We have studied the phase behavior of the poly(n-butyl acrylate)-b-poly(acrylic acid) block copolymer in a mixture of two miscible solvents, water and tetrahydrofuran (THF). The techniques used to examine the different polymers, structures and phases formed in mixed solvents were static and dynamic light scattering, small-angle neutron scattering, nuclear magnetic resonance and fluorescence microscopy. By lowering the water/THF mixing ratio X, the sequence unimers, micron-sized droplets, polymeric micelles was observed. The transition between unimers and the micron-sized droplets occurred at X = 0.75, whereas the microstructuration into core-shell polymeric micelles was effective below X = 0.4. At intermediate mixing ratios, a coexistence between the micron-sized droplets and the polymeric micelles was observed. Combining the different aforementioned techniques, it was concluded that the droplet dispersion resulted from a solvent partitioning that was induced by the hydrophobic blocks. Comparison of poly(n-butyl acrylate) homopolymers and poly(n-butyl acrylate)-b-poly(acrylic acid) block copolymers suggested that the droplets were rich in THF and concentrated in copolymers and that they were stabilized by the hydrophilic poly(acrylic acid) moieties.
We study theoretically using scaling arguments the behavior of polyelectrolyte gels in poor solvents. Following the classical picture of Katchalsky, our approach is based on single chain elasticity but it accounts for the recently proposed pearl-necklace structure of polyelectrolytes in poor solvents. The elasticity both of gels at swelling equilibrium and of partially swollen, non equilibrium, gels is studied when parameters such as the ionic strength or the fraction of charged monomers are varied. Our theory could be useful to interpret recent experiments performed in Strasbourg that show th…
We consider conformations and intra-molecular conformational transitions in amphiphilic starlike polymers formed by diblock copolymer arms with inner hydrophobic and outer polyelectrolyte blocks. A combination of an analytical mean-field theory with the assumption-free numerical self-consistent field (SCF) modeling approach is applied. It is demonstrated that unimolecular micelles with collapsed hydrophobic cores and swollen polyelectrolyte coronae are formed in dilute aqueous solutions at high ionic strength or/and low degree of ionization of the outer hydrophilic block. An intra-molecular co…
In this work, a comprehensive study about the influence on shear viscosity of polyelectrolyte concentration, persistence length, salt concentration and solvent quality is reported, using numerical simulations of confined solutions under stationary Poiseuille flow. Various scaling regimes for the viscosity are reproduced, both under good solvent and theta solvent conditions. The key role played by the electrostatic interactions in the viscosity is borne out when the ionic strength is varied. It is argued that these results are helpful for the understanding of viscosity scaling in entangled poly…
The collapse kinetics of strongly charged polyelectrolytes in poor solvents is investigated by Langevin simulations and scaling arguments. The rate of collapse increases sharply as the valence of counterions, z, increases from one to four. The combined system of the collapsed chain and the condensed counterions forms a Wigner crystal when the solvent quality is not too poor provided z >= 2. For very poor solvents the morphology of the collapsed structure resembles a Wigner glass. For a fixed z and quality of the solvent the efficiency of collapse decreases dramatically as the size of the count…
We present a set of molecular dynamics (MD) simulations of strongly charged, flexible polyelectrolyte chains under poor solvent conditions in a salt free solution. Structural properties of the chains and of the solutions are reported. By varying the polymer density and the electrostatic interaction strength we study the crossover from a dominating electrostatic interaction to the regime of strong screening, where the hydrophobic interactions dominate. During the crossover a multitude of structures is observed. In the limit of low polymer density strongly stretched, necklace like conformations …
We propose a mean-field model to explore the equilibrium coupling between micelle aggregation and lattice choice in neutral copolymer and selective solvent mixtures. We find both thermotropic and lyotropic transitions from face-centered cubic to body-centered cubic ordered phases of spherical micelles, in agreement with experimental observations of these systems over a broad range of conditions. Stability of the non-closed packed phase can be attributed to two physical mechanisms: the large entropy of lattice phonons near crystal melting and the preference of the inter-micelle repulsions for t…
We demonstrate in this paper the influence of solvent quality on the structure of the semi-dilute solution of a hydrophobic polyelectrolyte, partially sulfonated Poly-Styrene Sulfonate. Two solvents are used: (i) one mixture of water and an organic solvent: THF, which is also slightly polar; (ii) DMSO, a polar organic solvent. In case (i), it is shown by SAXS study that the structure - namely the scattering from all chains, characterised by a maximum ("polyelectrolyte peak"), of the aqueous hydrophobic polyelectrolyte solutions (PSS) depends on the solvent quality through the added amount of o…
The structure of spherical micelles of the diblock poly(styrene-block-acrylic acid) [PS-b-PA] copolymer in water was investigated up to concentrations where the polyelectrolyte coronal layers have to shrink and/or interpenetrate in order to accommodate the micelles in the increasingly crowded volume. We obtained the partial structure factors pertaining to the core and corona density correlations with small angle neutron scattering (SANS) and contrast matching in the water. The counterion structure factor was obtained with small angle X-ray scattering (SAXS) with a synchrotron radiation source.…
We explore the mechanical response of a single polyelectrolyte chain under tension in good and poor solvents using a combination of simulation and theory. In poor solvents, where the equilibrium state of the chain is a collapsed globule, we find that the chain undergoes a globule-coil transition, as the magnitude of the force is increased beyond a critical value. This transition, where the polymer size changes discontinuously from a small to a large value, is accompanied by release of bound counterions from the chain. We explain these results by adhering to a statistical mechanical theory of c…
We examined the kinetics of the transformation from the lamellar (LAM) to the hexagonally packed cylinder (HEX) phase for the triblock copolymer, polystyrene-b-poly (ethylene-co-butylene)-b-polystyrene (SEBS) in dibutyl phthalate (DBP), a selective solvent for polystyrene (PS), using time-resolved small angle x-ray scattering (SAXS). We observe the HEX phase with the EB block in the cores at a lower temperature than the LAM phase due to the solvent selectivity of DBP for the PS block. Analysis of the SAXS data for a deep temperature quench well below the LAM-HEX transition shows that the trans…
We study the properties of polyelectrolyte chains under different solvent conditions, using a variational technique. The free energy and the conformational properties of a polyelectrolyte chain are studied minimizing the free energy $F_N$, depending on $N(N-1)/2$ trial probabilities that characterize the conformation of the chain. The Gaussian approximation is considered for a ring of length $2^4<N<2^{16}$ and for an open chain of length $2^4<N<2^9$ in poor and theta solvent conditions, including a Coulomb repulsion between the monomers. In theta solvent conditions the blob size is measured an…
When polyelectrolyte-neutral block copolymers are mixed in aqueous solutions with oppositely charged species, stable complexes are found to form spontaneously. The mechanism is based on electrostatics, and on the compensation between the opposite charges. Electrostatic complexes exhibit a core-shell microstructure. In the core, the polyelectrolyte blocks and the oppositely charged species are tightly bound and form a dense coacervate microphase. The shell is made of the neutral chains and surrounds the core. In this paper, we report on the structural and magnetic properties of such complexes m…
The structure of spherical micelles of the diblock copolymer poly(styrene-block-acrylic acid) in water was investigated with small angle neutron scattering (SANS) and contrast matching. We have monitored inter-micelle correlation and the extension of the polyelectrolyte chains in the coronal layer through the overlap concentration. Irrespective of ionic strength, the corona shrinks with increasing packing fraction. Furthermore, at high charge and minimal screening conditions, the corona layers interpenetrate once the volume fraction exceeds the critical value 0.53.
A phenomenological mean-field theory is used to investigate the properties of solvent-diluted di-block copolymers (BCP), in which the two BCP components (A and B) form a variety of phases that are diluted by a solvent (S). Using this approach, we model mixtures of di-block copolymers and a solvent and obtained the corresponding critical behavior. In the low solvent limit, we find how the critical point depends on the solvent density. Due to the non-linear nature of the coupling between the A/B and BCP/solvent concentrations, the A/B modulation induces modulations in the polymer-solvent relativ…
We examine the condensed states of a simple semiflexible copolymer in which there are two monomer types that are immiscible with each other and with the solvent. Although this is similar to the well known problem of collapse for a semiflexible homopolymer we find that it gives rise to a much richer variety of condensed states. We predict the existence of these states using simple analytic arguments and also observe them directly using Brownian dynamics simulations.
In order to better understand the collapse of polyions in poor solvent conditions the effective charge and the solvent quality of the hypothetically uncharged polymer backbone need to be known. In the present work this is achieved by utilizing poly-2-vinylpyridine quaternized to 4.3% with ethylbromide. Conductivity and light scattering measurements were utilized to study the polyion collapse in isorefractive solvent/non-solvent mixtures consisting of 1-propanol and 2-pentanone, respectively, at nearly constant dielectric constant. The solvent quality of the uncharged polyion could be quantifie…
We present simulations on charged polymers in poor solvent. First we investigate in detail the dilute concentration range with and without imposed extension constraints. The resulting necklace polymer conformations are analyzed in detail. We find strong fluctuations in the number of pearls and their sizes leading only to small signatures in the form factor and the force-extension relation. The scaling of the peak in the structure factor with the monomer density shows a pertinent different behavior from good solvent chains.
We perform extensive molecular dynamics simulations of a highly charged flexible polyelectrolyte (PE) chain in a poor solvent for the case when the chain is in a collapsed state and the electrostatic interactions, characterized by the reduced Bjerrum length $\ell_B$, are strong. We detect the existence of several sub-regimes, $R_g \sim \ell_B^{-γ}$, in the dependence of the gyration radius of the chain $R_g$ on $\ell_B$. In contrast to a good solvent, the exponent $γ$ for a poor solvent crucially depends on the size and valency of counterions. To explain the different sub-regimes we generalize…
Using molecular dynamics simulations we study the behavior of a dilute solution of strongly charged polyelectrolytes in poor solvents, where we take counterions explicitly into account. We focus on the chain conformational properties under conditions where chain-chain interactions can be neglected, but the counterion concentration remains finite. We investigate the conformations with regard to the parameters chain length, Coulomb interaction strength, and solvent quality, and explore in which regime the competition between short range hydrophobic interactions and long range Coulomb interaction…
We have studied a bulk electrolyte, and polyelectrolyte solutions with surfactants or multivalent salt with the explicit presence of counterions and solvent molecules by means of the mesoscopic dissipative particle dynamics (DPD) method in the Grand Canonical ensemble. The electrostatic interactions are calculated using the Ewald sum method and the structure of the fluid is analyzed through the radial distribution function between charged particles. The results are in very good agreement with those reported in the literature using a different method for the calculation of the electrostatic for…
We use a standard bead-spring model and molecular dynamics simulations to study the static properties of symmetric linear multiblock copolymer chains and their blocks under poor solvent conditions in a dilute solution from the regime close to theta conditions, where the chains adopt a coil-like formation, to the poorer solvent regime where the chains collapse obtaining a globular formation and phase separation between the blocks occurs. We choose interaction parameters as is done for a standard model, i.e., the Lennard-Jones fluid and we consider symmetric chains, i.e., the multiblock copolyme…
We studied solvent-driven ordering dynamics of block copolymer films supported by a densely cross-linked polymer network designed as organic hard mask (HM) for lithographic fabrications. The ordering of microphase separated domains at low degrees of swelling corresponding to intermediate/strong segregation regimes was found to proceed significantly faster in films on a HM layer as compared to similar block copolymer films on silicon wafers. The ten-fold enhancement of the chain mobility was evident in the dynamics of morphological phase transitions and of related process of terrace-formation o…
Morphological transformations of amphiphilic AB diblock copolymers in mixtures of a common solvent (S1) and a selective solvent (S2) for the B block are studied using the simulated annealing method. We focus on the morphological transformation depending on the fraction of the selective solvent CS2, the concentration of the polymer CP, and the polymer-solvent interactions εij (i = A, B; j = S1, S2). Morphology diagrams are constructed as functions of CP, CS2, and/or εAS2. The copolymer morphological sequence from dissolved -> sphere -> rod-> ring/cage-> vesicle is obtained upon increasing CS2 a…
Molecular dynamics simulations are used to study the phase behavior of linear multiblock copolymers with two types of monomers, A and B, where the length of the polymer blocks $N_{A}$ and $N_{B}$ ($N_{A}=N_{B}=N$), the number of the blocks $n_{A}$ and $n_{B}$ ($n_{A}=n_{B}=n$), and the solvent quality varies. The fraction $f$ of A-type monomers is kept constant and equal to 0.5. Whereas at high enough temperatures these macromolecules form coil structures, where each block A or B forms rather individual clusters, at low enough temperatures A and B monomers from different blocks can join togeth…
Conformational properties of a single flexible polyelectrolyte chain in a poor solvent are studied using constant temperature molecular dynamics simulation. The effects of counterions are explicitly taken in to account. Structural properties of various phases and the transition between these phases are studied by tracking the values of asphericity, radius of gyration, fraction of condensed counterions, number of non-bonded neighbours and Coulomb interaction energies. From our simulations, we find strong evidence for a first-order phase transition from extended to collapsed phase consistent wit…
Using a continuum description, we account for fluctuations in the ionic solvent surrounding a Gaussian, charged chain and derive an effective short-ranged potential between the charges on the chain. This potential is repulsive at short separations and attractive at longer distances. The chemical potential can be derived from this potential. When the chemical potential is positive, it leads to a melt-like state. For a vanishingly low concentration of segments, this state exhibits scaling behavior for long chains. The Flory exponent characterizing the radius of gyration for long chains is calcul…
We investigate the dynamics of the collapse of a single copolymer chain, when the solvent quality is suddenly quenched from good to poor. We employ Brownian dynamics simulations of a bead-spring chain model and incorporate fluctuating hydrodynamic interactions via the Rotne-Prager-Yamakawa tensor. Various copolymer architectures are studied within the framework of a two-letter HP model, where monomers of type H (hydrophobic) attract each other, while all interactions involving P (polar or hydrophilic) monomers are purely repulsive. The hydrodynamic interactions are found to assist the collapse…
Although the volume transition of the polyelectrolyte gel has been studied for decades, little research on the effects of size of the mobile ions has been conducted. In the present paper, Tanaka classical theory of polyelectrolyte gel is extended to the cases of mobile ions of finite volume. In the salt free limit, the theoretical results show that the discontinuous volume transition of the polyelectrolyte gel will become a continuous one with an increase of the counter-ionic size. An increase in salt concentration can also make the polyelectrolyte gel in poor solvent collapse. Poorer solvent …
Poly(ethylene oxide)-$\textit{b}$-poly(butylmethacrylate) (PEO-$\textit{b}$-PBMA) copolymers have recently been identified as excellent building blocks for the synthesis of hierarchical nanoporous materials. Nevertheless, while experiments have unveiled their potential to form bicontinuous phases and vesicles, a general picture of their phase and aggregation behavior is still missing. By performing Molecular Dynamics simulations, we here apply our recent coarse-grained model of PEO-$\textit{b}$-PBMA to investigate its self-assembly in water and tetrahydrofuran (THF) and unveil the occurrence o…
Solvent vapor annealing (SVA) is known to be a simple, low-cost and highly efficient technique to produce defect-free diblock copolymer (BCP) thin films. Not only can the solvent weaken the BCP segmental interactions, but it can vary the characteristic spacing of the BCP microstructures. We carry out systematic theoretical studies on the effect of adding solvent into lamellar BCP thin films on the defect removal close to the BCP order-disorder transition. We find that the increase of the lamellar spacing, as is induced by addition of solvent, facilitates more efficient removal of defects. The …
A weakly charged flexible polyelectrolyte chain in a neutral spherical cavity is analyzed using self-consistent field theory (SCFT) within an explicit solvent model. Assuming the radial symmetry for the system, it is found that the confinement of the chain leads to creation of a charge density wave along with the development of a potential difference across the center of cavity and the surface. We show that the solvent entropy plays an important role in the free energy of the confined system. For a given radius of the spherical cavity and fixed charge density along the backbone of the chain, s…
We report on the structural properties of mixed aggregates made from rare-earth inorganic nanoparticles (radius 20 Angstroms) and polyelectrolyte-neutral block copolymers in aqueous solutions. Using scattering experiments and Monte Carlo simulations, we show that these mixed aggregates have a hierarchical core-shell microstructure. The core is made of densely packed nanoparticles and it is surrounded by a corona of neutral chains. This microstructure results from a process of controlled association and confers to the hybrid aggregates a remarkable colloidal stability.
We use molecular dynamics simulations to study the static properties of a single linear multiblock copolymer chain under poor solvent conditions varying the block length $N$, the number of blocks $n$, and the solvent quality by variation of the temperature $T$. We study the most symmetrical case, where the number of blocks of monomers of type A, $n_{A}$, equals that of monomers B, $n_{B}$ ($n_{A}=n_{B}=n/2$), the length of all blocks is the same irrespective of their type, and potential parameters are also chosen symmetrically, as for a standard Lennard-Jones fluid. Under poor solvent conditio…
We report on the formation of colloidal complexes resulting from the electrostatic self-assembly of polyelectrolyte-neutral block copolymers and oppositely charged surfactants. The copolymers investigated are asymmetric and characterized by a large neutral block. Using light, neutron and x-ray scattering experiments, we have shown that the colloidal complexes exhibit a core-shell microstructure. The core is described as a dense and disordered micro-phase of micelles connected by the polyelectrolyte blocks, whereas the shell is a diffuse brush made from the neutral chains. Colloidal complexes w…
We investigate the equilibrium charge distribution along a single annealed polyelectrolyte chain under different conditions. The coupling between the conformation of the chain and the local charge distribution is described for various solvent qualities and salt concentration. In salt free solution, we find a slight charge depletion in the central part of the chain: the charges accumulate at the ends. The effect is less important if salt is added to the solution since the charge inhomogeneity is localized close to the chain ends over a distance of order of the Debye length. In the case of poor …
Macromolecules based on dendritic or hyperbranched polyelectrolytes have been emerging as high potential candidates for biomedical applications. Here we study the charge and solvation structure of dendritic polyglycerol sulphate (dPGS) of generations 0 to 3 in aqueous sodium chloride solution by explicit-solvent molecular dynamics computer simulations. We characterize dPGS by calculating several important properties such as relevant dPGS radii, molecular distributions, the solvent accessible surface area, and the partial molecular volume. In particular, as the dPGS exhibits high charge renorma…
The charge of a polyelectrolyte (PE) controls myriads of phenomena in biology, biotechnology, and materials science, but still remains elusive from an understanding. Considering the adsorption of counterions on an isolated PE chain, an analytical expression for the effective charge - valid for all chain flexibility, for variable salt, in good solvents at all conditions, in poor solvents in the expanded (coil) state away from the critical point, and for gels, brushes and other PE systems in their expanded state - is derived. Phase boundaries and the critical point for the I-st order collapse tr…
Understanding the effect of inhomogeneity on the charge regulation and dielectric properties, and how it depends on the conformational characteristics of the macromolecules is a long-standing problem. In order to address this problem, we have developed a field-theory to study charge regulation and local dielectric function in planar polyelectrolyte brushes. The theory is used to study a polyacid brush, which is comprised of chains end-grafted at the solid-fluid interface, in equilibrium with a bulk solution containing monovalent salt ions, solvent molecules and pH controlling acid. In particul…
Nanostructured thermosets may be obtained by self-assembly of amphiphilic block copolymers in a reactive solvent and fixation of the morphologies by the cross-linking reaction. Nanostructuration requires the presence of a bock that remains miscible in the polymer during polymerization. The selection of the miscible block depends on the particular system and in some cases (e.g., for epoxy-amine network based on diglycidyl ether of bisphenol A, and 4,4'- diaminodiphenylsulfone) it is very difficult to find such a block. In this manuscript it is shown that random copolymers of methyl methacrylate…
Time-resolved small angle x-ray scattering (SAXS) was used to examine the kinetics of the transition from HEX cylinders to BCC spheres at various temperatures in poly(styrene-b- ethylene-co-butylene-b-styrene) (SEBS) in mineral oil, a selective solvent for the middle EB block. Temperature-ramp SAXS and rheology measurements show the HEX to BCC order-order transition (OOT) at ~127 oC and order-disorder transition (ODT) at ~180 oC. We also observed the metastability limit of HEX in BCC with a spinodal temperature, Ts ~ 150 oC. The OOT exhibits 3 stages and occurs via a nucleation and growth mech…
We study both experimentally and theoretically the self-assembly of binary block copolymers in dilute solution, where self-assembly is triggered by changing the solvent from the common good solvent THF to the selective solvent water, and where the two species on their own in water form vesicles and spherical micelles respectively. We find that in water the inter-micellar exchange of these block copolymers is very slow so that the self-assembled structures are in local but not global equilibrium (i.e., they are non-ergodic). This opens up the possibility of controlling micelle morphology both t…
We present a Ginzburg-Landau theory of inhomogeneous polyelectrolytes with a polar solvent. First, we take into account the molecular (solvation) interaction among the ions, the charged monomers, the uncharged monomers, and the solvent molecules, together with the electrostatic interaction with a composition-dependent dielectric constant. Second, we treat the degree of ionization as a fluctuating variable dependent on the local electric potential. With these two ingredients included, our results are as follows. (i) We derive a mass reaction law and a general expression for the surface tension.…
The dynamic behavior of polyelectrolyte chains in the oligomer range is investigated with coarse-grained molecular dynamics simulation and compared to data obtained by two different experimental methods, namely capillary electrophoresis and electrophoresis NMR. We find excellent agreement of experiments and simulations when hydrodynamic interactions are accounted for in the simulations. We show that the electrophoretic mobility exhibits a maximum in the oligomer range and for the first time illustrate that this maximum is due to the hydrodynamical shielding between the chain monomers. Our find…
We present a model for describing flexible polyelectrolytes in a good solvent a nd in the presence of monovalent salt . The molecule composed by $N$ monomers is characterized by the end to end distanc e $R_e=b (Z-1)^γ$ and the number of associated counterions $n$. At high tem peratures the polyelectrolyte behaves as a neutral polymer ($γ=0.588$). Dec reasing the temperature, the macromolecule changes from this extended configurat ion($γ=0.588$) to a stretched form ($γ\approx 1$). At even lower temp eratures, above the Manning condensation threshold, the polyelectrolyte collapse s ($γ\approx 0.…
We investigate the localization of a hydrophobic - polar (HP) - regular copolymer at a selective solvent-solvent interface with emphasis on the impact of block length $M$ on the copolymer behavior. The considerations are based on simple scaling arguments and use the mapping of the problem onto a homopolymer adsorption problem. The resulting scaling relations treat the gyration radius of the copolymer chain perpendicular and parallel to the interface in terms of chain length N and block size M, as well as the selectivity parameter χ. The scaling relations differ for the case of weak and strong …
When polyelectrolyte-neutral block copolymers are mixed in solutions to oppositely charged species (e.g. surfactant micelles, macromolecules, proteins etc), there is the formation of stable supermicellar aggregates combining both components. The resulting colloidal complexes exhibit a core-shell structure and the mechanism yielding to their formation is electrostatic self-assembly. In this contribution, we report on the structural properties of supermicellar aggregates made from yttrium-based inorganic nanoparticles (radius 2 nm) and polyelectrolyte-neutral block copolymers in aqueous solution…
We introduced a new concept for fabricating solvent stimulus polymer actuators with unprecedented sensitivity and accuracy. This was accomplished by integrating porous architectures and electrostatic complexation gradients in a poly(ionic liquid) membrane that bears ionic liquid species for solvent sorption. In contact with 1.5 mol% of acetone molecules in water, the actuator membrane (1 mm x 20 mm x 30 um) bent into a closed loop. While the interaction between solvents and the polymer drives the actuation, the continuous gradient in complexation degree combined with the porous architecture op…
Molecular dynamics simulations are used to study the phase behavior of a single linear multiblock copolymer with blocks of A- and B-type monomers under poor solvent conditions, varying the block length $N$, number of blocks $n$, and the solvent quality (by variation of the temperature $T$). The fraction $f$ of A-type monomers is kept constant and equal to 0.5, and always the lengths of A and B blocks were equal ($N_{A}=N_{B}=N$), as well as the number of blocks ($n_{A}=n_{B}=n$). We identify the three following regimes where: (i) full microphase separation between blocks of different type occu…
In this work we investigated the question of how the molecular nature of the dielectric media and the polymer-solvent dielectric mismatch affect the collapse of a polyelectrolyte chain in solution by means of dissipative particle dynamics simulations. First, we studied whether the explicit treatment of dielectric media as polar beads instead of homogeneous dielectric background results in a different system behavior. We showed that the explicit treatment of polar beads facilitates the chain collapse, i.e. it occurs at smaller values of the electrostatic strength parameter values $λ^2 \sim Q^2/…
We discuss in this paper the phase diagram of a diblock polyampholyte solution in the limit of high ionic strength as a function of concentration and charge asymmetry. This system is shown to be very similar to solutions of so-called charged-neutral diblock copolymers: at zero charge asymmetry the solution phase separates into a polyelectrolyte complex and almost pure solvent. Above a charge asymmetry threshold, the copolymers are soluble as finite size aggregates. Scaling laws of the aggregates radius as a function of pH of the solution are in qualitative agreement with experiments.
