively.

dN ðt Þ 1 ¼ − αT dt 2

rffiffiffiffiffiffiffiffiffi 8πε ð2a0 Þ3 Nðt Þ2 15ν

ð5Þ

Here, ε, ν and a0 are rate of energy dissipation per unit mass of fluid, kinematic viscosity and the radius of colloidal particles, respectively. αT is the collision efficiency taking into account the hydrodynamic interaction and Van der Waals attraction [94]. The value of αT without any electrostatic repulsion can be approximately expressed as [95]. 0 B αT ¼ @

rffiffiffiffiffiffiffiffi 3 8πε  a0 þ ap Np 15v

ð9Þ

3.2.2. Evaluation of the thickness of an adsorbing polymer layer from the collision frequency The effect of the application of polymer flocculants can be evaluated by simply comparing the progress of flocculation with and without polymer flocculants. Immediately after adding polymer to the dispersion, flocculation is enhanced, because the surfaces are still largely ‘starved’ (bare) and the attachment of polymers leads to an increase in effective collision radius of the particles. Moreover, the hydrodynamic interaction is weak because the particle surfaces do not have to come close for a bridge to form, which implies that the collision efficiency is unity in this case. After some time, the enhancement will be suppressed when the surface becomes saturated with polymers, because saturated surfaces lead to steric repulsions between polymer segments. As can be seen in Eq. (5), the rate of flocculation in the flow field is strongly dependent on the size of colloidal particles. The effective thickness of polymer layer, δH, can be obtained from the ratio of the initial

10:18 A 36πμa0

3

C qffiffiffiffiffiffiffiffiA 4ε 15πν

ð6Þ

Where A is the Hamaker constant and μ is the viscosity. At the early stages of coagulation, Eq. (5) can be approximately integrated, using the continuity condition Nðt Þa3 ¼ Nð0Þa30 ¼ Const:

ð7Þ

Then the progress of coagulation (number of particles) at the initial stage can be written as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nðt Þ 128πε ¼ −α T Nð0Þa30 t: ln Nð0Þ 15v

ð8Þ

Fig. 5. The end-over-end rotation mixing device to generate standardized mixing for small volume (10 ml) samples. The forked flask which is made by connecting two flat-bottomed cylindrical bottles 2.5 cm in diameter and 3.0 cm in height, and mounted on a rotation device. After mixing of a predetermined number of tumblings (mixing steps), the sample will be analyzed by particle counting, electrophoretic mobility, diffusion coefficient, etc.

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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L. Feng et al. / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

rate of flocculation, with and without polymer, by the following equation. η¼

ða0 þ δH Þ3 α T a0 3

ð10Þ

When the polymer concentration is high, the probability to find particles with freshly adsorbing polymers is also high. That is, the effective collision radius will increase with an increase of polymer concentration. However, the required time for saturation will then also decrease. Hence, at higher polymer concentration the initial flocculation rate is higher, but the stabilization also comes sooner. 3.2.3. Hydrodynamic layer thickness from diffusion coefficient We have developed a method to evaluate the hydrodynamic layer thickness from the Brownian diffusion coefficient obtained by tracking the motion of single colloidal particle. Trajectories are recorded by a CCD video camera mounted on the normal optical microscope [96]. The diffusion constants can be obtained on the basis of the Einstein relation [97]: Δx2 ¼ 2Dτ

ð11Þ

where Δx is the displacement of the colloidal particle in one dimension at the time interval of τ, calculated from the change in position of its center of mass. The hydrodynamic radius a is then obtained using the Stokes–Einstein equation: D¼

kT 6πaη

ð12Þ

where k is the Boltzmann constant, T the temperature and η is the viscosity of the solvent. When a polymer (polyelectrolyte) is adsorbed onto the surface of colloidal particle, the particle radius increases by an amount equal to the layer thickness of adsorbed polyelectrolyte [98]. That is, the diffusion constant becomes D¼

kT 6πða þ δH Þη

ð13Þ

where δH is the hydrodynamic thickness of the polyelectrolyte layer formed on the particle surface. In principle, one can obtain δH of course also by means of DLS (dynamic light scattering) [99,100], but as a rule, the data acquisition time is too long to observe the early stage of adsorption and in addition, the distinction of the decrease of diffusion constant by the formation of adsorbing polymer layer and by the flocculation of colloidal particles is not decisive.

3.2.4. Electrokinetics and flocculation rate Electrokinetics and flocculation rate are widely accepted techniques to study the flocculation properties of colloidal suspensions and they also provide information on the dynamic behavior of polymers on the surface of colloidal particles. The most fundamental index is the absolute rate of Brownian flocculation. The rate of flocculation can be determined by monitoring the temporal evolution of size distribution of flocs as a function of elapsed time, by means of a Coulter Counter or an alternative optical method. The charging behavior of a colloidal particle can be detected by the zeta potential, which is defined as the electric potential at the slipping plane, and determined by the electrophoretic mobility of particle in an electric field. Since in the case of adsorption of polymer chains, the location of slipping plane becomes a rather ambiguous concept, we directly compare the measured electrophoretic mobility. Monitoring the electrophoretic mobility along with the progress of flocculation will provide information on how the polyelectrolyte adsorption is followed by the flocculation. 3.3. Results and discussion 3.3.1. Application of SCM to the dynamics of flocculation at excess dosage Results of the progress of flocculation induced with PTMA4.9M (where M stands for million gram per mole) at high ionic strength are shown in Fig. 6 [41]. At such high ionic strength, the polyelectrolyte chains in solution take the conformation of an ideal, non-swollen random coil. The large enhancement of the initial flocculation rate followed by an abrupt stabilization is the common characteristics of flocculation process at excess dosage. The tendency will be weakened with a decrease of polyelectrolyte concentration. At moderate dosage, one indeed observes a less steep initial slope and a long duration of the initial stage. This tendency is also observed at lower ionic strength [34]. However, it should be noted that the estimated values of δH on the basis of Eq. (10) are now much larger than the size of the same polyelectrolyte estimated from viscosity and DLS (Dynamic light scattering). At this ionic strength, polyelectrolyte chains are expected to be very flexible and hence can be easily stretched by the fluid shear during the mixing operation. This may well explain why we detected such extremely thick layers (as illustrated in Fig. 7). In Fig. 8, we demonstrate the effect of molecular weight of PTMA on the rate of flocculation at low ionic strength (KCl10−4M). Fig. 8(a) represents the result at excess dosage (Cp = 0.5 mg/l) and (b) represents the result at moderate dosage (Cp = 0.075 mg/l). Polyelectrolytes of higher molecular weight enhance the initial rate of flocculation more effectively, but also bring it to an earlier halt. This tendency is the same as the bridging flocculation progress induced by neutral

Fig. 6. Progress of flocculation of PSL1356 with PTMA4.9 M at high ionic strength (KCl 1.0 M). (a) ln(N(t)/N(0)) vs. t, the number of steps(t) up to 200 steps. (b) Figure scaled up to 60 steps. ln(N(t)/N(0)) vs t for flocculation induced by added polyelectrolyte. N(0) = 5.11 × 107 cm−3. The solid line represents the regression line of the result of salt-induced rapid coagulation (◆). Cp = (+) 0.2 (mg/l), (×) 0.5 mg/l, (▲) 1 mg/l, (■) 3 mg/l. Taken from [41].

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Table 2 Values of δH (nm) for PTMA4.9M obtained from NM under overshooting conditions.

Fig. 7. Schematic drawing of a polymer chain attached to a colloidal particle at high ionic strength. (Taken from the graphical abstract of [41]).

polymer [26,36]. Hence, one can conclude that at excess dosage, the conformation of polyelectrolytes in solution is well correlated with the increment of the collision radius, and that bridging is the flocculation mechanism. In Table 2, the values of δH (nm) for PTMA4.9M obtained on the basis of Eq. (10) are summarized. As indicated in the table, δH has a tendency to increase with a decrease of ionic strength, reflecting the swelling behavior of polyelectrolyte chains in solution. δH also decreases with a decrease of Cp. This reflects that polyelectrolyte chains on the surface unfold in the course of time. The result of flocculation at moderate dosage is shown in Fig. 8(b). Two trends are noteworthy. One is that we can detect a time delay for the onset of flocculation. The delay becomes longer for shorter chains. The other is that the maximum rate of flocculation does not always appear immediately after the start of mixing. Moreover, the maximum rate is almost the same as, or slightly larger, than that of salt-induced rapid coagulation. This can be regarded as an implication of patchwise heterogeneous attractive interaction which will be more clearly discussed in 3.3.6.

3.3.2. The progress of polyelectrolyte adsorption at excess dosage In Fig. 9(a)–(d), the electrophoretic mobility of primary particles with adsorbed PTMA (different in molecular weight) was plotted as a function of the number of tumbling. As indicated in these figures, the negative charge of PSL particles reduces and eventually reverses with the continuous adsorption of oppositely charged polyelectrolytes. Interestingly, the electrophoretic mobility of particle-polyelectrolyte complexes converges to the same plateau value regardless of the polymer concentration, except some sort of overshooting seems to occur in the

Cp (mg/l)

10−4

10−2

100

3 1 0.5 0.2 0.13

– – 465 297 180

– – 333 222 50

701 274 84 – –

(KCl M)

case of high molecular weight chains at low ionic strength. These exceptions can be regarded as an appearance of kinetically controlled adsorption [90]. The polyelectrolyte with larger molecular weight can be regarded to be less mobile at lower ionic strength. The landscape is illustrated in Fig. 10.

3.3.3. Relaxation of polyelectrolyte layers after overshooting The layer observed under conditions of overshooting can be regarded as transient appearance of a non-equilibrium state. This raises the question how, and how fast the non-equilibrium state will relax to a more stable state. Here, we apply two tools. One is the estimation of the diffusion coefficient of one single colloidal particle, and the other the analysis of electrophoresis. Examples of Brownian trajectories of PSL particles are demonstrated in Fig. 11. As can be seen in the figure, Brownian motion is dramatically hindered by the adsorbed polyelectrolyte layer. Indeed, a relaxation process is observed; the adsorption experiments were carried out at two different ionic strengths. Layer thickness is calculated on the basis of Eq. (13) and shown as a function of elapsed time in Fig. 12. In the case of 10−4 M KCl, extremely large values were initially detected at polyelectrolyte concentration of 5.0 mg/l. The initial value of δH was found to correlate with the size of polyelectrolyte coil in bulk solution. With decreasing polyelectrolyte concentration at fixed ionic strength, a thinner layer seems to be formed. As “kinetically-controlled adsorption” predicted, the decrease of layer thickness can be attributed to two competitive mechanisms for layer formation: (1) polymer flux toward the particle surface and (2) spreading of adsorbed polyelectrolytes [90,101]. The increase of associated counter ion concentration has essentially two effects. One is to induce shrinkage of the polyelectrolyte in the bulk solution. This will enhance the mobility of polyelectrolyte chains on the surface as well as in the bulk. The other is the decrease the electrostatic attractive forces between the segment of polymer chain and the

Fig. 8. Progress of flocculation of PSL1956 with PTMA of different molecular weight [121]. ln(N(t)/N(0)) vs t (sec.), (a) Cp = 0.5[㎎/l],(b) Cp = 0.075 [㎎/l]. KCl = 10−4 [M], molecular weight of applied polymers: ■: 4.9 × 106, ◇:3.5 × 106, ▲:1.2 × 106, ○:5 × 105, ×: 1.6 × 105 [g/mol] (nominal value).

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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Fig. 9. Electrophoretic mobility of the primary PSL particle with adsorbed PTMA vs number of mixing steps [34]. (a) Mw = 490 kDa, KCl = 10−4 [M] (b) Mw = 1200 k Da, KCl = 10−4 [M],(c) Mw = 490 k Da, KCl = 10−2 [M],(d) Mw = 1200 k Da, KCl = 10.−2 ●:Cp = 2.0 ㎎/l △: Cp = 1.0 ㎎/l *: Cp = 0.5 ㎎/l.

oppositely charged surface. Strictly speaking, the mechanism of the latter in general is not so simple. Hesselink [102] theoretically predicted the adsorption of polyelectrolyte will be enhanced or reduced depending on the charge density of polyelectrolyte and surface, and non-electrostatic interaction between chain and surface. Actually, in many cases, an increase of ionic strength will enhance the adsorption of polyelectrolyte [103]. However, our experimental result obtained using PSL particles demonstrates opposite tendency. The contrary can be explained by the presence of specific interaction of the counter ions to the surface reported by the analysis of subtle force balance [104]. Here, we conclude our case corresponds to

Fig. 10. Expected conformation of a polyelectrolyte adsorbed on the surface of colloidal particles [34]. The conformation of polyelectrolyte with large molecular weight under the condition of low ionic strength is found to be affected by the adsorption process.

the case of the reduction of attractive force between the polyelectrolyte chain and the surface of PSL particles. The two effects mentioned above will reduce the amount of adsorption, but increase the rate of relaxation with the increase of ionic strength. As demonstrated in Fig. 12, the layer thickness indeed decreases dramatically over a typical timescale of one hour. Detachment of polymer chains and/or reconformation of initially adsorbed chains into more flat conformation are very likely to take place. Experimental results of electrophoresis of colloidal particles show the relaxation process of the adsorbed layer thickness, presented in Fig. 13. The initial value of electrophoretic mobility qualitatively confirms our previous results [34,37,38] summarized in Fig. 10; at low ionic strength the value of particles adsorbing polymers is largely dependent on the polyelectrolyte concentration, while, at high ionic strength, this difference disappears. As indicated in Fig. 13, the electrophoretic mobility of polyelectrolyte-coated particles remains almost constant over a time scale of several hours, whatever the ionic strength. However, surprisingly, the electrophoretic mobility remains constant even though the value of hydrodynamic layer thickness decreases dramatically. One can conclude that the ‘lost’ part of the adsorbed layer does not contribute to the electrophoretic mobility. So, the outer polyelectrolyte layer can be regarded as free-draining [105,106]. Essentially, it means charge reversal takes place in a contact layer very close to the substrate, i.e., less than 10 nm, thinner than the detection limit of hydrodynamic layer. Within this region, the competition between the spreading of pre-adsorbed chains and the arrival of newcomers will determine the electrokinetic properties. This kinetically controlled process is considered to take place at low ionic strength. The picture which we conclude as best expressing this situation is illustrated in Fig. 14. It should be noted that the concept of kinetically

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Fig. 11. Typical trajectory of PSL804 particles. Diffusive Brownian motion is slowed down by the adsorption of PTMA4.9M. Taken from [98].

controlled adsorption and overcharging [107] does not conflict with the concept of the formation of thin stagnant layer. 3.3.4. Colloidal stability near the isoelectric point In previous sections, we have analyzed the characteristics of adsorbed polymer layers formed at excess dosage. However, it is usually recognized that flocculation takes place effectively at the so-called optimum flocculation concentration (OFC), which is, close to the isoelectric point (iep) [8,12–14,16,17,91]. The analysis near the isoelectric point is important from practical viewpoint. In Fig. 15, the adsorbing behavior of polyelectrolytes differing in charge density is compared. Fig. 15(a) is the result of the electrophoretic mobility after adsorption of high-charge-density polyelectrolyte (PTMA4.9M) onto PSL1356 [15] and Fig. 15(b) is that of low-charge-density polyelectrolyte (PTMC5.2M) [108]. For both polyelectrolytes, charge reversal takes place, demonstrating the presence of an isoelectric point. At the iep, the value of the hydrodynamic layer thickness, δH, for the case of PTMA4.9M is found to be smaller than detection limit (10 nm), and for the case of PTMC5.2M is found to be 15–20 nm [10,20]. In Fig. 16, the rate of Brownian (perikinetic) flocculation is plotted at different ionic strengths and polymer concentrations. The rate was measured using a Coulter Counter. If one compares this result with corresponding data of electrophoresis (Fig. 15), one can confirm that the OFC corresponds to the iep. It is noticeable the rate of flocculation with high charge density polymer (PTMA4.9M) at OFC is faster than that of saltinduced rapid coagulation at low ionic strength. The rate of flocculation with low charge density polymer (PTMC5.2M) is always smaller than

the value of salt-induced rapid coagulation. Their adsorption and corresponding flocculation process are illustrated in Fig. 17. The approach of the high-charge-density polymer is considered to be strongly accelerated by the attractive electrostatic force between polyelectrolyte segments and oppositely charged surface. The adsorption will create a ‘hole’ in the diffuse electric double layer. The flocculation by high-charge-density polymer (PTMA4.9M) can be regarded as an example of charge neutralization. Taking hydrodynamic layer thickness into consideration, the data confirm the concept of heterogeneous patch-wise attraction proposed by Gregory [8]; the patchy layer is thin. In the case of low charge density polyelectrolyte, the adsorption can occur preserving the diffuse electric double layer of free counter ions. Although the flocculation takes places near the iep, low charge density polymer (PTMC5.2M) can be regarded as a typical example of bridging. The electrokinetically neutralized condition allows two colliding particles to approach each other closely. However, the presence of the polymer layer will disturb their closer approach. Bridging is successful when the protruding end of adsorbed polymer chain reaches the surface of another colloidal particle. However, the process is sterically hindered by other adsorbed polymers, and thus the capture efficiency is reduced to less than unity. 3.3.5. Isoelectric point shift versus ionic strength With respect to the isoelectric points attained by the two polyelectrolytes, two trends attract attention. One is that the isoelectric points of two polyelectrolytes shift in opposite directions as a function of ionic strength. The other is that the iep appears at lower concentration than predicted on the basis of stoichiometric analysis; this effect is

Fig. 12. Time dependence of the hydrodynamic thickness of polyelectrolyte layer on a single PSL particle after the mixing event. The core particle is PSL804 and the polyelectrolyte is PTMA4.9M. (a) At ionic strength10−4 M KCl. (b) At ionic strength10−2 M KCl. Taken from [98].

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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Fig. 13. Time dependence of electrophoretic mobility of PSL804 with adsorbing PTMA4.9M after mixing event. (a) 10−4 M KCl. (b) 10−2 M KCl. Taken from [98].

more apparent for the low-charge-density polyelectrolyte. It should be noted that the charge density of PTMC5.2M is 25 times smaller than that of PTMA4.9M. The isoelectric point of particle adsorbing PTMA4.9M is approximately the same for salt-free and for 10−3 M KCl, but gives an obvious shift to higher polyelectrolyte dosage with further increasing ionic strength. A similar shift has also been reported in the system composed of PSL particles and polycation [101,105]. The increase of isoelectric point as a function of ionic strength can be explained by a reduction in the effective charge of polyelectrolyte with increasing ionic strength resulting from an enhanced degree of counter ions binding to the chain of polyelectrolyte. The shift of the iep with an increase of ionic strength in the case of the application of PTMC5.2M can be explained by the formation of stagnant layer, as depicted in Fig. 18. Counter ions located in the stagnant layer will contribute to the neutralization of charge of the colloidal particles. Due to this contribution, the iep will decrease upon an increase of ionic strength, which will be accompanied by the compression

of electric double layer. When ionic strength is increased, the probability that the diffusive part of electric double layer accumulate in the region of stagnant layer, will increase leading to the observed downward shift of the iep. 3.3.6. Analysis of orthokinetic flocculation near the isoelectric point In the previous section, we have characterized the two main functions of polyelectrolyte: (1) charge neutralization enhanced by patchwise heterogeneous attraction, and (2) formation of polymer bridges between colloidal particles. In many practical applications, however, the flocculation is performed in a turbulent flow. Unfortunately, the data of flocculation under standardized (optimized) conditions are scarce. In Fig. 19, the flocculation by charge neutralization and by bridging at optimum dosage are compared under standardized mixing (SCM). In Fig. 19(a), the flocculation of PSL1356 (PSL particle with diameter of 1356 nm) with PTMA4.9M is enhanced by patch-wise heterogeneous attraction [15]. As demonstrated by the slope of data plot, the rate of flocculation is enhanced by a factor of 2.4, and this enhancement continues

Fig. 14. Schematic illustration of the relaxation behavior of polyelectrolytes adsorbed onto the surface of a colloidal particle. Relaxation may involve desorption and reconformation which may take place mainly from a hydrodynamically unstable layer. As illustrated at the bottom of the figure, an electrokinetically stable thin stagnant layer with thickness of δ is considered to be formed at the time of mixing. In the case of high molecular weight and low ionic strength, the formation of this layer is kinetically controlled. Above this thin stagnant layer, a hydrodynamically unstable layer is stacked.

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Fig. 15. Electrophoretic mobility of PSL1356 particles as a function of polymer dosage at different ionic strengths. (a) PTMA4.9M [11], (b) PTMC5.2M [10].

over a rather long period. In terms of Eq. (6), it implies the present attractive force is approximately twenty times larger than Van der Waals forces [15]. In Fig. 19(b), we show the progress of bridging flocculation of PSL1356 induced by PTMC5.2M without salt addition [20]. The flocculation immediately takes place upon the onset of mixing, and it is faster than salt-induced rapid coagulation. The initial fast progress can be explained supposing that the conformation of the attached polymer maintains its solution conformation. In the later stage, the initial fast rate decreases gradually, presumably reflecting the onset of steric effect of adsorbed polymer. That is, at this stage, the formation of bridge is disturbed by the presence of adsorbed polymer layer. It is interesting to examine the flocculation efficiency taking the adsorption of polyelectrolytes into account. When the process of adsorption is limited by transport of polymer chains to the reference particles, the rate of adsorption can be approximately estimated on the basis of Eq. (9). Although this method is crude, it will provide a reasonable firstorder estimation [15,20,34,38]. The electrophoretic mobility of PSL1356 adsorbed with PTMA4.9M is shown as a function of mixing time in Fig. 20. After the predetermined number of mixing, the sample was gently picked up and diluted into the cell in which the electrophoretic mobility was monitored. The absolute value of electrophoretic mobility decreases with adsorption of oppositely charged polymers. A longer time is needed at higher ionic strength, and its possible reasons are a smaller size of polyelectrolyte coil in solution and weaker attraction

for adsorption. In the latter case, we can say, however, that adsorption of PTMA4.9M is not complete in less than 20 s. That is, the initial fast rate of flocculation observed in Fig. 19(a) should be ascribed to the increase of effective collision radius by extending polyelectrolytes. This condition gradually changes to a situation of patch-wise attractive interaction between the colloidal particles as the adsorption of PTMA4.9M proceeds. The adsorption behavior of PTMC5.2M onto PSL1356 is shown in Fig. 21. Without salt addition, the adsorption takes place almost at the same rate as that of PTMA4.9M. Although the adsorption behavior of PTMC5.2M is remarkably enhanced by the presence of a small amount of salt (see the result for 100 mM KCl), we prefer the alternative explanation that the appearance of zero mobility is attained by the formed stagnant layer (Fig. 18). As we have seen so far, the adsorption of both polyelectrolytes (PTMA4.9M and PTMC5.2M) at the very initial stage contributes to the increase of collision radius. This behavior is much more pronounced for no-salt addition. That is, the coil size of the polyelectrolyte chain in solution is considered to have direct influence on this behavior. This is consistent with the picture that the persistence length of a chain in solution is proportional to the Debye length as proposed by De Gennes (see Section 2). The folding rate of high-charge-density polymer (PTMA4.9M) is significantly enhanced by the electrostatic attraction between charged segments and oppositely charged surface. However, even though the decrease of protruding length decreases the collision

Fig. 16. (a). Rate of flocculation as a function of concentration of PTMA4.9M. KCl concentration; 0 M (△),10−3 M (◆), 0.5 M (●). Horizontal line represents the maximum rate observed with KCl [15]. (b) PTMC5.2M, KCl concentration; 0 M (◇), 10−3M (■), 10−2M (▲) [20].

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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Fig. 17. Comparison of adsorption and resulting flocculation process of PSL particles with high-charge-density polymer [11] and with low-charge-density polymer [10].

radius, the rate of collision does not, because the inter-patch attraction comes to appear in later stage after t = 20 s., compensating the effect. In the case of PTMC5.2M without added salt, the folding rate of the chain is not so fast, so that the attached chain will preserve more or less bulky structure. This will eventually bring about steric repulsion resulting in a decline of the rate of flocculation after approximately 20 s. It is interesting that this point almost corresponds to the measurement of electrophoretic mobility in Fig. 21. 4. Conclusions, remarks and perspectives In view of the significance of practical applications, understanding the events which take place in the beginning of any flocculation

operation is critically important. In this review, we have discussed an approach on the basis of the theory of transport phenomena which was mainly developed in our group during the last two decades. This provides us with a window on the key processes of adsorption and reconformation or folding. The folding process of high-charge-density polymers is obviously enhanced by electrostatic attraction, while for low-charge-density polymers, competition between steric repulsion and bridging was found to be significant. We also introduced the idea of a stagnant layer which seems to be consistent with electrokinetic data. Most of the data obtained so far were for model systems, addressing rather fundamental questions. However, these data together with the way of analysis will be useful when we consider practical cases like enhanced polyelectrolyte flocculation with two oppositely charged polymers (dual systems) or polyion complexes [109–112]. In the biological system, the flexibility of chain is recognized to be very important to the appearance of optimum flocculation [24,113,114]. Usually, the objective of flocculation studies is to obtain process control. However, if we can use well-defined materials, the flocculation itself becomes a good sensor for the dynamic processes of polyelectrolytes at interfaces. There is quite some recent progress in the understanding of polyelectrolytes, covering topics ranging from surface modification [115], complex formation [111], and soft matter [116], to multilayer production [115,117]. Among reported physicochemical aspects we find not only solution properties but also the behavior at interfaces. Nevertheless, a simple theory of polyelectrolyte solutions which explains what is needed to make a normal floc has not yet been established. Classical problems regarding electrokinetics and surface properties are still worth being considered [118,119]. More efforts are necessary to identify the key factors of polymers with respect to flocculation, such as size, charge density, charge distribution along the chain, stiffness and interaction with hydrophobic materials. Other important factors are the role of hydrodynamics, effects of shear, the interaction with permeable porous structures, and dynamics of phase separation. Together with floc morphology, all such aspects should form a basis for the ‘soft matter science’ of flocs [120]. A lot still remains to be done. Acknowledgments

Fig. 18. Schematic illustration of weakly charged polyelectrolyte (PTMC5.2M) adsorbed onto the colloidal surface [10].

This work is funded by a Grant-in-Aid for Scientific Research (A22248025) from JSPS. L.F. expresses her thanks to the Chinese government for the scholarship (A22248025). Useful discussions with Dr. Motoyoshi Kobayashi, dedicated experimental efforts of Dr. Tetsuhiro Matsumoto and Dr. Kenji Aoki during their PhD student period are also acknowledged. Professor Sandor Barany is acknowledged for his critical reading and helpful suggestions to this manuscript.

Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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Fig. 19. ln(N(t)/N(0)) vs t. for flocculation with polyelectrolytes in the absence of salt. (a) PTMA4.9M [11], (b) PTMC5.2M [10].

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Fig. 20. Electrophoretic mobility measurements of PSL particles with polyelectrolyte (PTMA4.9M) as a function of mixing time in the absence of KCl, and in the presence of 10 mM KCl [15].

Fig. 21. Electrophoretic mobility measurements of PSL particles with polyelectrolyte (PTMC5.2M) as a function of mixing time in the absence of KCl, and in the presence of 100 mM KCl [20].

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Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004

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Please cite this article as: Feng L, et al, Dynamics of polyelectrolyte adsorption and colloidal flocculation upon mixing studied using monodispersed polystyrene latex particles, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/j.cis.2015.09.004