18804
J. Phys. Chem. B 2004, 108, 18804-18814
ARTICLES Pearl-Necklace Structures in Annealed Polyelectrolytes Sahin Uyaver† and Christian Seidel*,† Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung, Am Mu¨hlenberg, D-14476 Potsdam, Germany ReceiVed: August 9, 2004
Using (semi-)grand canonical Monte Carlo simulations of a polyelectrolyte chain of length N where the charges are in contact with a reservoir of constant chemical potential µ given by the solution pH, we study the behavior of annealed polyelectrolytes in a poor solvent. We focus on the conformational properties in the close-toΘ-point regime τ < τ* ∼ N-1/5u-3/5, which can be reached either by reducing electrostatic interaction strength u ) λB/b or by improving solvent quality, i.e., by reducing τ ) (Θ - T)/Θ. We investigate the conformations with regard to parameters u, τ, and µ and explore in which conditions pearl-necklace-like structures are stable. Most of the pearl-necklace parameters are found to obey the scaling relations predicted for quenched polyelectrolytes. However, similarly to the behavior known for this class of polyelectrolytes, we obtain large fluctuations in pearl number and size. In agreement with theoretical predictions we find a nonuniform charge distribution between pearls and strings. For τ , 1 and moderate interaction strength we demonstrate that a cascade of pearl-necklace transitions can be initiated by changing µ, i.e., by tuning the solution pH.
1. Introduction Polyelectrolytes (PELs) are macromolecules that contain subunits having the ability to dissociate charges in polar solvents as, e.g., water. In recent years, PELs have received a lot of attention because of their importance in materials science, soft matter research, and molecular biology. The combination of polymer properties and long-range electrostatic interactions, which causes competing interactions and gives rise to quite different length scales, results in a wide variety of phenomena that makes PELs interesting also from a fundamental point of view. However, despite considerable effort, the understanding of PELs constitutes still a challenging area for both fundamental and applied studies.1-4 Many polymers are based on a hydrocarbon backbone, which is known to be highly hydrophobic. Therefore the solubility of PELs in water is often only given by charged side groups. Typical examples of polyelectrolytes are sulfonated polystyrene (PSS), poly(methacrylic acid) (PMA), poly(vinylamine) (PVA) and DNA. The competition between the attractive interaction due to the poor solubility of the backbone and the repulsive Coulomb interaction between the polymer charges gives rise to a rather complex phase behavior of PELs in poor solvents. To minimize the interfacial area with the solvent, an uncharged hydrophobic polymer immersed in water forms a collapsed globule.5,6 After adding charges to the polymer the globule losses its spherical shape and can form so-called pearl necklaces, locally collapsed structures (pearls) connected by elongated strings.7-9 The particular structure is a result of the Rayleigh instability which is known to cause a splitting of charged droplets into smaller ones.10,11 Already in ref 9 the results of * To whom correspondence should be addressed. E-mail: seidel@ mpikg.mpg.de. † Mailing address: D-14424 Potsdam, Germany.
the scaling approach were supported by Monte Carlo simulations that demonstrated a cascade of transitions from one to two to three globules/pearls. Later the theoretical understanding of pearl-necklace-like structures has been extended12-16 and various aspects of the theoretical picture have been confirmed by Monte Carlo simulations within Debye-Hu¨ckel models17,18 and by molecular dynamics simulations using the full Coulomb interaction with explicite counterions.19-23 However, direct experimental evidence for such pearl-necklace-like structures in polyelectrolyte chains has been lacking up to recent years. For a long time, there was only some indirect signature based on the observation of conformational chain properties that seem to be consistent with the necklace picture.24-28 Except fluorescence images of large DNA molecules modified by synthetic polymers,29-31 only very recently was the first direct experimental evidence given by AFM studies on conformational changes of PELs in poor solvents upon adsorption onto mica.32-34 With respect to different dissociation behaviors, one can distinguish between strong and weak PELs, a classification widely used in the chemistry community,2 or between quenched and annealed PELs, the common classification in the physics community.3 Strong polyelectrolytes, poly-salts as, e.g., NaPSS, dissociate completely in the total pH range accessible by experiment. The total charge as well as its particular distribution along the chain is solely imposed by chemistry, i.e., by polymer synthesis. Thus in the language of statistical mechanics of disordered systems, the charge distribution is a quenched variable. That is why such polyelectrolytes are also called “quenched”. On the other hand, weak polyelectrolytes represented by poly-acids and poly-bases dissociate in a limited pH range only. The total charge of the chain is not fixed but can be tuned by changing the solution pH. The number of charges as well as their positions are fluctuating thermodynamic
10.1021/jp0464270 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/13/2004
Pearl-Necklace Structures in Annealed Polyelectrolytes variables. The imposed quantity is the pH of the solution which is, up to trivial additive constants, the chemical potential of the charges. In this case, the distribution of charges is an annealed variable. Therefore such polymers are also called “annealed” PELs. Due to dissociation and recombination of ion pairs the charges can move along the chain. This extra degree of freedom gives rise to new and nontrivial features. The specific behavior of weak PELs has attracted considerable interest in experimental,35-38 theoretical,39-43 and simulation studies.44-50 In particular, for PELs in a poor solvent, the annealing of charges can have strong effects. A first-order phase transition between a weakly charged globule and a strongly charged stretched chain predicted by theory39 was recently confirmed by simulations.50 Actually, the first-order transition into a stretched conformation not occurring for strong polyelectrolytes is the most evident fingerprint of the so-called polyelectrolyte effect. For sufficiently poor solvent quality, the behavior of the polymer is dominated by the discontinuous transition, which is reflected on pronounced anomalies of the titration curves of, e.g., PMA35 and copolymers of maleic acid.36 If the solvent quality is not very poor but the system is close to the Θ-point, the phase transition is suppressed and theory predicts that intermediate structures as, e.g., pearl necklaces can be stable.4,39 Recently, first direct experimental evidence for the existence of pearl necklaces in annealed PELs was given by Kirwan et al.34 Investigating conformational changes of PVA caused by tuning of the polyelectrolyte charge through the solution pH, they observed segmental collapse through pearl-necklace structures. Hence, a detailed understanding of pearl necklaces in annealed PELs in the “close-to-Θpoint” regime is desired. This is a great challenge, in particular for simulation studies. However, such investigations are lacking in the literature so far. The outline of the paper is as follows: In section II we give a short overview over the theory of pearl necklaces and annealed PELs. The necessary selection is focused on points that are essential to discuss the simulation results presented in section IV. Simulation model and method are introduced in section III. Finally, a brief summary and our conclusions can be found in section V. II. Theory A. Pearl-Necklace Structure. To minimize its interfacial energy, uncharged polymers in a poor solvent collapse into a dense globule.5,6 Consider a chain of length N and monomer size b. The monomer density Fglob inside the globule is defined by the balance between two-body attraction, τb3NFglob, and threebody repulsion, b6NFglob2, giving Fglob = τ/b3, with τ ) (Θ T)/Θ being the normalized distance from the Θ-point, which is a measure of the solvent quality. On length scales smaller than the thermal correlation length ξT ∼ b/τ the attractive interactions are not relevant but are dominated by Gaussian fluctuations of the polymer chain. On the other hand, at larger length scales density fluctuations are suppressed and the globule can be thought to consist of densely packed thermal blobs of size ξT. The total size of the globule becomes Rglob ∼ (N/Fglob)1/3 ∼ bτ-1/3N1/3. For most purposes the globule can be viewed as a liquid droplet and the connectivity of the chain does not play any important role. For example, the free energy of the globule is dominated by an interfacial free energy with the poor solvent with a surface tension γ ∼ kBT/ξT2 ∼ τ2. If the polymer is charged, the Coulomb repulsion between charged monomers is expected to change the shape of the globule, but not significantly to affect its volume.51 The volume
J. Phys. Chem. B, Vol. 108, No. 49, 2004 18805
Figure 1. Schematic view of a pearl-necklace structure. Pearls are spherical with diameter dp and consist of gp monomers each. Strings are cylindrical with length lstr and diameter dstr and consist of gstr monomers each.
occupied by the polymer is still defined by the solvent quality. A charged globule will spontaneously deform if the Coulomb repulsion inside the globule FCoul ∼ e2f2N2/4π0Rglob ∼ kBT f2N2λB/Rglob becomes comparable to its surface energy Fsur ∼ γRglob2, where f is the degree of charging and λB ) e2/4π0kBT is the Bjerrum length that determines the strength of the Coulomb interaction. The corresponding dimensionless interaction parameter is given by u ) λB/b. Finally, one obtains that a deformation occurs if the total charge of the globule fN becomes larger than (Nτ/u)1/2, which appears at
f > fc ∼ (τ/uN)1/2
(1)
The first attempt to describe polyelectrolyte chains in a poor solvent was by Khokhlov.51 He has shown that the globule can lower its energy by elongating into a cylinder or cigar-like shape. This introduces the electrostatic blob size ξel ∼ b/(uf2)1/3, which is determined by the balance between the electrostatic energy of the blob and its surface energy. At length scales smaller than the size of these electrostatic blobs ξel the chain statistics is only slightly perturbed by the charges. On large length scales, however, electrostatics dominates the structure: The blobs repel each other to form an elongated chain of width D ) ξel. Applying a volume constraint, its length becomes Lcyl ∼ bN(uf2)2/3/τ. Later, however, it was realized by Dobrynin, Rubinstein, and Obukhov (DRO)9 that the cylinder conformation is unstable and changes to the so-called pearl necklaces. As first discussed by Rayleigh,10 with increasing charge an oil droplet in water undergoes an instability when its electrostatic energy reaches the order of the interfacial energy. It splits into smaller drops. For a drop of radius R, the critical charge at which the instability occurs is QR ∼ e(γR3/kBTλB)1/2. In a first approximation a neutral collapsed polymer chain can be considered as a liquid drop that undergoes the Rayleigh instability when it becomes charged.7,9 However, contrary to simple liquids, the presence of chemical bonds along the polymer chain prevents infinite separation between daughter drops, but they remain linked by stretched polymer strands. Thus, the picture obtained for polyelectrolytes in a poor solvent is that of a necklace of collapsed globules, the pearls, connected by narrow strings that are stretched by the electrostatic repulsion between the pearls (see Figure 1). Following refs 9 and 13 the pearls are just at the Rayleigh instability threshold, their density is that of a collapsed globule and their size obtained from the Rayleigh charge is
dp ∼ ξel ∼
b (uf 2)1/3
(2)
The number of monomers in a pearl reads
gp ∼ Fglobξel3 ∼
τ uf 2
(3)
18806 J. Phys. Chem. B, Vol. 108, No. 49, 2004
Uyaver and Seidel
and the number of pearls becomes
np ∼
N uf 2 ∼N gp τ
(4)
i.e., in agreement with the discussion above the first splitting occurs at fc ∼ (τ/uN)1/2. The thickness of a string is of the order of the thermal blob size dstr ∼ ξT. Balancing the Coulomb interaction between two beads with the surface energy of the connecting string, the length of a string becomes
lstr ∼
( ) τ uf 2
1/2
b
(5)
calculation is rather tedious. To get a first picture of the charge distribution, a two state model has been proposed by Castelnovo et al.:41 The monomers either belong to pearls and have a probability 〈f〉 - δf of being charged or they belong to strings and have a probability 〈f〉 + δf ′ of being charged. The grand canonical free energy that determines the distribution is very similar to eq 9, but the polymer contribution includes both the electrostatic and the interfacial free energies of the pearlnecklace structure. In the limit far from the transition to the stretched Gaussian chain behavior where all the pearls disappear, the reduction of the fraction of charged monomers in pearls was found to be41
( )
〈f〉2u δf ≈ 〈f〉 τ3
and the number of monomers in a string is
gstr ∼ Fgloblstrdstr2 ∼ (τuf 2)-1/2
(6)
Finally, the scaling behavior of the total length of pearl-necklace structures becomes
( )
uf 2 Lnec ≈ nplstr ∼ Nb τ
1/2
(7)
Note that at τ ∼ (uf 2)1/3, Lnec crosses over to the end-to-end distance of a stretched Gaussian polyelectrolyte; i.e., pearlnecklace structures are stable in the range (uf 2)1/3 < τ < uf 2N, where the upper limit comes from the transition into a single globule discussed above. In terms of the degree of charging the stability condition reads
( ) τ uN
1/2
τ*) has been studied extensively in I. On one hand the first-order phase transition between weakly charged globules and a strongly charged extended conformations theoretically predicted by Raphael and Joanny39 was clearly confirmed; on the other hand there were no indications of pearlnecklace-like structures recently widely discussed for quenched polyelectrolytes. The latter result was not surprising because theory predicts that, for annealed polyelectrolytes, substructures such as pearl necklaces are only stable in a region close to the Θ-point τ < τ*.39,41 Obviously, there are two ways to study this region: (i) by increasing τ* at constant τ, i.e., fixing solvent quality, but enlarging the width of the specific region close to the Θ-point, or (ii) by reducing τ at constant τ*, i.e., improving solvent quality and coming closer to the Θ-point. Here we have
18808 J. Phys. Chem. B, Vol. 108, No. 49, 2004
Uyaver and Seidel
Figure 3. (a) Simulation results at constant chemical potential vs coupling strength. (b) Mean square end-to-end distance vs coupling strength.
Figure 4. Typical simulation snapshots demonstrating a varying conformation when the coupling strength is increased at constant τ ) 0.26, µ ) 9.0. The different coupling strengths are 0.008 (I), 0.016 (II), 0.05 (III), 1.52 (IV), and 2.0 (V). Charged monomers are colored red, uncharged ones yellow.
used both approaches to study the structure of annealed polyelectrolytes in the close-to-Θ-point regime. Both routes have their advantages and disadvantages. Choosing a relatively large distance from the Θ-point, in the uncharged state at small µ one has well-defined globular conformations with weak fluctuations only. According to eq 13, τ* can be enlarged by reducing both chain length N and coupling strength u. Because the use of shorter chains would obviously restrict the search for pearl necklaces, the favorable way to succeed with the search for these particular configurations at large τ* is to reduce u. Following this route, we obtain well-defined pearl necklaces that are analyzed by using a cluster-recognition algorithm known from literature.21 Due to the weak coupling strength u , 1, however, there appear certain peculiarities. In particular, pearl necklaces occur in the strongly charged regime with f ≈ 1. Besides that, the range u , 1 is far from the parameter region known for polyelectrolytes usually considered in experiments. Note that a realistic way to reduce u in experimental studies is to enlarge the distance between ionizable groups by synthesizing appropriate block copolymers. Nevertheless, performing simulations at reduced τ enables a perspective to study pearl necklaces in the parameter range λB ≈ 1, which is close to typical experimental situations. Below we will see, however, that then the uncharged globules become much more floppy and pearl necklaces are strongly effected by fluctuations. But, following the second route, in fact, we are able to tune the number of pearls by changing the chemical potential, i.e., by changing the solution pH. A. Large τ* at Weak Electrostatic Interaction. Figure 3 shows the behavior of both degree of charging and polymer size over a wide range of interaction strength (0.008 < u0 < 2.0) at constant solvent quality τ ) 0.26. (Note that in the simulation model we use the average bond length b, which is slightly affected by the average degree of charging 〈f〉. To determine clearly the simulated system, henceforth we use the bare coupling strength u0 ) λD/b0, where both λD and b0 are given input parameters.) For weak and intermediate interaction strenth, the polyelectrolyte is completely charged due to the large chemical potential µ ) 9.0. At u* ≈ 1.5, however, the polymer undergoes a first-order phase transition into a nearly uncharged, collapsed globule.50 At that point the Coulomb energy per charged monomer becomes larger than the penalty for neutralizing a charge. Using the corresponding parameters (N ) 256),
in fact the stability condition (13) yields that pearl necklaces become unstable at u > u* ∝ N-1/3τ-5/3 ≈ 1.5. In other words, they might exist in the highly charged region u < 1.5. Typical snapshots, showing in a schematic way the configurational changes occurring over the total range of u considered in our simulations, are presented in Figure 4. From the snapshots, one can clearly see that pearl necklaces with a varying number of pearls occur indeed. Although the chain is completely charged at very weak interaction strength, it collapses into a globule (point I). With increasing u, the globule becomes deformed and splits into several pearls. Pearl-necklace structures are formed between points II (u0 ) 0.016) and III (u0 ) 0.05). Beyond point III, all the pearls are pulled out, and the chain exhibits a stretched but more or less homogeneous configuration. Following de Gennes et al.52 the chain extension is expected to grow as R ∝ (uf2)1/3, which is nicely confirmed by the simulation result (see Figure 3). Note that at u < u* there is no difference in the behavior compared to quenched polyelectrolytes, which is reasonable because, for a completely charged polymer at large pH, one cannot distinguish between annealed and quenched behavior. Only at u g u*, where the polyelectrolyte minimizes its energy by neutralizing almost all charges and collapsing back into a nearly uncharged globule (point V, u0 ) 1.53), does the annealed nature become relevant and do we observe a discontinuous transition in the degree of both charging and chain stretching known from I. The pearl-necklace structures occurring between points II and III, i.e., at 0.016 e u0 e 0.040, are analyzed by means of the cluster-recognition algorithm, which has been optimized to analyze configurations up to five pearls. The results are collected in Table 1, where only conformations are included that have a weight of 1% at least. The behavior of np, gp, and dp is found to be in a qualitative agreement with the scaling predictions given in eqs 2-4. The number of pearls increases with the coupling parameter u. For np > 2 we observe a large fluctuation not only in the number of pearls but also in their size. This is a feature known from all simulation studies on quenched polyelectrolytes in a poor solvent.23 Both gp and dp show a tendency to decrease with growing coupling strength, which is in agreement with theoretical predictions. Disagreement appears, however, for the string parameter gstr and lstr. Probably the strings are too short for applying asymptotic scaling relations as done in theory. Nevertheless, considering the total necklace length,
Pearl-Necklace Structures in Annealed Polyelectrolytes
J. Phys. Chem. B, Vol. 108, No. 49, 2004 18809
TABLE 1: Globules and Pearl Necklaces at Different Interaction Strength u0 ) λB/b0 (τ ) 0.26, µ ) 9.0) u0 0.008 0.016 0.023 0.030 0.040
np wt (%) 1 2 3 4 3 4 5 2 3 4 5 6
100 99.9 70.9 27.8 10.3 65.0 24.0 2.5 20.7 38.8 28.6 8.2
gp
gstr
256 124 ( 8 77.3 ( 5.1 58.8 ( 3.8 62.5 ( 4.6 51.9 ( 3.6 43.4 ( 2.9 40.3 ( 4.5 37.6 ( 3.5 34.5 ( 2.9 32.0 ( 2.5 29.9 ( 2.2
dp
lstr
2.17 ( 0.02 7.73 ( 0.08 3.36 ( 0.04 11.2 ( 0.1 3.16 ( 0.04 6.95 ( 0.07 2.90 ( 0.03 34.2 ( 0.7 3.30 ( 0.05 15.9 ( 0.1 2.95 ( 0.05 9.72 ( 0.07 2.72 ( 0.02 175 ( 1 3.62 ( 0.02 71.6 ( 0.2 3.22 ( 0.02 39.3 ( 0.1 2.97 ( 0.02 23.9 ( 0.1 2.79 ( 0.02 15.2 ( 0.1
4.87 ( 0.03 5.61 ( 0.07 3.45 ( 0.03 10.5 ( 0.1 6.11 ( 0.09 4.11 ( 0.04 36.5 ( 0.1 17.9 ( 0.1 11.2 ( 0.1 7.75 ( 0.06
Results of the cluster-recognition algorithm: np is the number of pearls, gp the number of monomers per pearl and gstr the number of monomers per string. dp is the pearl size and lstr the length of a string. The third column gives the weight of the various peal-necklace structures.
Figure 5. Scaling of pearl-necklace conformations: average mean square end-to-end distance vs u01/2 (τ ) 0.26, f ) 1).
which according to theory is basically given by the total string length, one obtains a rather nice agreement between theory and simulation data. Following eq 7 the pearl-necklace length should grow as Lnec/Nb ∼ (uf2/τ)1/2. Plotting the average mean square end-to-end distance vs u1/2, which is shown in Figure 5 for the range of u0 given in Table 1 (τ, f ) const.), one obtains a nearly linear behavior indeed. This is rather surprising because the higher order structures at larger u0 exhibit strong fluctuations. For a more detailed examination of the structure at all length scales, we calculate the spherically averaged form factor
S(q) )
〈〈
1
N+1
|∑ N
j)0
exp(iq·rj)
|〉 2
|q|
〉
(18)
where rj is the position vector of monomer j. In Figure 6 S(q) is plotted for the five points discussed above: (I) at vanishing interaction where the polymer forms a compact globule, (II) at very small coupling strength showing a stable dumbbell structure, (III) at intermediate interaction giving higher order pearl necklaces, (IV) just before the discontinuous coil-globule transition where the polyelectrolyte exhibits a highly stretched conformation, and (V) at any coupling strength beyond point IV with a chain collapsed back into a compact globule that is now almost uncharged in contrast to (I). From (I) to (IV) the polymer is almost fully charged. Note that the effect of electrostatic interactions that can be measured by the scaling variable uf2 vanishes at both point I with u ≈ 0 and point V with f ≈ 0. The form factor of these globular structures is almost
Figure 6. Spherically averaged structure factor at τ ) 0.26, µ ) 9.0 and varying coupling strength. The particular values of u0 are the same as in Figure 4: Circles (I), squares (II), diamonds (III), triangles (IV), and stars (V). Straight lines indicate limiting scaling laws. Arrows point to additional peaks: solid line for globules, dashed for dumbbell.
identical, showing an additional peak at q ≈ 3.3 that is related to globule size and yields Rglob ≈ 1.9. The corresponding mean square end-to-end distances directly given by the simulations are R ) 2.2 (I), which is in agreement with the results of the cluster-search algorithm (see Table 1), and R ) 2.1 (V). For point II, one observes a pronounced shoulder at q ≈ 0.75, which is related to the string length of the dumbbell structure (for a detailed consideration see below). Considering the local degree of charging of pearls and strings, we observe that strings are almost always completely charged, but there occurs a finite contribution of configurations where a few monomers located in pearls are uncharged. Thus, there is a qualitative agreement with the theoretical predictions given by Castelnovo et al.41 Due to the weak interaction strength the characteristic energy (see eq 12) becomes (τ3u/〈f〉)1/3 , 1. Hence, the charge distribution is almost uniform and eq 11 yields O(δf)) 10-3. Thus, we have shown that pearl-necklaces exist indeed in annealed PELs provided the solvent quality is not too poor. Analyzing the structure of the polymer at a given solvent quality τ, one can shift the system from “rather poor” behavior to a “close-to-Θ-point” one by enlarging τ*. In simulations this can be easily done by reducing u; in experiments, however, an insitu continuous variation of u is impossible. Therefore the interesting question is, will globule-necklace transitions also occur in a kind of titration “experiment”, i.e., simulating the system at constant u and τ < τ*, but varying µ? This problem is addressed in the remaining part of the subsection. At τ ) 0.26 we study two coupling strengths that both fulfill τ < τ*: (i) u0 ) 0.016 and (ii) u0 ) 0.03. Figure 7 shows average degree of charging and mean square end-to-end distance as a function of chemical potential. Typical snapshots are given in Figure 8. Figures 7 and 8 show several unusual and interesting features: At small µ the degree of charging increases linearly with the chemical potential (pH). The larger the interaction strength u, the smaller is the slope. Obviously, the rather strong growth of f occurs due to the weak interaction strength. Note that in I at u0 ) 1 the slope was found to be very small. As the degree of charging increases from zero to about 0.5 (at u0 ) 0.03) or even almost 0.8 (at u0 ) 0016), respectively, the shape of the chains is not changed. It remains a globule, only the size of which is very weakly expanded with growing f. However, reaching a certain degree of charging fc the globule becomes unstable. It splits into several pearls, the number of which depends on coupling strength u. In contrast to quenched PELs,
18810 J. Phys. Chem. B, Vol. 108, No. 49, 2004
Uyaver and Seidel TABLE 2: Position of Globule-to-Necklace Transition u0
µc
〈fc〉
0.016 0.03
2.19 2.76
0.77 0.54
Chemical potential µc and averag degree of charging 〈fc〉 at the coupling strengths u0 ) 0.016 and u0 ) 0.03 (τ ) 0.26, N ) 256).
Figure 7. Close-to-Θ-point behavior at τ ) 0.26 and weak interaction strengths u0 ) 0.016 (circles), 0.03 (squares): (a) average degree of charging vs chemical potential; (b) mean square end-to-end distance vs chemical potential.
Figure 8. Close-to-Θ-point behavior at τ ) 0.26 and weak interaction strengths: typical simulation snapshots.
for annealed PELs, the structure instability is coupled with a charge instability. At µ ) µc(fc) the chain undergoes a noncontinuous transition into a completely charged state. This behavior is very similar to the first-order transition discussed in I. However, now the interaction is too weak to stretch the chain completely, but it stays in the multipearl state corresponding to complete charging at the specific coupling strength. Thus, at u0 ) 0.016, one obtains a dumbbell whereas a higher-order pearl-necklace appears at u0 ) 0.03. The positions of the globule-to-necklace transition are given in Table 2. According to eq 1 the instability is expected to occur at fc ∼ u-1/2. Hence, the critical degrees of charging should obey
()
fc,1 u2 ) fc,2 u1
1/2
≈ 1.4
(19)
which is fulfilled quite well indeed. Because complete charging is reached at µ ) µc, no further changes occur at µ > µc. Note the strong fluctuations on the number of pearls in the case u0 ) 0.03.
Figure 9. Time evolution of a pearl-necklace-like structure (number of pearls np and mean square end-to-end distance R) with different configurations (u0 ) 0.03, µ ) 4.0).
In most of the systems where we obtain pearl necklaces, we find coexistence of configurations with different numbers of pearls (see Table 1). A typical time evolution of a pearl-necklace structure is shown in Figure 9. In this case the chain mainly fluctuates between configurations with four, five, and three pearls, which occur with a weight of 65%, 24%, and 10%, respectively. The end-to-end distance exhibits a reasonable correlation with the number of pearls. The strong fluctuations of the structure caused by the small differences in free energy between the different structures are a common feature of pearl necklaces, which is already known from the study of quenched PELs.23 On the basis of our simulation data, we are not able to decide whether the annealing of the charge distribution gives a damping of structural fluctuations. The mean square end-to-end distances R measured directly from the simulation data are 9.6 and 26.7 at u0 ) 0.016 and 0.03, respectively. Theoretically the total length of a straight pearl necklace is given by
Lnec ) np × dp + (np - 1) × lstr
(20)
which yields Lnec ) 10.7 and 30.2 for u0 ) 0.016 and 0.03, respectively. Basically, the measured size deviates from the theoretically estimated one because in general pearl-necklace chains do not exhibit a completely elongated conformation as assumed in eq 20. Therefore the real end-to-end distance obeys R < Lnec. To get more direct information about the average structure of pearl necklaces, again we calculate the form factor. Figure 10 shows the spherically averaged single chain structure factors just before the globule undergoes a pearl-necklace instability (solid line) as well as after the transition (dashed and dot-dashed lines). Considering S(q) for different chemical potentials, but at constant interaction strength, we obtain almost unchanged behavior as long as one stays below the transition (µ < µc) or above (µ > µc), respectively. Beyond that, on the globular side µ < µc, S(q) is almost identical even at different u. The form factor of globular structures exhibits a pronounced peak at q ≈ 3.3 that is related to a globule size Rglob ≈ 1.9. Directly calculated from simulations one obtains R ≈ 2.1. The well-developed oscillations at large q indicate that the globule
Pearl-Necklace Structures in Annealed Polyelectrolytes
Figure 10. Close-to-Θ-point behavior at τ ) 0.26 and weak interaction strengths. Shown is the single chain structure factor at (i) u0 ) 0.03, µ ) 2.7585 (solid line), (ii) u0 ) 0.016, µ ) 2.20 (dashed), and (iii) u0 ) 0.03, µ ) 2.767 (dot-dashed). (i) represents globular structure before the transition. Note that an almost identical S(q) is obtained at the corresponding point u0 ) 0.016, µ ) 2.175. The solid arrow points to the peak position that is related to globule size. (ii) and (iii) exhibit fingerprints of pearl necklaces after the transition, i.e., dumbbell at u0 ) 0.016, µ ) 2.20 (dashed arrows point to the shoulders that correspond to string length and pearl size) and a multi-pearl structure at u0 ) 0.03, µ ) 2.767, respectively.
has sharp boundaries and does not fluctuate strongly. The structure factor of the dumbbell occurring at u0 ) 0.016, µ ) 2.20 (dashed line in Figure 10) shows two shoulders. The first one at small q (q ≈ 0.74) is related to the pearl-pearl distance and gives lpp ≈ 8.5. Identifying lpp as the distance between the center of mass of both pearls, the results of the clusterrecognition algorithm given in Table 1 yield lpp ) lstr + 2((dp/ 2)) ) 4.87 + 3.36 ≈ 8.23, which is in reasonable agreement with the form factor. The second shoulder seen at large q (q ≈ 3.8) is related to pearl size dp ) 4π/q ≈ 3.3 which is in a good agreement with Table 1. Thus, the different analysis methods are found to give a consistent picture of pearl-necklace-like structures. Due to strong fluctuations of the number of pearls the form factor plotted for u0 ) 0.03, µ ) 2.767 (dot-dashed line in Figure 10) shows only a broad shoulder. Therefore, in this case, one cannot obtain any specific information on the pearl necklace from the structure factor. Note that this is a typical problem also known from scattering experiments on pearlnecklace structures. B. Small τ at Moderate Electrostatic Interaction. In the preceding subsection we demonstrated that pearl necklaces occur in annealed polyelectrolytes provided they are in the close-toΘ-point regime with τ < τ*. If existing as a stable conformation, pearl necklaces obey the scaling predictions as good as it is known for quenched PELs. However, to reach the region τ < τ* above, we had substantially to reduce the interaction strength, which causes particular effects. In the following we will show that pearl necklaces may also exist at moderate interaction strength u ≈ 1, which is comparable to typical experimental situations. To reach this parameter range while fulfilling condition τ < τ*, one has to choose solvent qualities rather close to the Θ-point. Henceforth we use τ ) 0.07 (β ) 2.80). At this distance from the Θ-point, neutral chains are still collapsing into a compact globule; however, the globule exhibits much stronger fluctuations than seen above at the larger τ-values. Figure 11 shows both average degree of charging f and mean square end-to-end distance R in dependence of chemical potential µ, i.e., dependences that are related to titration experiments. In contrast to the behavior shown in Figure 7 there is no indication of any discrete transition. (Note that Figure 11
J. Phys. Chem. B, Vol. 108, No. 49, 2004 18811
Figure 11. Close-to-Θ-point behavior at τ ) 0.07 and moderate interaction strengths u0 ) 1: (a) average degree of charging vs chemical potential; (b) mean square end-to-end distance vs chemical potential.
Figure 12. Close-to-Θ-point behavior at τ ) 0.07 and moderate interaction strength u0 ) 1: scaled end-to-end distance vs average degree of charging. Straight lines indicate scaling predictions for pearl necklaces (ν ) 1) and elongated PEL chains (ν ) 2/3).
is a log-log plot.) Although 〈f〉 is monotonically growing with µ, in the titration curve one can clearly distinguish four different regimes: (A) At 0 e µ < 2 we obtain a rather weak exponential increase. But, due to 〈f〉 , 1 there appear almost no changes in the globule size. (B) At 2 < µ < 3 both 〈f〉 and R exhibit a strong increase where 〈f〉 grows from about 0.02 to about 0.1. Below we will see that this is the region where pearl necklaces occur. (C) At 3 < µ < 5.5 we find an increase of 〈f〉 up to full charging. Both 〈f〉 and R obey a power law. (D) For µ > 5.5, saturation is reached. The chain is stretched to an extension of about two-thirds of its contour length, which is clearly beyond the range where Gaussian elasticity can be assumed. According to eq 7 the pearl-necklace length Lnec is expected to scale linearly with f. In Figure 12 we plot the scaling behavior of end-to-end distance R that follows from the data shown in Figure 11. Indeed, in region (B) there is a range of 〈f〉 where we observe a scaling of R that is in agreement with the prediction for pearl necklaces. Although the particular region is rather narrow the observed result gives an indication on the existence of pearl necklaces. On the other hand, at large 〈f〉 we find a clear power-law behavior with exponent 2/3, which is in perfect agreement with the scaling prediction for a stretched PEL chain R ∝ (uf2)1/3 given by de Gennes et al.52
18812 J. Phys. Chem. B, Vol. 108, No. 49, 2004
Uyaver and Seidel
TABLE 3: Globules and Pearl Necklaces at Different Chemical Potentials (τ ) 0.07, λB ) 1.0, N ) 256) µ 0.0 2.0
2.20
2.30
2.40
2.50
wt np (%) 1 1 2 3 4 1 2 3 4 5 1 2 3 4 5 2 3 4 5 2 3 4 5
95.0 43.5 26.4 15.0 10.5 19.9 19.9 26.4 22.5 8.6 7.8 13.8 30.7 31.6 13.0 16.2 34.7 33.9 12.2 15.9 37.0 33.3 11.7
gp 256 256 117 ( 8 65.8 ( 4.7 43.8 ( 3.3 256 101 ( 7 53.8 ( 4.2 38.7 ( 3.1 32.3 ( 2.5 256 82.3 ( 6.4 45.4 ( 3.9 36.9 ( 3.0 32.1 ( 2.5 53.1 ( 5.2 39.5 ( 3.6 32.9 ( 3.6 30.8 ( 2.5 39.8 ( 4.5 35.4 ( 3.4 31.8 ( 2.8 30.0 ( 2.5
gstr
20.8 ( 1.4 29.3 ( 1.6 26.8 ( 1.0 52.8 ( 3.7 47.2 ( 1.3 33.7 ( 0.7 23.5 ( 0.6 91.0 ( 5.5 59.8 ( 1.2 36.1 ( 0.6 23.8 ( 0.5 150 ( 5 68.8 ( 1.0 40.0 ( 1.0 25.5 ( 0.5 176 ( 3 74.9 ( 0.8 42.9 ( 0.4 26.4 ( 0.5
dp 5.36 ( 0.41 5.22 ( 0.61 5.88 ( 0.13 4.51 ( 0.11 3.89 ( 0.11 5.42 ( 0.32 6.07 ( 0.12 4.64 ( 0.10 3.84 ( 0.09 3.47 ( 0.09 5.48 ( 0.51 5.98 ( 0.10 4.42 ( 0.09 3.91 ( 0.09 3.57 ( 0.08 5.83 ( 0.10 4.49 ( 0.09 3.89 ( 0.09 3.50 ( 0.08 5.83 ( 0.10 4.51 ( 0.09 3.93 ( 0.08 3.49
lstr
4.38 ( 0.14 5.97 ( 0.05 6.01 ( 0.13 8.96 ( 0.26 9.07 ( 0.11 7.42 ( 0.15 5.65 ( 0.13 15.2 ( 0.3 11.6 ( 0.2 7.98 ( 0.15 5.92 ( 0.12 24.3 ( 0.3 13.4 ( 0.2 8.98 ( 0.19 6.37 ( 0.13 29.5 ( 0.3 15.1 ( 0.2 9.82 ( 0.18 6.68 ( 0.14
fp 0.01 0.03 0.03 0.05 0.06 0.03 0.05 0.07 0.07 0.08 0.03 0.07 0.08 0.08 0.08 0.10 0.09 0.09 0.09 0.12 0.10 0.10 0.10
fstr
0.05 0.06 0.07 0.06 0.08 0.08 0.08 0.08 0.09 0.09 0.09 0.10 0.10 0.10 0.10 0.12 0.12 0.11 0.11
a n is the number of pearls, g is the number of monomers per pearl, p p and gstr is the corresponding number per string. dp is the pearl size and lstr is the length of a string. The third column gives the weight of the various peal-necklace structures. fp and fstr are the average degrees of charging of pearls and strings, respectively.
In Table 3, we collect the results of the analysis by means of the cluster-recognition algorithm. The corresponding simulation snapshots are shown in Figure 13. In the range 2.2 e µ e 2.5 or 0.03 e 〈f〉 e 0.12 we find a wide population of different pearl necklaces. Again only conformations are included that have a weight larger than 1%. Compared to pearl necklaces at rather poor solvent quality but weak electrostatic interaction (see Table 1), here we see much stronger fluctuations in the number of pearls. The modified behavior is indicated, however, already by the size fluctuations of the underlying globule at vanishing electrostatic interaction (uf2)1/2 ) 0. Although at τ ) 0.26 we observe Rglob ) 2.17 ( 0.02, on the basis of simulation trajectories of the same length, at τ ) 0.07 we find Rglob ) 5.36 ( 0.41. Thus, due to the reduced distance to the Θ-point not only the globule size is enlarged by more than a factor of 2, but also its relative fluctuation width is increased from 0.9% to 7.6%. In Figure 14 we plot the single chain structure factor S(q) calculated at the various chemical potentials for which snapshots are shown in Figure 13. There is a continuous transition from globular behavior to that of stretched chains, but there appears almost no signature of pearl necklaces. For small τ, enhanced fluctuations between pearl necklaces of different order broadens all characteristic features in the form factor. Remember that at τ ) 0.26 we obtain the most pronounced signature for a dumbbell conformation that appears with a weight of 99.9%. For τ ) 0.07, however, there is no corresponding state. The dumbbell with the largest weight of about 26% appears at µ ) 2.0. But, there occurs also a globule with almost twice the weight. Now, at τ ) 0.07 and u0 ) 1, the difference between the average degree of charging of chain sections that belong to pearls or strings, respectively, is measurable, at least in simulations. For 〈f〉 ) 0.05, following eq 12 the electrostatic energy of a counterion at the surface of a pearl becomes FCoul,ci(dp)/kBT ∼ 0.2, which is large enough to cause substantial charge inhomogeneity. Using eq 11, one obtains δf ≈ 0.01 at
Figure 13. Close-to-Θ-point behavior at τ ) 0.07 and moderate interaction strength u0 ) 1. Simulation snapshots taken at different chemical potentials, shown are the conformations with largest weight: globular structure (0 e µ e 2); pearl necklaces (2.2 e µ e 2.5); stretched chain (3 e µ e 10).
〈f〉 ) 0.03 and δf ) 0.03 at 〈f〉 ) 0.05, which are in good or fair agreement, respectively, with the charge inhomogeneity obtained in the simulations (see Table 3). For larger 〈f〉, obviously the first order theory41 used to receive relation (11) fails. Recently, the lack of direct experimental evidence for pearlnecklace structures was successfully reduced. One of the first contributions was that by Kirwan and co-workers,34 who studied conformational changes of PVA, which is actually a weak cationic PEL. Its charge density can be tuned through the solution pH from uncharged (pH > 10) to fully ionized (pH < 3). Hence, this study is in fact the first one dealing with pearl necklaces in annealed PELs. When the conformations are imaged by AFM upon adsorption onto mica, extended conformations are found at small pH and globules at high pH. At an
Pearl-Necklace Structures in Annealed Polyelectrolytes
Figure 14. Close-to-Θ-point behavior at τ ) 0.07 and moderate interaction strength u0 ) 1: single chain structure factor for varying µ. Straight lines indicate the asymptotic behavior of a collapsed globule and a stretched chain, respectively.
intermediate pH range 4.9 e pH < 7, where a significant decrease in the degree of ionization of PVA sets in, the AFM pictures clearly confirm the existence of pearl necklaces. Obviously the conformation of the chains is highly sensitive to small changes in the pH, i.e., in the average charge density on the polymer. In general, the set of AFM images taken at varying pH (see Figures 2 and 3 in ref 34) is very similar to the cascade of different structures, including pearl necklaces, shown as simulation snapshots at varying µ in Figure 13. V. Conclusions Performing extensive grand canonical Monte Carlo simulations of annealed PELs in poor solvents, in agreement with theoretical predictions4,39 we have shown that pearl-necklacelike structures can be stable in weak PELs provided the solvent quality is not too poor. Although at a rather poor solvent quality τ > τ* the behavior of the polymer is dominated by the firstorder phase transition between a weakly charged collapsed globule and a strongly charged stretched chain,39,50 at τ < τ* the conformational transition becomes almost continuous. Although there occurs a cascade of discrete pearl-necklace transitions embedded in the continuous crossover from globule to stretched chain, due to strong fluctuations in the number of pearls as well as in their size, the transition as a whole appears to be continuous. To study the relevant region τ < τ*, we explore two different routes: either enlarging the width τ* of the close-to-Θ-point region or reducing the distance τ to the Θ-point. Both routes allow us to investigate stable pearl-necklace-like conformations. Besides the effect of the solvent quality τ, we have studied the chain conformation with regard to electrostatic interaction strength u ) λB/b and to chemical potential µ, which equals the solution pH except a trivial additive constant. Following the first route, we enlarge τ* by choosing rather small interaction strengths. Such a setting allows us to work at rather poor solvent qualities where quite stable globular as well as pearl-necklacelike structures occur. However, because of the relative strength of the two interactions, given by the relation u/τ , 1, there appear some peculiarities: (i) Due to the weak Coulomb interaction a stretched chain is not accessible, but the final conformation at complete dissociation is a pearl necklace. Because of f ) 1, at a given solvent quality τ, the number of pearls depends only on interaction strength u. (ii) Due to the rather strong attractive interaction there appear some remains of the first-order phase transition. When globules reach the
J. Phys. Chem. B, Vol. 108, No. 49, 2004 18813 critical degree of charging fc, in contrast to quenched PELs, the structural Rayleigh instability is coupled to a charge instability. Thus, at µc the chain undergoes a noncontinuous transition into a completely charged state. Following the second route, i.e., improving the solvent quality by reducing τ, one can study the behavior at intermediate interaction strengths u ≈ 1. Such a parameter setting is comparable to experiments with typical model polyelectrolytes. Note, however, that the weak coupling regime could be investigated by synthesizing block copolymers including nonionizable monomers. When the strength of the attractive volume interaction is reduced, naturally all the completely or partially collapsed conformations, including the pearl-necklace-like one, exhibit stronger fluctuations. However, here we find weakly charged pearl necklaces that are very similar to the corresponding conformations of quenched PELs. Changing the degree of ionization by tuning the chemical potential, i.e., by tuning the solution pH, we obtain a cascade of pearl-necklace transitions that is in fact very similar to the results of a recent experimental study of poly(vinylamine).34 Comparing the structure parameters of the pearl necklaces with theoretical scaling predictions, independently on the route by which the pearl necklaces are obtained, we find good agreement for all pearl parameters as well as for the total necklace length. The single strings are obviously too short to obey asymptotic scaling relations. Note that this result is in agreement with former simulation studies on quenched PELs. Hence, pearl necklaces in annealed PELs can be described by the theoretical models developed for the quenched case. However, the region where they are stable is rather restricted. Beyond that, the average degree of charging can be affected both by external conditions and by instabilities in the charge density. In this study the screening length of the electrostatic interaction was fixed at λD ) 10. Doubtless a variation of the screening by tuning the concentration of additional salt can have a strong influence on the formation of specific structures of charged chains. For annealed PELs, particular features were predicted by theory.53 Therfore a detailed analysis of the specific dependences is required. For strongly charged chains, from simulations studies on quenched PELs with explicit counterions, it is known that a delicate interplay between counterion distribution and chain conformation influences the structure of PELs in a poor solvent, in particular if they are strongly charged.23 Counterion penetration into pearls and charge renormalization can be important. For the systems considered in this study, however, the effect is expected to be weak because strongly charged pearl necklaces occur only at weak electrostatic interaction. Nevertheless, the influence of nonhomogeneous counterion distributions on the conformational properties of annealed PELs in a poor solvent can be important, in particular at finite polymer concentration. Acknowledgment. We thank M. Borkovec for useful comments. We gratefully acknowledge partial funding of this work by the Deutsche Forschungsgemeinschaft within the priority program SPP 1009. References and Notes (1) Hara, M., Ed. Polyelectrolytes: Science and Technology; Marcel Dekker: New York, 1993. (2) Dautzenberg, H.; Jaeger, W.; Ko¨tz, J.; Philipp, B.; Seidel, C.; Stscherbina, D. Polyelectrolytes: Formation, Characterization and Application; Hanser Publishers: Munich, Vienna, New York, 1994. (3) Barrat, J.-L.; Joanny, J.-F. AdV. Chem. Phys. 1996, 94, 1-66.
18814 J. Phys. Chem. B, Vol. 108, No. 49, 2004 (4) Schmidt, M., Ed. Polyelectrolytes with defined molecular architecture; Advances in Polymer Science; Springer: Heidelberg, 2003; Vol. 165/166. (5) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1985. (6) Grosberg, A. Y.; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994. (7) Kantor, Y.; Kardar, M.; Li, H. Phys. ReV. E 1994, 49, 1383-1392. (8) Kantor, Y.; Kardar, M. Phys. ReV. E 1995, 51, 1299-1312. (9) Dobrynin, A. V.; Rubinstein, M.; Obukhov, S. P. Macromolecules 1996, 29, 2974-2979. (10) Rayleigh, L. Philos. Mag. 1882, 14, 184-186. (11) Taylor, G. Proc. R. Soc., London Ser. A 1964, 280, 383-397. (12) Solis, F. J.; de la Cruz, M. O. Macromolecules 1998, 31, 55025506. (13) Dobrynin, A. V.; Rubinstein, M. Macromolecules 1999, 32, 915922. (14) Dobrynin, A. V.; Rubinstein, M. Macromolecules 2001, 34, 19641972. (15) Pickett, G. T.; Balazs, A. C. Langmuir 2001, 17, 1511-1517. (16) Migliorini, G.; Lee, N.; Rostiashvili, V.; Vilgis, T. Eur. Phys. J. E 2001, 6, 259-270. (17) Lyulin, A. V.; Du¨nweg, B.; Borisov, O. V.; Darinskii, A. A. Macromolecules 1999, 32, 3264-3278. (18) Chodanowski, P.; Stoll, S. J. Chem. Phys. 1999, 111, 6069-6081. (19) Micka, U.; Holm, C.; Kremer, K. Langmuir 1999, 15, 4033-4044. (20) Micka, U.; Kremer, K. Europhys. Lett. 2000, 49, 189-195. (21) Limbach, H. J.; Holm, C. Comput. Phys. Commun. 2002, 147, 321324. (22) Limbach, H. J.; Holm, C.; Kremer, K. Europhys. Lett. 2002, 60, 566-572. (23) Limbach, H. J.; Holm, C. J. Phys. Chem. B 2003, 107, 80418055. (24) Carbajal-Tinoco, M. D.; Williams, C. E. Europhys. Lett. 2000, 52, 284-290. (25) Waigh, T. A.; Ober, R.; Williams, C. E. Macromolecules 2001, 34, 1973-1980. (26) Aseyev, V. O.; Klenin, S. I.; Tenhu, H.; Grillo, I.; Geissler, E. Macromolecules 2001, 34, 3706-3709. (27) Lee, M.-J.; Green, M. M.; Mikes, F.; Morawetz, H. Macromolecules 2002, 35, 4216-4217.
Uyaver and Seidel (28) Baigl, D.; Sferrazza, M.; Williams, C. E. Europhys. Lett. 2003, 62, 110-116. (29) Starodoubtsev, S. G.; Yoshikawa, K. J. Phys. Chem. 1996, 100, 19702-19705. (30) Ueda, M.; Yoshikawa, K. Phys. ReV. Lett. 1996, 77, 2133-2136. (31) Yoshikawa, K.; Yoshikawa, Y.; Koyama, Y.; Kanbe, T. J. J. Am. Chem. Soc. 1997, 119, 6473-6477. (32) Minko, S.; Kiriy, A.; Gorodyska, G.; Stamm, M. J. Am. Chem. Soc. 2002, 124, 3218-3219. (33) Kiriy, A.; Gorodyska, G.; Minko, S.; Jaeger, W.; Sˇ taˇpa´nek, P.; Stamm, M. J. Am. Chem. Soc. 2002, 124, 13454-13462. (34) Kirwan, L. J.; Papastavrou, G.; Borkovec, M. Nano Lett. 2004, 4, 149-152. (35) Nagasawa, M.; Murase, T.; Kondo, K. J. Phys. Chem. 1965, 69, 4005-4012. (36) Dubin, P.; Strauss, U. P. J. Phys. Chem. 1973, 77, 1427-1431. (37) Ko¨tz, J.; Philipp, B.; Pfannemu¨ller, B. Makromol. Chem. 1990, 191, 1219. (38) Tirtaatmadja, V.; Tam, K. C.; Jenkins, R. D. Macromolecules 1997, 30, 3271-3282. (39) Raphael, E.; Joanny, J.-F. Europhys. Lett. 1990, 13, 623-628. (40) van der Schoot, P. Langmuir 1997, 13, 4926-4928. (41) Castelnovo, M.; Sens, P.; Joanny, J.-F. Eur. Phys. J. E 2000, 1, 115-125. (42) Borukhov, I.; Andelman, D.; Borrega, R.; Cloitre, M.; Leibler, L.; Orland, H. J. Phys. Chem. B 2000, 104, 11027-11034. (43) Netz, R. R. J. Phys.: Condens. Matter 2003, 15, 239-244. (44) Reed, C. E.; Reed, W. F. J. Chem. Phys. 1992, 96, 1609-1620. (45) Sassi, A. P.; Beltra´n, S.; Hooper, H. H.; Blanch, H. W.; Prausnitz, J. M.; Siegel, R. A. J. Chem. Phys. 1992, 97, 8767-8774. (46) Ullner, M.; Jo¨nsson, B.; So¨derberg, B.; Peterson, C. J. Chem. Phys. 1996, 104, 3048-3057. (47) Ullner, M.; Jo¨nsson, B. Macromolecules 1996, 29, 6645-6655. (48) Berghold, G.; van der Schoot, P.; Seidel, C. J. Chem. Phys. 1997, 107, 8083-8088. (49) Zito, T.; Seidel, C. Eur. Phys. J. E. 2002, 8, 339-346. (50) Uyaver, S.; Seidel, C. Europhys. Lett. 2003, 64, 536-542. (51) Khokhlov, A. R. J. Phys. A 1980, 13, 979-987. (52) de Gennes, P.-G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. J. Phys. (Paris) 1976, 37, 1461-1473. (53) Higgs, P. G.; Raphael, E. J. Phys. (Paris) I 1991, 1, 1-7.
