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Physical Chemistry of Polyelectrolytes. Surfactant Science Series. Volume 99 Edited by Tsetska Radeva (Bulgarian Academy of Sciences).
Table of contents
- CRC Press Online - Series: Surfactant Science
- 1st Edition
- Handbook of Detergents, Part A - Properties (Surfactant Science Series)
Their combined citations are counted only for the first article. Merged citations. This "Cited by" count includes citations to the following articles in Scholar. Add co-authors Co-authors. Upload PDF. Follow this author. New articles by this author. New citations to this author. New articles related to this author's research.
Email address for updates. My profile My library Metrics Alerts. Sign in. Articles Cited by. The Journal of chemical physics 19 , , Journal of Physics: Condensed Matter 14 9 , , The Journal of chemical physics 18 , , Physica A: Statistical Mechanics and its Applications , , The Journal of chemical physics 12 , , They predict that while the chains in dilute salt-free solutions are stretched beyond the good solvent structure, they do not attain a rigid rod structure.
Chains do not change appreciably its conformation at low c but shrink appreciably at higher c before overlap or eventually even entanglement occurs. The shrinking is not uniform; chains become more coiled at longer length scales while at shorter length scales several bond lengths the structure is not altered [51,52]. This prediction is difcult to prove by light scattering due to the experimental requirement of low c. Relatively more powerful in this respect is SANS, which can utilize the contrast variation method to obtain form factors at conditions of strong interpolyion correlations.
On the other hand, SANS is limited to high concentrations due to weak scattering signal . In this case higher polymer concentrations yielding higher and. Results can be extrapolated from these polymer concentrations to innite dilution, to assure that a pure form factor is obtained. The most frequently treated problem regarding the polyion conformation is the dependence of Rg on cs.
Figure 17 shows data on NaPSS collected from several works [30,31,57]. The general feature is that Rg decreases with cs approxi mately by a power law Rg c with exponent 0. Results on other s polyelectrolytes such as quaternized poly vinylpyridine , ionized poly acrylic acid , and bacterial hyaluronate  show similar behavior with ranging from 0.
The chains are appreciably coiled at higher cs. For comparison with results in Figure 17, we can calculate Rg of chains upon assumption of a fully extended rodlike conformation. The chain conformation is determined by the interaction between neighboring segments and the interaction between distant segments along a polymer which, via chain exibility, are located in each others vicinity. The former effect determines the local chain stiffness. The latter is referred to as the excluded volume effect and inuences the overall conformation.
Both types of interaction can be of electrostatic and nonelectrostatic origin. In the absence of excluded volume effects exible polyions in a theta state or. The local stiffening of the chain can be expressed in terms of a persistence length distance along the chain over which the polymer maintains a stiff rodlike structure. The intrinsic part L0 is due to the exibility of the backbone without charges and is given by xed bond angles, rotameric states, helical structures, etc.
The electrostatic part Le is given by the contribution of charge interactions to increased stiffness of the chain. The issue of Le was a subject of intense theoretical work . Light scattering measures the perturbed radius of gyration, and it is practically impossible to distinguish clearly between particular contributions of the local stiffness and the excluded volume effect to the polyion size solely on an experimental basis. Several approaches were used to interpret SLS results on the polyion radius of gyration.
A different approach to the interpretation of experimental data was based on the combination of both excluded volume and persistence length effects [44,58]. A good agreement between calculation and experiment was achieved for several systems [44,58]. On the basis of substantially lower Rg0 calculated values compared to experimentally obtained Rg , it was concluded that the electrostatic excluded volume effect dominates over the electrostatic persistence length effect at higher ionic strengths cs M to 1 M.
Another approach to the interpretation of experimental data was based on the assumption of a uniform expansion of the chain, where the interaction energy between the segments was given by two terms: the usual short-range excluded volume and the electrostatic interaction via DebyeHu ckel potential . The resulting formula for the expansion factor r was used to t experimental dependencies of Rg with two tting parameters: the degree of ionization and the intrinsic excluded volume pseudopotential proportional to the FloryHuggins interaction parameter.
This method was applied to a large set of experimental data on Rg of quaternized poly vinylpyridine as a function of added salt concentration, backbone hydrophobicity, solvent dielectric permittivity, and the chemical nature of coions and counterions . Although the physical signicance of the obtained effective degree of ionization f and the intrinsic excluded volume w0 as t parameters was critically discussed , reasonable conclusions were drawn from such analysis of the data.
The main results can be summarized as follows: Rg decreases with the chain hydrophobicity and increases with the solvent dielectric permittiv ity, and the exponent in the relation Rg c varies from 0. Above cs M, a downward curvature in the dependence of Rg vs. This curvature determines the solubility limit salting out effect at high cs. In terms of the tting parameters f and w0 , f dominates at low cs , while w0 controls the curvature and the precipitation at high cs.
Recent molecular dynamics simulations on the effect of added salt on polyion conformation represent a new challenge. They are currently restricted to shorter chains, but they enable us to use full Coulomb interactions without approximations . In the case of intrinsically rigid polyelectrolytes, such as DNA, experimental results  show that electrostatic persistence length calculated from the data shows no unique power law dependence on cs. Compared to the OSF theory [60,61], a much better agreement with these data was achieved later by the calculation of Le via numerical solution of the PoissonBoltzmann equation for a toroidal polyion geometry [59,62].
The last comment concerns polyions which are intrinsically rigid not exible or semiexible. In this case, light scattering can provide information on the size and partly also on the shape of the polyions by comparing. The estimation of shapes requires monodisperse samples .
This requirement is met usually for biological polymers. A common feature of scattering from polyelectrolyte solutions is the existence of peak s in angular dependencies of scattering intensity. Figure 18 shows an example of such a result obtained by static light scattering . The sample is the same as the one as in Fig. The position of the peak maximum qm changes as qm c 0. Similar angular maxima were reported by SLS for NaPSS by different authors , and also for a variety of other polyelectrolytes with different conformations architectures , e.
Apparently, qualitatively the same phenomenon is seen by various scattering techniques at different length scales according to the polyelectrolyte concentration. The existence of the peak is clearly due to electrostatic interactions. The addition of salt decreases the height of the peak and eventually, at high cs , a complete disappearance occurs. The position of the peak either stays constant within experimental accuracy or a slight decrease of qm is observed upon increasing cs [70,72].
In the case of weak polyelectrolytes with variable charge densities, the peak gradually develops upon increase of charge density, and qm moves towards higher q . No change is observed upon stabilization of charge above the Manning Oosawa counterion condensation limit. A shift of qm towards higher q upon increasing charge density on variably sulfonated NaPSS was observed, too .
The presence of peaks in angular dependencies of scattering intensity can have, in general, several physical origins. Multiple sharp peaks appear in scattering from highly ordered crystalline lattices. Peaks appear also in scattering from simple liquids; the rst and largest peak is followed by several others with gradually decreasing peak height. In all cases peaks arise due to interparticle correlations. These correlations are reected in the solution structure factor S q , which was dened by Eq.
We remind the reader that. At high exceeding 0. The rst peak is followed by several smaller peaks reecting the positional correlation between the central particle and the outer shells of more and more distant neighbors. These are gradually less and less correlated leading to gradually smaller peaks with g r 1 for r.
The corresponding S q function is shown in Figure 19, too. Figure 19 shows also g r corresponding to the so-called correlation hole effect. It is assumed that there is a shell hole around each particle, from which other particles are excluded. This assumption leads to a step function g r with the discontinuity at r equal the effective particle size or, more realistically, to g r of a sigmoidal shape with inection point at r given by the effective particle size. This shape of g r does not lead to the presence of peaks in S q.
For completeness, we note that g r has multiple delta peaks in the case of ideal crystal lattices leading to sharp multiple peaks in S q , that for less perfect crystals lattice defects, paracrystals a broadening of peaks occurs, especially for higher order peaks, which decrease in height, and that for liquidlike order of polydisperse spheres, the resulting S q is strongly modied the main peak shifts to lower q and decreases in height while the oscillations after the main peak are washed out .
In the correlation hole concept, the S q function does not exhibit a maximum. However, upon combination with a monotonically decreasing P q. The idea of an effective volume of polyions, very similar to the correlation hole concept, was originally created by Doty and Steiner in early studies on proteins . De Gennes et al. The chain is considered to be a random walk of correlation blobs of size , where on length scales smaller than the dilute solution scaling applies and on length scales larger than all interactions are screened.
The correlation blobs correlation volumes are space lling. The correlation volume can be considered as a space around each segment, from which other segments are excluded. The correlation hole concept was used to calculate q dependencies of scattering intensities for spheres and rods  and for coils .
In the latter case the calculation is focused on semidilute solutions and therefore involves the radial distribution function of the coil segments instead of that of the centers of gravity of the coils. In order to interpret experimental data, the knowledge of S q is required. While no problems occur in the case of spheres, the spatial distribution and mutual orientation may be correlated for rods. In solutions of linear exible polyelectrolytes, the chain conformation reected in P q may depend on the spatial distribution reected in S q.
These facts must be taken into account in the interpretation of scattering peaks. Several approaches were made to separate S q from experimental data. In both cases S q itself displays a peak .
In the case of light scattering data on TMV rodlike particles , the separation of S q from I q was justied by the absence of birefringence, which assures that the orientation distribution of rods is isotropic on a macroscopic scale. Again, S q itself displays a peak. S q is obtained from the extrapolation of data acquired upon increasing the fraction of visible chains from zero to one .
S q obtained in this way contains a clear maximum. The second method is the so called zero-average contrast method, which enables us to obtain partial structure factors monomermonomer,. The monomermonomer partial structure factor clearly exhibits a peak . Another indication that S q itself contains a peak comes from DLS data, where a minimum in diffusion coefcient appears concurrently with the presence of a maximum in scattering intensity. V for more details. All these experimental observations contradict the correlation hole concept.
The liquidlike correlations between polyions or polyion segments appear to be the plausible explanation of the presence of peaks in S q. However, the question of the exact determination of the dilutesemidilute transition in linear exible polyelectrolytes is still an open question in contrast to neutral polymers. Similarly an open question is also what is the range of correlations and whether the presence of only one broad peak sometimes a feeble secondary maximum means just the nearest neighbor correlations or whether some kind of longer range order may be present, too.
It is known that concentrated solutions of highly charged colloids latex or silica particles form colloidal crystals with long-range order. It was shown in a long series of experimental work [71,83] that a two-state structure forms in solutions of highly charged colloids at low concentrations, where ordered domains of particles coexist with less ordered or disordered regions and eventually voids particle-free regions may appear and grow. This requires the presence of long-range attractive interactions between like-charged particles [71,83].
Because of the large size of these particles, they can be photographed by ultramicroscopy and their trajectories monitored by video. Experimental mean separation distances of particles dexp calculated from the position of scattering peaks are systematically lower than the theoretical mean separation distances d0 calculated upon assumption of homogeneously dispersed particles. More work on polyelectrolytes in this area is clearly needed. It was shown that formation of mesophases microscopic dense regions originating as a result of the aggregation of poorly solvated parts of chains between charges, is possible.
The structure of polyelectrolyte solutions on large length scales is frequently not homogeneous. Polyion domains clusters with dimensions appreciably exceeding the size of single chains are present. They were found in a wide variety of synthetic and biological polyelectrolytes, e. They have no relation to specic chemical structure s , which would lead to chemically specic type of aggregation. It appears that it is a universal property typical for charged systems as a group of macromolecular compounds.
The origin of these structures, the mechanism by which macromolecules of like charge associate into larger structures, and detailed knowledge of their properties are still puzzling. We present in this section the current status of our understanding of this phenomenon while further investigation is in progress. The presence of multichain domains is reected in both static and dynamic light scattering results. A slow diffusive mode is observed in spectra of relaxation times and is clearly separated from other modes over a wide range of experimental conditions see Figure 2.
The diffusion coefcient ascribed to this mode is referred to as the slow diffusion coefcient Ds. The associated scattering amplitude is marked as As. Both As and Ds are angularly dependent Figure The characterization of Rg as apparent means that it is not sure whether the scatterers giving rise to the angular dependence of As are independent scatterers and that the Guinier approximation is fully applicable. Values of Rg,app estimated from As 0 evidently exceed dimensions of individual chains and range from 30 to nm, according to ex-.
The plot in Figure 20 also shows that domains are polydisperse in size, giving a curvature in the Guinier plot. Thus the Rg,app from the 0 extrapolation is rather an upper estimate of the size. Assuming a specic shape of the scatterer, a more detailed analysis regarding polydispersity in sizes can be performed. Relatively broad distribution of sizes are obtained from As if the scatterers have simple spherical or close-to-spherical shape .
The distribution function of sizes D Rp may range in extreme cases more polydisperse than the example shown in Figure.
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As is expressed in arbitrary units. Very large domains are usually accompanied by a whole spectrum of more numerous smaller ones. The angular dependence of Ds can be explained by two effects: 1 as a consequence of the size 1 polydispersity due to Ds R and 2 due to the possibility of internal p dynamics. In the presence of internal dynamics of scatterers, Eq. The C parameter is determined by the slowest internal mode of motion in the object and is in the range from 0 to 0. Thus for Rg comparable to or larger than q1, a measurable q dependence of D can be obtained.
Although the applicability of Eq. A semiquantitative agreement between Rg,app and Rh,app is obtained upon using solution not solvent viscosity for. The presence of polyelectrolyte domains is reected also in small-angle x-ray and small-angle neutron scattering data as an upturn at low q. This behavior was observed qualitatively in many cases, but only a more sophisticated instrumentation capable of working at q almost as low as the light scattering q range made a reliable investigation of the low g upturn possible [,]. This means that the number of polyions per domain may vary from a few chains in the case of high-molecular-weight polyions Mw to thousands of chains in the case of low-molecular-weight polyions Mw There is currently available no detailed information on the domain internal structure.
These objects, in order to be visible by light scattering, have to have scattering contrast, i. Inhomogeneities in refractive index may arise due to a different arrangement or, more probably, simply due to a different concentration of polyions inside the domain compared to the rest of the solution. Taking into account large sizes of domains and the fact that the scattering contribution from domains As is in some cases comparable to the contribution from polyions Af , it can be concluded. The slow diffusive mode associated with the existence of domains is clearly separable from the fast diffusive mode over a wide range of molecular weights and polymer concentrations.
Figure 21 summarizes the results obtained on a set of NaPSS molecular weight standards at salt-free conditions [13,14]. It can be seen that Ds decreases with increasing polymer concentration and that this decrease is more pronounced the higher the molec ular weight is. At very low concentrations the fast and slow diffusion coefcients merge and the domain. Sodium poly styrene sulfonate NaPSS in water, no added salt. All diffusion coefcients were calculated from scattering data obtained at Adapted from Refs. Due to a broad interval of concentrations and molecular weights used, it can be concluded that both dilute and semidilute regimes were covered by these experiments [13,14].
No signature of a dilutesemidilute transition was found in the data concerning the slow diffusive mode, similarly to the data on the fast diffusive mode see Sec. The behavior of the domains does not signicantly differ in the dilute and semidilute regimes, respectively. Domains were found also in binary polyelectrolyte mixtures of NaPSS with different molecular weights .
As can be seen from Figure 21, binary solutions at higher polymer concentrations yield appreciably different values of Ds peaks at appreciably different time scales according to the polymer molecular weight. Only one narrow peak corresponding to the slow diffusion appears in the spectrum upon mixing of binary solutions. This is consistent with the concept of the domain effect and rules out very early speculations that the slow mode could be ascribed to some kind of single-chain slow diffusive motions. In this case, two modes would be present upon mixing.
Detailed data on polyion self-diffusion by the pulsed eld gradient NMR technique  or forced Rayleigh scattering  show that polyion self-diffusion coefcients are substantially smaller than Df but evidently larger than Ds. The origin of domains the mechanism by which like-charged polyions associate is not clear yet, although it has been the subject of numerous experimental investigations.
It was concluded that these structures are due to electrostatic interactions, not only because of the presence of charges as a necessary requirement for their occurrence but also because the slow mode appeared to be sensitive to charge interaction parameters like ionic strength and polyion charge. On the other hand, it was shown that the formation of domains is not due to the poor solvation of hydrophobic chain backbones in polar solvents hydrophobic-type interactions. The possible role of the backbone solvation in the mechanism of the domain formation was investigated by two approaches: 1 Polymersolvent pairs with different solvent quality for the uncharged chain were investigated.
This was the case of poly acrylic acid and poly methacrylic acid , which can be investigated in both the uncharged and the charged state . In both cases no correlation between the strength of the domain effect and the backbone solvation was found. In addition, the slow mode was found also in many systems with weakly hydrophobic backbones.
It was also reported in quaternized poly 2-vinylpyridine in ethyleneglycol, which is a good solvent for both neutral and quaternized charged P2VP . It should be noted at this point that the polyelectrolyte domain effect was found in a. The response of the slow mode to charge interaction parameters like ionic strength and polyion charge appears to be essential for the understanding of the domain effect.
As already discussed in Sec. III, it is necessary to decompose the total scattering intensity into contributions from particular modes and then to analyze them separately. Such analysis  shows that both fast and slow mode respond to charge interaction parameters, each in a different way. In general, the fast mode is much more sensitive to charge interaction parameters than the slow mode. The fast mode amplitude Af increases dramatically upon increase of added salt or ionic strength.
This increase is caused mainly by the increase of osmotic compressibility because the intrinsic scattering power of polyions scattering contrast is not changing dramatically. Sedlak, unpublished results. The increase of Af over the range of available cs is proportional to the polyion molecular weight Mw. This follows from the fact that Af is independent of Mw at saltfree conditions, while at high cs close to theta conditions Af is directly proportional to Mw. Therefore the higher the polyion molecular weight, the larger the change of Af over a given interval of cs .
The slow mode is also responsive to ionic strength, though in the opposite way. The slow mode amplitude As becomes weaker upon an increase of cs.
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Figure 22 shows the dependence of both scattering amplitudes Af and As on ionic strength. As can be seen from Figure 22, As smoothly decreases over all accessible ionic strengths, and no sharp transition at specic ionic strength occurs. It can be concluded that the presence of domains slow mode is rather a universal phenomenon that can be found in a very broad range of ionic strengths.
Only the amplitude strength of the effect is varying upon changing ionic strength. The appearance of the slow mode is not connected with any kind of sharp transition similar to phase transitions with a discontinuity in some physical property at strictly dened conditions. The slow mode cannot be characterized as a sign of the existence of the system in some strictly dened special regime.
The apparent sharp transition classied in literature as the ordinaryextraordinary transition is given mostly by a rapid decrease of the fast mode scattering amplitude upon lowering cs , which results in a sharp increase of the. Both added salt NaCl and free counterions are included in the calculation of ionic strength.
Amplitudes represent excess scattering and are expressed in units of the scattering intensity of a benzene standard. Very clear is the situation in the case of low-molecular-weight samples such as those in Figure 22, where Af does not exceed As over all cs investigated and so As can be very precisely measured until very high cs.
There is no splitting of the diffusion coefcient at specic cs in the sense that diffusion coefcients Df and Ds are well separated over all cs . At xed cs , the amplitude of the slow mode increases with polyion concentration. This change is also rather smooth. The size of the domains as calculated from angular dependencies of As gradually decreases with ionic strength .
Another important result of the latest investigation on the nature of the domain effect is a conclusion that domains are metastable structures with very long lifetimes. In the following we discuss briey the main qualitative aspects of the work, which is still in progress. It was found that domains are mechanically disrupted when solution passes through membrane lters with pores smaller than the natural dimensions of the domains.
Such ltration is routinely used in light scattering to avoid dust particles in the scat-. Domains become smaller and scatter less light after ltration. Importantly, the system does not spontaneously return back after ltration as expected for a system in a true equilibrium. Instead it stays in another metastable state smaller domains, eventually also less dense, although the change in density is difcult to measure.
An opposite process, i. The resistance against mechanical disruption is strongly dependent on the ionic strength. The smaller the ionic strength, the stronger the domains and vice versa. The ionic strength dependence of the slow mode amplitude as shown in Figure 22 can be therefore viewed also as an ionic strength dependence of the stability strength of domains. After a detailed study on the mechanical resistivity where metastable states were inuenced by force, a question arose whether these nonequilibrium structures can deviate spontaneously from a metastable state.
Therefore systematic experiments were carried out when samples were investigated over long periods of time up to 30 months. Indeed, a long-term time dependence was observed in some cases. The scattering amplitude of the slow mode gradually decreased from an initial value after the sample preparation to another value.
The latter value was in some cases slightly smaller than the initial one; in other cases it was substantially smaller. In many cases no long-term drift decrease in the amplitude of the slow mode is measurable even over the above-mentioned long interval of investigation, indicating virtually innite lifetime of the domains without changes. The metastability can be therefore understood mainly as the ability of the system to exist in different states at given physical and chemical conditions.
The metastability explains also the observation [97,99] that the slow mode amplitude depends on the sample history whether the sample is prepared directly at given cs or indirectly by dissolving the polymer at different cs and subsequently changing cs. In connection with above-mentioned results, the question arises whether the presence of domains is a result of the inability of the polyelectrolyte bulk material to disperse homogeneously in solution or whether there is also a mechanism by which the heterogeneous domain structure can be generated once the material is dispersed homogeneously in solution.
In order to answer this question, careful experiments were performed.
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An example is shown in Figure Under such conditions the polymer material is dispersed completely homogeneously and there is no sign of any inhomogeneities at large length scales and no modes present at long times Figure 23 above. There is one main mode corresponding to polyion diffusion, and another weak mode corresponding to salt diffusion as described in Sec. Then the solution was dialyzed to decrease the salt. The polymer was rst dissolved in 3 M NaCl above. Spectra are normalized so that peak areas correspond to scattering amplitudes of particular modes normalized to benzene scattering.
The response of the polyion diffusion is that it becomes faster and the amplitude Af is smaller. This is consistent with the expectation based on results already presented in Secs. V and XI. A pronounced slow mode with angularly dependent amplitude As appears after dialysis. Both spectra in Figure 23 are normalized so that peak areas correspond to scattering amplitudes of particular modes normalized to benzene scattering as described in Sec. It is clear from Figure 23 that the scattering contribution of the slow mode after dialysis is so large that it could not be hidden overscattered by the fast mode in the sample before dialysis.
This is important. In order to be very careful and to rule out the possibility that domains in the dialyzed sample may form just during the ltration prior to measurement when polyions are pushed under pressure into micropores of the lter membranes, an alternative method of sample purication by centrifugation was used, too. In conclusion, it can be stated that polyions spontaneously form domains upon lowering ionic strength by dialysis. The experimental results presented indicate that conditions can be met in polyelectrolyte solutions upon which attractive interactions between polyions become operative.
The idea of attractive electrostatic interactions between objects of like charge is relatively old, but historically it was mostly related to charged colloids, which are bigger and thus enable more powerful experimental approaches to be used, such as direct observation by microscopy and videomicroscopy [71,83,] or even a direct measurement of force between charged particles .
A very intense discussion on the attraction was initiated mainly by a long series of experimental work [71,83], where ordered domains of charged particles coexisting with less ordered or disordered regions and voids particle-free regions were observed. The experimental knowledge on the domain effect for polymeric macroions is currently much more limited and in our opinion still in a state of collecting fundamental experimental information, which is in many cases rather surprising and puzzling. Therefore we would not like at this stage to proceed into a deep discussion of how current concepts of charge attraction t the picture in the case of polymeric macroions.
This would also go beyond the scope of this chapter. Regarding the concepts developed for charged colloids [20,71,83], the question is whether similar behavior can occur on signicantly smaller length scales polymeric macroions and intermacroion distances are much smaller compared to colloids. Nevertheless several concepts of attraction between polymeric macroions were created. This includes a qualitative concept of charge uctuation forces given by the dynamics of counterions shared by several polyions , analytical calculation  and Brownian dynamics simulations [,] of the attraction between charged rods given by charge uctuations along rods due to the condensed multivalent counterions, attraction by expansion of the condensed layer between charged rods or rodlike segments with a decrease of free energy of the condensed layer with approach of polyions [,], and attraction due to excluded volume screening by topologically correlated objects .
An important aspect is also the violation of the summability of pairwise interactions in certain cases, which means that the overall interaction may be attractive even when the pairwise interaction is repulsive [ ]. A dumbbell model for the structure of charged rodlike macromolecules in dilute solution. Macromolecules ; Zernike F, Prins JA. Die beugung von ro ntgenstrahlen in u ssigkeiten als effekt der moleku larordnung. Z Physik ; Chu B. Laser Light Scattering. San Diego: Academic Press, Schmitz KS. Akcasu AZ. Dynamic scattering from multicomponent polymer mixtures in solution and in bulk.
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Interactions, dynamics, and elasticity in charge-stabilized colloidal crystals. Polyelectrolytes are polymers bearing ionizable groups, which, in polar solvents, can dissociate into charged polymer chains macroions and small counterions ; see Figure 1. The combination of macromolecular properties and long-range electrostatic interactions results in an impressive variety of phenomena.
It makes these systems interesting from a fundamental as well as a technological point of view. Some of the relevant questions primarily motivated by scientic interest are the following. How is the size of a polyelectrolyte affected by molecular weight, intrinsic stiffness, solvent quality, or ionic strength? Which observables are well characterized by coarse-grained quantities such as a linear charge density, and which depend on chemical details?
How are dynamic quantities like viscosity or electrophoretic mobility related to static properties of polyelectrolytes? A thorough understanding of polyelectrolytes has become increasingly important in biochemistry and molecular biology. The reason is that virtually all proteins, as well as DNA, are polyelectrolytes. Their interactions with each other and with the charged cell membrane are still shrouded in mystery to a high degree. For instance, a puzzling question is why two equally charged objects should attract each other in the rst place.
Unfortunately, theoretical understanding of polyelectrolytes is less developed than the understanding of the properties of neutral polymers. Some reasons are that the presence of long-range interactions renders the application of renormalization group techniques and scaling ideas much more difcult than in the neutral case. The reason is that many new length scales Constitution formula for sulfonated polystyrene with sodium counterions left and a physicists picture right.
The degrees of freedom related to the counterions contribute largely to the entropy. In contrast, the macroion itself is usually poor in entropy. Hence, there are effects which, despite the strong electrostatic interactions, are in fact entropy driven. For a given chain geometry, the competition between energy minimization and entropy maximization results in a particular equilibrium counterion distribution around the polyelectrolyte.