Adsorption of polyelectrolytes on colloidal latex particles, electrostatic interactions and stability behaviour

The stabilization and flocculation behaviour of colloidal latex particles covered with cationic polyelectrolytes (PE) is studied with photon correlation spectroscopy and zetapotential measurements. Diffusion coefficients, flocculation rate constants and zetapotentials have been determined as a function of adsorbed amount of cationic poly-(diallyl-dimethyl-ammoniumchloride) (PDADMAC) of different molar masses and of statistic copolymers of DADMAC and N-methyl-N-vinyl-acetamide (NMVA) of various compositions in water and at high ionic strength. Flocculation by van der Waals attraction can be observed if the zetapotential is low. This occurs, if the surface charge is screened by the oppositely charged cations. Furthermore, in the case of adsorption of high molecular polycations mosaic flocculation occurs if the adsorbed amount is low. At high ionic strength, flocculation takes place if the adsorbed amount is below the adsorption plateau. If the adsorption plateau is reached the suspensions become stabilized. In water the charge reversal at full coverage leads to electrosteric stabilization both with low and high molar mass polycations. At high ionic strength only polycations with high molar mass are able to stabilize the suspension. If a certain molar mass of the polycation is exceeded, steric stabilization of the suspension occurs due to the formation of long adsorbed PE tails and their osmotic repulsion. The layer thicknesses are determined as a function of the molar mass. Received: 4 July 2000/Accepted: 18 August 2000     

Asymmetric layer-by-layer polyelectrolyte nanofiltration membranes with tunable retention

Nanofiltration (NF) membrane processes are attractive to remove multivalent ions. As ion retention in NF membranes is determined by both size and charge exclusion, negatively charged membranes are required to reject negatively charged ions. Layer-by-layer assembly of alternating polycation (PC) and polyanion layers on top of a support is a versatile method to produce membranes. Especially the polyelectrolyte (PE) couple polydiallyldimethylammoniumchloride and poly(sodium-4-styrenesulfonate) (PDADMAC/PSS) is extensively investigated. This PE couple cannot form highly negatively charged membrane surfaces, due to interdiffusion and charge overcompensation of PDADMAC into the PSS layers, which limits the operational window to tailor membrane properties. We propose the use of asymmetric layer formation and show how combining two charge densities of one PC can produce negatively charged NF membranes. Starting from hollow fiber ultrafiltration supports coated with base layers of PDADMAC/PSS, they are coated with PDADMAC/PSS or poly(acrylamide-co-diallyldimethylammoniumchloride), P(AM-co-DADMAC)/PSS layers. P(AM-co-DADMAC) has a charge density of only 32% compared to 100% for PDADMAC. The particular novel membranes coated with P(AM-co-DADMAC) have a highly negatively charged surface and high permeabilities (7–19 L/[m²hbar]), with high retentions for Na₂SO₄ of up to 95%. These values position the developed membranes in the top range compared to commercial and other layer-by-layer membranes.     

Counter-ion activity and microstructure in polyelectrolyte complexes as determined by osmotic pressure measurements

We have investigated the activity of counter-ions at 60 degrees C through the osmotic coefficient K in solutions of anionic and cationic polyelectrolyte complexes of variable compositions. For excess of polyanion in the complexes (molar fraction of polycation f < 0.5), K increases as the polyanion is neutralized by the polycation (f getting closer to 0.5). By contrast, for an excess of polycation (f > 0.5), K stays constant or even slightly decreases as the polycation is getting neutralized by the polyanion. This asymmetric behavior depending on the charge of the complexes indicates that the globally negatively charged complexes are homogeneous and can be treated as a single polyelectrolyte of reduced linear charge density. On the other hand, the positively charged complexes show a micro-phase separation between neutral fully compensated microdomains and domains where the excess polycation is locally segregated. These two different microstructures are reminiscent of the coacervation and segregation regimes observed at higher concentrations and salinities, and also of polyelectrolyte complexes with oppositely charged surfactants. This interpretation is supported by two simple predictive models.     

Polyelectrolyte adsorption on charged surfaces: study by electrokinetic measurements

In order to describe the influence of cationic polyelectrolytes on flocculation of disperse systems the adsorption of poly (diallyldimethylammonium chloride) (PDADMAC) onto silica, mica and acidic polymer latex was investigated. The plateau value of the adsorption isotherms grows with increasing surface charge density of the substrates and electrolyte concentration. The adsorbed layer of the polycation was characterized by zeta potential measurements with KCl solutions of constant ionic strength and varied pH. The zero point of the charge as well as the shape of the zeta potential–pH plot depends on the coverage of the surface with polycations. For fully covered substrates the zero point of the charge as well as the pK_A and pK_B values calculated by a stochastic search programme are independent of the substrate. Maximum flocculation was observed at about 30% of the plateau value of the adsorption isotherms.     

Preparation and characterization of different polyelectrolyte complexes and their application as flocculants

 Polyelectrolyte complexes (PEC) from poly(diallyl-dimethyl-ammoniumchloride) PDADMAC and two different polyanions, formed in aqueous solution, were characterized by different methods (zeta-potential, net content, turbidity) and applied as flocculants. The flocculation of clay was investigated by sedimentation measurement as well as by a dynamic method, using a Fibre Optical Flocculation Sensor. The results of both methods showed that the most important advantages of PEC were the high velocity of sedimentation and a very broad range of the optimum flocculation concentration. In spite of the differences in the complex-forming behavior of the two polyanions used, no significant differences between complexes of the same composition but different polyanions are obtained. In contrast, the ratio of anionic to cationic charges is of great importance for the mechanism of flocculation. Received: 30 May 1997 Accepted: 22 September 1997     

Flocculation of kaolin particles by two typical polyelectrolytes: A comparative study on the kinetics and floc structures

The flocculation kinetics of kaolin particles induced by two polyelectrolytes is studied by using small-angle laser light scattering (SALLS). Two different methods, image analysis and SALLS, are used to calculated the fractal dimensions of flocs formed under different flocculation mechanisms. For a high charge density of polydiallyldimethylammonium chloride (PDADMAC), the initially flocculation rates are slow due to the quite low molecular weight. Smaller and more compact flocs are in the particle–particle connections, and restructuring of the flocs occurs in the flocculation process. With cationic polyacrylamide C498 of very high molecular weight and low charge density, however, the initially flocculation rates are much higher due to its rapid adsorption on kaolin particles, but it will take the adsorbed polymer a much longer time to reach equilibrium due to re-conformation. High potentialities of adsorption prevent the particles from entering the interior of the floc structure or rearrangement, which results in a more open floc structure. Different underlying flocculation mechanisms are evident for these two kinds of polyelectrolytes, in which charge neutralization is mainly involved for the low molecular weight and high charge density polymer of PDADMAC while polymer bridging is suggested to be the dominant mechanism for the high molecular weight polyelectrolyte of C498.     

Monomodal polyelectrolyte complex nanoparticles of PDADMAC/Poly(styrenesulfonate): preparation and protein interaction

The binding of the model proteins HSA, LYZ and MYO to PEC nanoparticles is reported. PEC particles were prepared by mixing solutions of PDADMAC either with PSS or PMA-MS, followed by consecutive centrifugation. Monomodal anionic (PEC-1.50) and cationic (PEC-0.66) PEC particles were obtained using non-stoichiometric mixing ratios. PEC/protein conjugates were prepared by adding charged protein solutions to dispersions of respective like charged PEC particles, followed by one centrifugation step. Mixing proteins and PEC particles under attractive conditions led to flocculation of the dispersion. From CD, DLS and AFM the following trend for protein binding at PEC particles under repulsive conditions was obtained: HSA/PEC-1.50 > MYO/PEC-1.50 > LYZ/PEC-0.66. Protein uptakes up to 0.33 g x g(-1) (protein/PEC) (CD) and particle diameter enlargements up to 13 nm (AFM) were obtained at c(PROT) = 0.091 mg . mL(-1). Furthermore, novel spin coated films of PEC particles were interacted with proteins under both repulsive and attractive conditions. In-situ ATR FT-IR spectroscopy revealed that the adsorbed amount of HSA and LYZ under attractive conditions was significantly higher than under repulsive ones, which is analogous to protein adsorption at polyelectrolyte multilayers terminated either by polycation or polyanion. Similarly to the dispersed PEC/protein conjugates, under repulsive conditions the uptake of HSA was higher compared to LYZ. The shown protein uptake under repulsive conditions is related to concepts of mild enzyme or protein binding at nonbiogenic substrates.     

Polyelectrolyte adsorption and stabilization of silica-suspensions

The adsorption of cationic polyelectrolytes on colloidal silica-particles is investigated. The polyelectrolytes poly(diallyl-dimethyl-ammoniumchloride) PDADMAC of different molar mass have been used. The adsorbed amount is influenced by the ionic strength and pH of the suspension and the molar mass of the macromolecule. The adsorption determines the zetapotential of the covered particles. The electrostatic interaction between the particles as well as the structure of the adsorbed polyelectrolytes play an important role in the stabilization and flocculation behaviour of the polyelectrolyte covered suspensions.     

Optimization of the polyelectrolyte dosage for dewatering sewage sludge suspensions by means of a new centrifugation analyser with an optoelectronic sensor

The flocculation of sewage sludge with poly(diallyldimethylammonium chloride) (PDADMAC) as well as cationic, poly[acrylamide-co-(N,N,N-trimethylammoniumethylacrylate chloride)] (PTAC), and anionic acrylamide copolymers, poly(acrylamide-co-acrylate) (PAAM/AA), was investigated by using a new method to evaluate the velocity of centrifugal separation and the packing behaviour of the sediments. By means of a centrifugation analyser with an optoelectronic measuring system the position of the centrate/sediment interface of the conditioned suspensions can be determined as a function of the separation time, which allows the polyelectrolyte dosage to be optimized for various rotational speeds. At high centrifugal accelerations it was found that using the anionic polyelectrolyte PAAM/AA does not result in better dewatering performance, whereas both polycations recorded high degrees of separation. Furthermore, the measurements with the high-molar-mass PTAC indicate a high resistance to centrifugation which correlates with floc stability. In contrast, addition of the low-molar-mass PDADMAC gives more fragile flocs, leading to faster collapse of the sludge cake in the compression zone of the centrifuge at lower gravitational forces. These experimental findings agree well with the much higher filtration capacity of sludges conditioned with PTAC which was measured by means of a pressure filtration device. Comparisons of the optimum polyelectrolyte concentrations obtained by pressure filtration or centrifugation at low gravitational forces with the zeta potential of the flocculated particles appear to be consistent with bridge formation for PTAC and with the charge patch mechanism for PDADMAC. The centrifugation experiments at high centrifugal accelerations show an optimum separation efficiency at relatively low degrees of coverage and a high level of agreement with the large-scale process. Received: 24 November 1998 Accepted in revised form: 12 February 1999     

Preparation of Langmuir/Blodgett-cellulose Surfaces by Using Horizontal Dipping Procedure. Application for Polyelectrolyte Adsorption Studies Performed with QCM-D

A method of preparing model cellulose surfaces by the Langmuir–Blodgett (LB) technique with horizontal dipping procedure has been developed. The primary aim for the use of these surfaces was adsorption studies performed with the quartz crystal microbalance with dissipation (QCM-D) instrument. Hydrophobised cellulose (trimethylsilyl cellulose, TMSC) was deposited on the hydrophobic, polystyrene-coated QCM-D crystal. After 15 dipping cycles, the TMSC film fully covers the crystal surface. TMSC can easily be hydrolysed back to cellulose with acid hydrolysis. With this method a smooth, rigid, thin and reproducible cellulose film was obtained. Its morphology, coverage, chemical composition and wetting was further characterised using atomic force microscopy (AFM), X-Ray photoelectron spectroscopy (XPS), and contact angle measurements. The swelling behaviour and the stability of the cellulose film in aqueous solutions at different ionic strengths were studied using the QCM-D instrument. The swelling/deswelling properties of the cellulose film were those expected of polyelectrolytes with low charge density; some swelling occurred in pure water and the swelling decreased when the ionic strength was increased. No significant layer softening was detected during the swelling. The effect of electrolyte concentration and polymer charge density on the adsorption of cationic polyelectrolytes on the cellulose surface was also investigated. At low electrolyte concentration less of the highly charged PDADMAC was adsorbed as compared to low charged C-PAM. The adsorbed amount of PDADMAC increased with increasing ionic strength and a more compact layer was formed while the effect of electrolyte concentration on the adsorption of C-PAM was not as pronounced.     

Polyelectrolyte complexes. I. Synthesis and characterization of some insoluble polyanion-polycation complexes

The synthesis of some water-insoluble synthetic polyelectrolyte complexes formed between a weak polyanion and a strong polycation was followed. Sodium salts of poly(acrylic acid) and of some copolymers of acrylic acid with itaconic acid or maleic acid were used as anionic polymers. Cationic polyelectrolytes with quaternary ammonium salt groups in the main chain were used as strong polycations. The cationic polymers were different as concerns both the content of quaternary nitrogen atoms and the degree of branching. The complex formation was followed by the variation of the conductivity and of the specific viscosity of the reaction medium as well as by the turbidimetric titration versus the unit molar ratio polyanion/polycation. The deviation of the endpoint from stoichiometry was influenced mainly by the structure of the complementary polymers and by their molecular weights. The greater the structural differences, the higher the endpoint deviation from stoichiometry. Only insoluble polyelectrolyte complexes (PEC) were obtained in all the polyanion/polycation systems taken into account. The PECs were separated and characterized by elemental and spectral analyses as compared with the complementary polymers. © 1996 John Wiley & Sons, Inc.     

Flocculation and stabilization of colloidal silica by the adsorption of poly-diallyl-dimethyl-ammoniumchloride (PDADMAC) and of copolymers of DADMAC with N-methyl-N-vinyl- acetamide (NMVA)

 The stabilization and flocculation behavior of colloidal silica-particles with cationic polyelectrolytes (PE) is investigated. The zetapotentials, diffusion coefficients and flocculation rate constants of silica particles have been measured as a function of the adsorbed amount of cationic polyelectrolytes poly(diallyl-dimethyl-ammoniumchloride) (PDADMAC) of different molar masses and of statistic copolymers of DADMAC and N-methyl-N-vinyl-acetamide (NMVA) of various compositions at different salt concentrations and pH-values. Very fast flocculation due to van der Waals attraction occurs if the zetapotential is small. At low ionic strength this condition occurs just below the plateau of the adsorption isotherms where the surface charges are screened by adsorbed polycations. Additionally with high molecular polycations slow mosaic flocculation is observed at lower PE concentrations. At high ionic strength fast flocculation takes place at low macroion concentration due to the screening of the surface charges by adsorbed polycations and salt ions. At medium concentrations of polycations below plateau adorption slow bridging flocculation is observed. At plateau adsorption the suspensions become stabilized up to high ionic strength. At low salt concentration charge reversal at full coverage with polycations results in electrostatic repulsion. At high ionic strength the particles are stabilized sterically due to the osmotic repulsion of the long adsorbed PE tails. Therefore macroions of high molar mass are necessary to stabilize the suspension at high ionic strength. Received: 27 January 1998 Accepted: 23 March 1988     

DNA and surfactants in bulk and at interfaces

Recent investigations of the DNA interactions with cationic surfactants and catanionic mixtures are reviewed. Several techniques have been used such as fluorescence microscopy, dynamic light scattering, electron microscopy, and Monte Carlo simulations.

The conformational behaviour of large DNA molecules in the presence of cationic surfactant was followed by fluorescence microscopy and also by dynamic light scattering. These techniques were in good agreement and it was possible to observe a discrete transition from extended coils to collapsed globules and their coexistence for intermediate amphiphile concentrations. The dependence on the surfactant alkyl chain was also monitored by fluorescence microscopy and, as expected, lower concentrations of the more hydrophobic surfactant were required to induce DNA compaction, although an excess of positive charges was still required.

Monte Carlo simulations on the compaction of a medium size polyanion with shorter polycations were performed. The polyanion chain suffers a sudden collapse as a function of the concentration of condensing agent, and of the number of charges on the polycation molecules. Further increase in the concentration increases the degree of compaction. The compaction was found to be associated with the polycations promoting bridging between different sites of the polyanion. When the total charge of the polycations was lower than that of the polyanion, a significant translational motion of the compacting agent along the polyanion was observed, producing only a small-degree of intrachain segregation, which can explain the excess of positive charges necessary to compact DNA.

Dissociation of the DNA–cationic surfactant complexes and a concomitant release of DNA was achieved by addition of anionic surfactants. The unfolding of DNA molecules, previously compacted with cationic surfactant, was shown to be strongly dependent on the anionic surfactant chain length; lower amounts of a longer chain surfactant were needed to release DNA into solution. On the other hand, no dependence on the hydrophobicity of the compacting agent was observed. The structures of the aggregates formed by the two surfactants, after the interaction with DNA, were imaged by cryogenic transmission electron microscopy. It is possible to predict the structure of the aggregates formed by the surfactants, like vesicles, from the phase behaviour of the mixed surfactant systems.

Studies on the interactions between DNA and catanionic mixtures were also performed. It was observed that DNA does not interact with negatively charged vesicles, even though they carry positive amphiphiles; however, in the presence of positively charged vesicles, DNA molecules compact and adsorb on their surface.

Finally Monte Carlo simulations were performed on the adsorption of a polyelectrolyte on catanionic surfaces. It was observed that the mobile charges in the surface react to the presence of the polyelectrolyte enabling a strong degree of adsorption even though the membrane was globally neutral. Our observations indicate that the adsorption behaviour of the polyelectrolyte is influenced by the response given by the membrane to its presence and that the number of adsorbed beads increases drastically with the increase of flexibility of the polymer. Calculations involving polymers with three different intrinsic stiffnesses showed that the variation is non-monotonic. It was observed also that a smaller polyanion typically adsorbs more completely than the larger one, which indicates that the polarisation of the membrane becomes less facilitated as the degree of disruption increases.     

Polyelectrolyte complex dispersions with a high colloidal stability controlled by the polyion structure and titrant addition rate

Three ionic/nonionic random copolymers of sodium 2-acrylamido-2-methylpropanesulfonate (AMPS) with either t-butyl acrylamide (TBA) or methyl methacrylate (MM), were used in the preparation of some polyelectrolyte complex dispersions (PCD) with two strong polycations of integral type, poly(diallyldimethylammonium chloride) (PDADMAC), and an ionene type polycation, containing 95 mol% N,N-dimethyl-2-hydroxypropyleneammonium chloride repeat units (PCA₅). The novelty of the paper is that PCDs with a high colloidal stability, both before and after the complex stoichiometry, were obtained even with hydrophilic/hydrophobic polyanions, with a high titrant addition rate (TAR), a less explored parameter, which allows a fine control of PCDs aggregation level, in salt-free aqueous solutions. The characteristics of PCDs were also correlated with the polyanion average charge density and the structure of the nonionic comonomer, at a constant TAR, the polyanion with the lowest charge density leading to the highest turbidities and the lowest colloidal stabilities. A mechanism of the PCDs colloidal stabilization as a function of TAR was suggested in the paper.     

Flocculation and reflocculation of clay suspension by different polymer systems under turbulent conditions

The focused beam reflectance measurement (FBRM), also known as scanning laser microscopy (SLM), was used as a real-time monitor to study the flocculation and reflocculation of clay suspensions under different shear conditions in the presence of single polymer, dual polymer, microparticle and poly(ethylene oxide)/phenolformaldehyde (PEO/PFR) flocculation systems. For initial flocculation, the high molecular weight PEO and cationic polyacrylamide (CPAM) produced larger flocs than others. However, reflocculation of clay suspensions formed by these non- or low-charged polymers was insignificant after the initial flocs were broken under high shear force. In contrast, high charge density polymers, such as poly(diallyldimethylammonium chloride) (PDADMAC), do not form large initial flocs, but they showed significant reflocculation ability under a continuous shear condition. It is concluded that high flocculation can be obtained by effective polymer bridging, but high reflocculation can only be induced by high electrostatic attractive forces between suspended particles.     

聚丙烯酸钠与聚(苯乙烯-co-4-乙烯基吡啶)阳离子复合作用的研究

聚电解质复合物 (Ｐｏｌｙｅｌｅｃｔｒｏｌｙｔｅｃｏｍｐｌｅｘ)是指带有相反电荷的两种聚电解质之间通过库仑力而结合形成的一类特殊的高分子材料[1 ] .由于生物体内的很多反应以及生物化学合成过程都是通过高分子复合物进行的 ,因此对高分子间相互作用及其聚集体形成的研究受到了人们的极大重视 .目前研究得较多的体系是聚苯乙烯衍生物 ,如Ｉｏｐｌｅｘ 1 0 1即由聚苯乙烯磺酸钠和聚氯化乙烯基苄基三甲基铵反应而得[2 ,3] .本文报道了不同电荷密度及相对分子质量的聚苯乙烯 ｃｏ 4 乙烯基吡啶的硫酸甲酯盐 ,与不同分子质量的聚丙烯酸钠…     

Polymer‐Templated Perylene‐Probe Noncovalent Self‐Assembly: A New Strategy for Label‐Free Ultrasensitive Fluorescence Turn‐On Biosensing

A new label‐free fluorescence turn‐on strategy for highly sensitive biosensing has been developed. A negatively charged perylene probe was synthesized. Polycations could induce aggregation of the perylene probe through noncovalent interactions and the fluorescence of the probe’s monomer was efficiently quenched. Upon addition of a single‐stranded nucleic acid, competitive binding of the negatively charged nucleic acid (a polyanion) to the cationic polymer resulted in the release of a monomer and thus a turn‐on fluorescence signal was detected. Without the use of any amplification techniques, a detection limit of 2 pM DNA was obtained. Based on these results, an assay strategy for the highly sensitive detection of alkaline phosphatase (ALP) activity has been demonstrated. λ Exonuclease (λ exo) could degrade 5′‐phosphorylated single‐stranded DNA. However, when the DNA sample was treated with ALP, the phosphate functional group was removed by ALP and it could no longer be degraded by λ exo. Binding of the DNA to the perylene probe–polycation complex resulted in a turn‐on fluorescence signal, which could be used for sensing of ALP. The method is highly sensitive, a limit of detection as low as 0.02 mU mL^?1 ALP was obtained. Our method is simple, convenient, highly sensitive, and inexpensive.     

PS胶体粒子表面逐层自组装固定化SOD及其生物活性

通过逐层自组装技术成功地把超氧化物歧化酶(SOD)吸附在聚苯乙烯(PS)胶体粒子表面.zeta电位和TEM证明了聚阳离子或聚阴离子型SOD与相反电荷的聚电解质在PS胶体粒子表面的交替吸附.通过测定SOD被胶体粒子吸附后上清液的生物活性,得到聚阴离子型SOD(pH=8.0)和聚阳离子型SOD(pH=4.3)在PS胶体粒子表面的吸附量分别为12和51IU,相对活性分别为23.4%和2.9%.聚阴离子型SOD在PS胶体粒子表面能形成平滑规整的膜,导致较高的相对活性.研究结果表明,通过调节pH值,可以优化自组装固定化酶的聚集状态和生物活性     

Preparation and characterization of polymer composite multilayers on SiO2

Poly(diallyldimethylammonium chloride) (PDADMAC) and poly(sodium 4-styrenesulfonate) (PSS) have been consecutively adsorbed onto 1.5-microm charged silica (SiO2) particles. Time-dependent adsorption studies indicate that, due to the strong ionic charge of the dissociated polycation in water, adsorption is complete in less than 30 min. Indications of the maximum adsorption density, changes in surface charge, and stability of the layered particles are demonstrated through adsorption isotherms and electrophoretic mobility (EPM) measurements. Further stability of the PDADMAC layer is demonstrated through multiwashing with ultra pure deionized water. Preliminary desorption studies of the PSS layer also illustrate a stabilized two-layer system. Due to the nature of the electrostatic charges on the surface of the SiO2 core particles and both polyelectrolytes in aqueous media, the use of polyelectrolytes as layering elements serves as a model for the assembly of time-released drug delivery particle systems.     

Polymer colloids formed by polyelectrolyte complexation of vinyl polymers and polysaccharides in aqueous solution

The polyelectrolyte complex formed from the polyanion and polycation was studied by turbidimetry, static and electrophoretic light scattering, and elementary analysis. Sodium salts of polyacrylate (PA) and heparin (Hep) were chosen as the polyanion, and hydrochloric salts of poly(vinyl amine) (PVA) and chitosan (Chts) as the polycation. Although these vinyl polymers and polysaccharides have remarkably different backbone chemical structures and linear charge densities, all the four combinations PA-PVA, PA-Chts, Hep-PVA, and Hep-Chts provide almost stoichiometric polyelectrolyte complexes which are slightly charged owing to the adsorption of the excess polyelectrolyte component onto the neutral complex. The charges stabilize the complex colloids in aqueous solution of a non-stoichiometric mixture, and the aggregation number of the complex colloids increases with approaching to the stoichiometric mixing ratio. The mixing ratio dependence of the aggregation number for the four complexes is explained by the model proposed in the previous study.