Hydrogels in Poor Solvents: A Molecular Dynamics Study

Transcription

1 Full Paper Hydrogels in Poor Solvents: A Molecular Dynamics Study Bernward A. F. Mann, a Kurt Kremer, Olaf Lenz, Christian Holm* The equilibrium swelling behavior of a crosslinked polyelectrolyte gel under poor-solvent conditions is examined. To this end, MD simulations of a coarse-grained network model with a diamond-like topology are employed where the counterions are explicitly taken into account. A large range of different parameter combinations in the fraction f of charged monomers of the network, the strength of electrostatic interactions measured via the Bjerrum length B, and various strand lengths are studied. The results agree with data on single PE chains, in particular the theoretically predicted pearl necklace conformations can be identified as well as a sausage-like regime in the strong-coupling region. Introduction Hydrogels are a particular class of gels well adapted to numerous applications, for example, as superabsorbants (hygiene products, health care, and environmental cleanup operations), in drug delivery, agriculture, and as actuators in microfluidic devices. [1 6] Polyelectrolyte (PE) gels, or PE networks, form a subclass of hydrogels, that are dominated by electrostatic effects. Experimentally those have been extensively studied for a long time, [7 9] where initial work concentrated on the swelling behavior in solutions of simple salt, [10,11] or, more recently, the influence of oppositely charged surfactants [12 16] and microgel particles. [17,18] PE networks are composed of crosslinked PEs, polymer chains which dissociate ions in solution. Due to the macroscopic requirement of electroneutrality the counterions released from the chains are confined inside the gel and exert an osmotic pressure which leads to the B. A. F. Mann, K. Kremer Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128, Mainz, Germany O. Lenz, C. Holm Institut für Computerphysik, Universität Stuttgart, Pfaffenwaldring 27, Stuttgart, Germany holm@icp.uni-stuttgart.de a Deceased. swelling of the network against the elastic response of the network strands. The delicate balance between long range interactions, counter ion entropy, and network elasticity renders the magnitude of the swelling and the phase behavior of PE networks hard to predict. [19] In real systems, the composition of the gel and its surrounding solvent plays also an important role, though usually only the relative amount of crosslinkers can be well-controlled experimentally. The topology is much harder to control, and almost impossible to determine. Also the distribution and type of strand lengths, the role of entanglements, and the presence and unknown number of dangling ends introduces many unknown/controllable features. Despite recent progress in that respect, allowing to synthesize well-defined model PE networks based on the photodimerization of monodisperse star polymers from poly(tert-butyl methacrylate) (PtBMA) star precursor polymers with subsequent photo-crosslinking, [20] equilibrium properties in general do depend on the exact preparation process of the gel (e.g., active ends and N-functional crosslinking agents). wileyonlinelibrary.com DOI: /mats

2 B. A. F. Mann, K. Kremer, O. Lenz, C. Holm While both single-chain PEs and neutral polymer gels have been thoroughly examined over the past decades, charged polymer networks or hydrogels have been the subject of much fewer analytical [10,19,21 26] or computational studies. [27 40] Also most hydrocarbon chains such as sulfonated polystyrene (PSS) have a hydrophobic hydrocarbon backbone, meaning that for this PE water is more or less, depending on the charge fraction f,apoor solvent, causing the macromolecule to try to minimize its surface contacts with the surrounding water by attempting to collapse into a globular state. The same effect occurs if the system temperature T is lowered below a critical threshold, the u temperature; there, monomers are more likely to be found close together due to an effective net attraction. An environment exhibiting such features is generally called a poor solvent because the macromolecule prefers contact with itself over the solvent molecules, and consequently falls out of solution eventually, if the system is neutral or the temperature too low. Adding charges to the system gives rise to a whole new set of structures. For the most intriguing one, a pearlnecklace-like shape of the chains, theoretical models for weakly [41 43] and strongly [44 47] charged chains have been developed. Computer simulation studies [47 53] were successful in proving the existence of such conformations within single PE chain systems, providing valuable insight into their basic properties. Although the theory was developed for single chains, similar conformations of strands within the gel can be expected for a PE gel under poor-solvent conditions [24] where the scaling predictions change due to the network topology. In this paper, we will study PE gels in equilibrium with the surrounding poor solvent, and present molecular dynamics (MD) results on structural aspects and geometrical observables of such gels. The system parameters and solvent conditions have been used in an earlier study of single chains in poor solvent, [53] while the hydrogel network setup and the analysis methods were used in our previous work on hydrogels in good solvent, [31,33,34] so that we can directly compare our results to the findings from those articles. To our knowledge, this simulation study is the first to reveal pearl-necklace structures in charged hydrogels, and it also corroborates findings from real-world experiments regarding scattering function observations which might indicate direct evidence for the existence of pearls-necklace structures in poor-solvent PEs. Poor-Solvent Theories This section introduces the theoretical concepts for treating charged polymers under poor-solvent conditions, as found in refs. [43,54]. We first discuss the scaling picture for PEs in dilute solution (Section The Scaling Behavior of Poor- Solvent Polyelectrolytes ) and its predictions for the pearlnecklace structures, and subsequently we transfer these findings to the network regime (Section Treatment of Poor- Solvent Gels ). The Scaling Behavior of Poor-Solvent Polyelectrolytes In a solution of monomers in a poor solvent, they are attracting each other. For a charged polymer, however, the repulsion of the likewise charged repeat units along the backbone introduces an additional effect counteracting the collapsing tendencies induced by the solvent, such that within certain parameter regimes a fragile balance can emerge, creating a sequence of locally collapsed globules (called pearls) interconnected by elongated strings. This structure is called a pearl necklace. The formation of this structure can be seen as an equilibrium between the attraction caused by the poor solvent and the electrostatic repulsion. Let us consider an uncharged chain consisting of N m monomers with bond length b at temperature T. Such a chain will form a globule whose density r is determined by the balance between three-body repulsion (b 6 N m r 2 ) and two-body attraction (BN m r), where B is the second virial coefficient B tb 3. The reduced temperature t is defined by t ¼ðu TÞ=u where u is the temperature of the u-point, or alternatively by t ¼ ð" u LJ " LJÞ=" u LJ, where " u LJ ¼ 0:34 k BT [49,50] is the Lennard-Jones (LJ) value that has been determined to lead to u conditions for an infinitely long uncharged chain with our model potentials. Summing up, the density is given by r t=b 3, and the size R of the globule is [43] R N 1=3 m N 1=3 m j r T bt 1=3 Nm 1=3 (1) g T This equation contains an important length scale, namely the thermal blob size j T, indicating the size of the density fluctuations. Below j T the chain statistics is unperturbed by the volume interactions, resembling a random walk of g T monomers with j T bg 1=2 T ; on length scales larger than j T, the attraction between monomers is stronger, causing the thermal blobs in the globule to be space-filling with r g T =j 3 T. While the number of monomers in a thermal blob is g T 1=t 2 and its size is j T b=t, the surface tension g of the globule is of the order of k B T per thermal blob at the globular surface, that is, g k B Tj 2 T. When adding electrostatic interactions to the system, a new length scale is introduced. The strength of the electrostatic interactions is commonly defined by the 722

3 Hydrogels in Poor Solvents:... so-called Bjerrum length B, which is defined by extension [43] B ¼ e 2 4p" 0 " r k B T (2) R E N m b Bf 2 1=2 t 1=2 (4) b (where e is the elementary charge and e r is the dielectric constant of the solvent), that is, B denotes the distance at which the electrostatic energy of two unit charges is comparable to the thermal energy. In the case of the pearlnecklace structure, this leads to the formation of Coulombic blobs. Their size j PE is determined by the balance between the electrostatic interactions within the blob and the surface tension g ffi k B Tj 2 T ffi k B Tðt=bÞ 2 : B ðfg PE Þ 2 j PE ffi gj 2 PE (3) where g PE is the number of monomers in the blob and f is the fraction of charged monomers. Below length scales of j PE electrostatic interactions are negligible, and the Coulombic blob is composed of densely packed thermal blobs; for strong electrostatic interactions, the poorsolvent effects are negligible, and the PE forms a linear sequence of Coulombic blobs. Starting from a short chain and increasing the chain length consequently increases the size of the globule, until it is energetically more favorable for them (Equation 3) to split into two smaller pearls, reducing the surface tension on the one hand (where j PE enters quadratically), but reducing the electrostatic repulsion on the other hand even more (where, through j PE ffi bðg PE =tþ 1=3, the blob size enters with j 5 PE ). Considering the shape of a charged globule turns out to be similar to the classical Rayleigh instability of a charged droplet [43,55,56] where it was shown that a spherical object with radius R and charge Q > eðgr 3 =ðk B T B ÞÞ 1=2 is locally unstable and deforms spontaneously. The equilibrium state of charged droplets, a set of smaller droplets each charged lower than the critical value and placed at infinite distances, cannot be directly related to a PE due to the latter s chemical bonds between each monomer. Still, the analogy allows to conclude that the system can reduce its energy by splitting into a set of smaller charged globules connected by narrow strings the pearl-necklace conformation. Overall, a long PE chain in a poor solvent, therefore, adopts an elongated shape determined by the balance between surface tension and electrostatic self-energy. Minimizing the free energy of the respective contributions from the pearls and the strings (both consisting of their surface energy and the Coulombic energy of the charges contained therein) and the total electrostatic energy of the pearl-necklace structure then leads to a chain which exhibits a different scaling exponent than for a good solvent. Treatment of Poor-Solvent Gels Not much can be said about the behavior of PE networks under poor-solvent conditions due to the many competing contributions, except for some simple scaling arguments [24] which are summarized in the following. There the authors considered a c -gel that consists out of disentangled meshes and without the complication of additional salt, where at equilibrium the total pressure in the system vanishes. Thus, the pressure contributions arising from the counterions and from the strand elasticity cancel and P ¼ P C þ P E ¼ 0. At the scaling level, the elastic energy density of a gel in a poor solvent can be computed via P E ¼ k B T t R 2 E b (5) This is compensated by the osmotic pressure of the counterions at the gel equilibrium concentration c, given by t 3 P C ¼ k B Tfc ¼ k B T ðfn m Þ 2 (6) b 3 Hence, the equilibrium mesh size follows as R E ¼ bfn m t With respect to a corresponding free pearl-necklace chain consisting of N m monomers, such a gel is consequently stretched. Also note that this description requires the necklace to be shorter than a string of thermal blobs as the pearl-necklace picture remains valid only if most monomers belong to the pearls, that is, if R E < N m bt. Therefore Equation 7 holds for t > f 1/2, whereas, closer to the u-point the solvent quality becomes irrelevant and the structure is that of a u-gel. For very poor solvents the single-chain results [49,52,53] showed that the counterions cannot all be considered to be free, since they start to condense onto the pearls. The same is expected for the network, where the gel is then to be expected to macroscopically collapse if t > f 1/3. Concluding, poor-solvent conditions are irrelevant close to the u-point for as long as t < f 1/2 ; the gel shrinks continuously for intermediate solvents f 1=2 < t < f 1=3 and (7) 723

4 B. A. F. Mann, K. Kremer, O. Lenz, C. Holm undergoes a collapse at t f 1/3 due to counterion condensation. Simulation Methods We have performed MD simulations of charged hydrogels under poor-solvent conditions. To be able to compare the results of our simulations with the results of a previous work that focused on simulations of PE chains in poor solvent, [53] we used the polymer parameters from that work. Furthermore, we reused the diamond-like network model and method from previous publications on hydrogels in good solvent. [31,33,34] In the simulations, we employ a perfect and defect-free network of N p PE chains connected at their ends to N nodes tetrafunctional crosslinking sites (nodes); this network topology is conserved at all times. The polymer chains are modeled as bead-spring chains of N m beads. For being able to compare the studies, the chain and solvent parameters were chosen to match the parameters of our previous simulations of single PE chains under poor-solvent conditions. [49,50,52,53]. The van-der-waals interaction between chain beads is modeled by the standard LJ potential: 0 U LJ ðrþ ¼ 4" s 12 s 6 c B LJ ; for r < r r r (8) 0; otherwise 12 6 s where the shift c ¼ is chosen such that the s r cut potential value is zero and continuous at the cutoff radius r ¼ r cut. Poor solvent conditions are modeled by setting the cutoff radius to r cut ¼ 2.5s and the Lennard-Jones parameter " LJ to " LJ ¼ 1:75k B T. By including the attractive part of the Lennard-Jones potential, we effectively model the attractive hydrophobic interaction between the monomers. Furthermore, we set the LJ parameter s ¼ 1. By doing so, we define the length unit of the system as the diameter of a bead s and the energy unit to k B T. The bonds between adjacent beads in a chain and the crosslinks between the ends of the chains were representedby an added finite extension non-linear elastic (FENE) potential: r cut 0 " U FENE ðrþ ¼ 1 2 k FrF 2 ln 1 r # 2 ; for r r F B F (9) 1; otherwise where k F ¼ 7 k BT and r s 2 F ¼ 2.0s. To model charged PE chains, every f th monomer of each chains carried a unit charge. To neutralize the system, a corresponding number of oppositely charged counterions was introduced into the system. Besides the electrostatic interaction, the counterions also employed an excluded volume interaction with other counterions and the chain monomers, which is modeled via the purely repulsive core of the Lennard-Jones potential with the parameters s ¼ 1, " LJ ¼ 1, and r cut ¼ 2s 1 6. The electrostatic interactions in the fully periodic system were taken into account using the P 3 M algorithm [57,58] for a given Bjerrum length B, tuned to an absolute accuracy of 10 3 k B T using the error formulas derived in ref. [59] Note that even in the case where the electrostatic interaction vanished ( B ¼ 0), the system still contained a number of counterions to counter the charge of the chains. This was done to ensure consistency and completeness in the parameter space rather than modeling a physical meaningful system. We investigated systems with a chain length of N m ¼ 199 (respectively, N m ¼ 200 for f ¼ 1/3 to ensure that N CI 2N) and charge fraction f 2f0:125; 0:25; 0:3333; 0:5; 1:0g for multiple Bjerrum lengths B 2f0s; 0:125s; 0:25s; 0:5s; 0:75s; s; 2s; 3s; 5s; 6s; 9sg, calling these parameter sets the C-series because it covers the regime of C1 C3 described extensively in ref. [53] We looked at additional systems with N m 2f200; 239; 287; 335; 383; 431; 479g for charge fraction f ¼ and Bjerrum length B ¼ 1:5s, corresponding to A1 in ref. [53] and consequently called A-series. Then, we also investigated very long chains at N m ¼ 479 for charge fractions f 2f0:0625; 0:125; 0:25; 0:3333; 0:5; 1:0g and B ¼ 1:5s, which have no counterpart in the previous study; for better reference it shall be named E-series. Table 1 gives a summary of these settings. To ensure that the gels were in thermodynamical equilibrium, we had to make sure that the systems were at a pressure of p 0. Unfortunately, we did not succeed to utilize any common barostat to simulate the gels in the (N, p, T)-ensemble, since the fluctuations of the pressure were too large and some of the systems were too close to a volume collapse, so that no good equilibrium could be found. Therefore we employed (N, V, T)-ensemble simulations at different values of the volume V, and determined the equilibrium swelling volume V eq by measuring p(v)- diagrams, from which we estimated via a spline fit the Table 1. The parameters used in our simulation study. Name N m f B " LJ A-series 200,..., s 1.75k B T C-series 199 (or 200) 0.125,..., 1.0 0s,..., 9.0s 1.75k B T E-series ,..., s 1.75k B T 724

5 Hydrogels in Poor Solvents:... equilibrium swelling volume V eq at p ¼ 0. Then we performed an additional long (N, V, T) production run at the determined equilibrium swelling volume. [60] Results In this section we present our results for the conformations under different parameter combinations, and present the data analysis performed regarding several experimentally accessible observables. In general, the parameter space for PEs is high-dimensional, including degrees of freedom such as the solvent quality " LJ, the valency of monomers or counterions, and added salt in the system, which complicate the analysis further. Here, we focus on the effects of the Bjerrum length B as a measurement of the strength of electrostatic interactions and the charge fraction f. Conformations The simulations revealed a great variety of conformations to be found at different points in the parameter space of charge fraction f and Bjerrum length B. Figure 1 summarizes the observed conformations into a structure diagram and shows small simulation snapshots, and it compares the structure diagram of hydrogel networks to the structure diagram of single PE chains from ref. [53] One can identify five regimes of similar structures which are described in greater detail below: collapsed conformations for small charge fractions or very strong electrostatics, pearl-necklaces for moderate to high charge fractions, and not too strong Coulombic coupling, stretched structures for large f and moderate B, and the sausage regime for larger Bjerrum lengths. Within the pearl-necklace regime, we distinguish two subregimes. The Case of Vanishing Electrostatic Interactions Starting in the limit of vanishing electrostatics, Figure 2 shows conformations for B ¼ 0s. At low charge fraction f ¼ 0.125, depicted in the top left of Figure 2, we find a collapsed structure. The system tries to minimize its density as much as possible, and the snapshot indicates a clear phase separation between the mutually attractive monomers and the counterions with their purely repulsive excluded volume interactions. Due to the periodic network topology, however, the latter are trapped within the hydrogel, and their entropic force counteracts the total collapse of the polymer into a space-filling globule. That is also the reason why for an increasing charge fraction f we no longer observe collapsed structures at all, even for the case of no electrostatics: The growing amount of free N CI ¼ N nodes þ N p fn m counterions exerts a larger, entropy-driven osmotic pressure against the short-range attraction between the monomers on smaller length scales, and enforces swelling of the hydrogel. The two snapshots on the right in Figure 2 illustrate this competition, having the network nodes act as condensation nuclei where, due to the larger monomer density there, onsets of globular collapse remain. Between the intermediate f ¼ in the upper right and the maximum f ¼ 1.0 (bottom right), these become smaller as more counterions enter the system and swell the hydrogel. Without the network connectivity, no such swelling mechanisms exists, allowing the counterions to obtain their entropic freedom without stretching the chains into energetically and entropically unfavorable Figure 1. Structure diagrams of the equilibrium swelling conformations of charged hydrogels (left, from ref. [34] ) and of single polymer chains (right, from ref. [53] ) in a poor solvent for varying charge fractions f and Bjerrum lengths B. 725

6 B. A. F. Mann, K. Kremer, O. Lenz, C. Holm Figure 2. Snapshots of C-series equilibrium conformations at Bjerrum length B ¼ 0s for charge fraction f ¼ (top left), f ¼ (top right), and f ¼ 1.0 (bottom right); its position in the parameter space is illustrated by the ellipsoid. positions. Therefore, the single PE structure diagram shows globular states for vanishing electrostatics at all values of f. On the other hand, neutral networks (i.e., without additional counterions ) would collapse despite their topology, since no counteracting force prevents them to do so. Therefore, it is the unique combination of balancing counterion and network chain properties which determines the hydrogel s behavior, making it distinct from single PEs and neutral networks. two pearls at the respective ends of the single chain. In the parameter range that was used, each pearl is estimated to contain g P ¼ 78 4 monomers. Since in the networks, the monomers at the nodes are actually shared with the neighboring chains, therefore, only around monomers have to be contributed by each one, such that an intermediate pearl and two adjacent strings of 40 monomers could exist. Figure 4 shows this rough estimate to be correct, as we find the expected number of pearls in the snapshots. Comparing the case of f ¼ and B ¼ 0:25s in the lower left to the other two examples, the trends discussed so far can be seen again: For a larger charge fraction of f ¼ 0.5 but the same strength in the electrostatics, equilibrium is reached at a larger simulation box volume due to the additional osmotic pressure caused by the the higher number of counterions; as a consequence, the network chains in the snapshot on the lower right are elongated further, the pearls become smaller since the stronger strain on the strands pull some of the monomers out of the globules. Increasing electrostatics to B ¼ 0:75s instead, while keeping the charge fraction constant at f ¼ , shifts the equilibrium toward smaller volumes of Pearl-Necklace Regime After the pearl-necklace regime was identified in single PE chains, theorists started to suspect its existence for charged hydrogels in poor solvents as well. The inset of Figure 3 shows such a prediction, taken from ref. [24], and relates it to an actual snapshot from a simulation of the E-series (see Table 1) with relatively long chains (N m ¼ 479) to emphasize that formation of more than one pearl is actually possible and is only dependent on the available chain length. From ref. [53] it is known that for a chain length of N m ¼ 200, all monomers can be expected to be in either of Figure 3. Snapshot of the equilibrium conformation N m ¼ 479 monomers per chain segment, charge fraction f ¼ , and Bjerrum length B ¼ 1:5s, which shows the predicted pearlnecklace structure envisioned in the theoretical sketch in the inset (taken from ref. [24] ). 726

7 Hydrogels in Poor Solvents:... Figure 4. Snapshots of C-series equilibrium conformations at Bjerrum length B ¼ 0:25s and charge fraction f ¼ (bottom left), B ¼ 0:25s and f ¼ 0.5 (bottom right), B ¼ 0:75s and f ¼ (top left); their position in the parameter space are illustrated by the ellipsoid; the structure diagram in the upper right is discussed in the text. the box, because the counterions are pulled closer to the chains, as one realizes upon visual inspection of the snapshot in the upper left of Figure 4, where the counterions are more commonly found close to the monomer beads; consequently, the chain extension decreases and more monomers may enter the pearls, favoring the ones at the nodes due to the locally increased concentration there. Transitions Between the Regimes The transitions between the five regimes are smooth and continuous, in good agreement with the results from single PEs, but contrary to what scaling theories [43,44] have predicted there. According to those scenarios, for example, a collapse of the pearl-necklace structures in a first-order transition should occur with the onset of counterion condensation. Particularly the first four regimes cannot be precisely separated as they smoothly cross-over from one to the other: starting out as a collapsed conformation, an increase in the amount of counterions (i.e., a larger f) and higher electrostatics (i.e., larger B ) slowly pushes the monomers apart and dissolves the globules around the network nodes, which separate once the required volume increase invokes the Rayleigh instability. The emerging structure has features of the dumbbell-like PEs, except that the dumbbell ends are joined and shared with the three neighboring network strands. Increasing f and B further exerts more pressure on the hydrogel and enforces larger box volumes, which in turn causes elongation of the chains and reduces the number of monomers in the globules at the nodes. As this is an external constraint, sufficiently long strands can form additional pearls, and the pearl-necklace regime is reached. In this regime, it is the delicate balance between electrostatic repulsion and short-range attraction which dominates the shape of the network strands, that is well known from the investigation of single PEs. [53] In hydrogels, however, the osmotic pressure of the counterions comes in as an additional factor. Even stronger electrostatics shifts the balance in favor of the Coulombic repulsion, the pearls dissolve, and as the short-range attraction becomes less important the network approaches the behavior of strongly charged hydrogels in good solvent. From this description, the continuous nature of the transitions between these regimes becomes apparent. Supplemented by fluctuations and anisotropic effects in the high density limit, no sudden phase transition can be expected. It nevertheless makes sense to distinguish regimes with similar conformations as good as possible, because they are characterized by intrinsically different physical mechanisms that originate in the dominating effects of either shortrange attraction, entropic degrees of freedom, electrostatic repulsion, surface tension, or network topology. Sausage Regime When the strength of the Coulomb coupling continues to grow, this gives rise to effects which do not so much depend on the charge fraction f, but instead on the Manning parameter j M ¼ Bf b, a characteristic combination of system properties and indicator for the onset of counterion condensation. Stronger electrostatics pulls the ions toward the chains, decreasing the effective osmotic pressure that swells the network against elasticity and short-range attraction. This allows more monomers to grow the globules, and it also screens the electrostatic repulsion separating the pearls. Consequently, the intermediate 727

8 B. A. F. Mann, K. Kremer, O. Lenz, C. Holm strings become shorter until neighboring clusters of monomers merge and render the chains to appear more cigar-like or sausage -shaped. Figure 5 displays three example conformations which do no longer have distinct pearls but rather exhibit massive monomeric structures along the network backbone. These are thicker for lower charge fractions, because the larger ratio of monomers to counterions makes the (attractive) contacts between beads more likely, while their repulsive electrostatic interaction is screened by the (almost) completely condensed counterions. In the case of f ¼ 0.25 and B ¼ 3:0s (lower left of Figure 5), the Manning parameter j M 7.5 indicates a very strong Coulombic coupling, and the counterions are indeed localized on the chains; the snapshot shows thick cylindrical or sausage - like clustering of the monomers, where the radius naturally varies with the distance from the nodes with their locally increased monomeric density. At even higher electrostatics with f ¼ , B ¼ 5:0s, and j M 16.7, more counterions condense onto the chain monomers, which prevents part of their mutual contacts and increases the node-node separation while shrinking the sausage size. The uniformity of such structures is disrupted further for larger amounts of counterions firmly bound to the chains due to the Coulombic coupling, as in the case of the upper right snapshot with f ¼ 0.5 and B ¼ 6:0s, where the strand in the center of the picture is still resembling the sausage -like clusters from the other examples, while some of the outer chains, for example, the one in the top left of the simulation box, almost seem to display another pearl-necklace conformation, clearly having a locally varying monomer density profile. Collapsed Regime If the Bjerrum length B is increased further, the counterions will be closer distributed around the polymer charges and screen most of the remaining repulsion, and induce also attractive interaction due to Wigner-crystal like arrangements, [44,47] helping the remaining elastic and the shortrange poor-solvent attraction to collapse the network. As Figure 5 inferred, lower charge fractions will aid in this process because monomer/monomer interactions are already stronger there, initiating the transition earlier as compared to systems with many counterions. Nevertheless, collapsed structures in the strong electrostatics limit are fundamentally different from the ones for vanishing electrostatics. At strong electrostatic coupling, one finds an almost dipolar structure, where pairs of ions and monomers are maximally mixed. In contrast, when the electrostatic coupling vanishes, there is a clear phase separation between the gas of counterions and the monomers. In that case the counterions try to maximize their distance to other particles but are nonetheless trapped within the network topology, while the monomers try to minimize their mutual separation due to the short-range attraction between them. There is a smooth cross-over in between, where the phase separation slowly gives rise to the total mixing of the two particle types for increasing Coulomb coupling; this, however, does not change the general trend of being collapsed in the swelling equilibrium. Comparison to Single-Polyelectrolyte Structures Figure 5. Snapshots of C-series equilibrium conformations at Bjerrum length B ¼ 3:0s and charge fraction f ¼ 0.25 (bottom left), B ¼ 5:0s and f ¼ (top left), and B ¼ 6:0s and f ¼ 0.5 (top right); their position in the parameter space is illustrated by the ellipsoid. Compared to the structure diagram of single PEs (right subfigure of Figure 1) we notice that the diagram for PE networks contains corresponding regimes, albeit at shifted positions in the parameter space, despite the fact that we used the very same set of parameters for our simula- 728

9 Hydrogels in Poor Solvents:... tions. The regime where the networks strands form pearlnecklace-like structures extends down to the case of vanishing electrostatics ( B ¼ 0s). For not too low charge fractions, we do not find globular conformations due to the network topology trapping the counterions. Once electrostatics becomes stronger and chains are sufficiently long, the charged hydrogels behave similarly to the single PEs, including the collapse for very strong Coulombic coupling, where the remaining dependence on the network topology becomes negligible. To determine the behavior at low charge fractions, we have studied fractions f ¼ 0.25 and 0.125, which were not considered in the single chain study. From these simulations, we can conclude that in the low charge fraction limit the hydrogel network is always collapsed, as the low amount of counterions is insufficient to generate a strong enough osmotic pressure to push the attractive monomers apart. One would consequently expect single PEs with f ¼ to also form globules for all values of B, with an increasing amount of counterion condensation and subsequent mixing between ions and monomers for growing electrostatics. Geometrical Properties Now that we have identified and discussed the respective regimes of equivalent conformations in the parameter space of charge fraction and Bjerrum length, we can try to find hints to that behavior in plots of the geometrical properties at swelling equilibrium. Node Separation R E The first observable that we have studied is the node separation R E plotted in Figure 6, that corresponds to the end-to-end distance from single polymer chains (hence the symbol R E ). While the single PEs from ref. [53] form compact globules at Bjerrum length B ¼ 0s with a size independent of the charge fraction, and returning to only slightly larger conformations in the high electrostatics limit, the networks have clearly f-dependent values of R E at vanishing electrostatics, as the osmotic pressure of the trapped counterions prevents the total collapse. This is corroborated by the fact that higher densities are possible for B! 0 if the charge fraction drops. In the other limit, the networks are following the example of the single PEs, as the counterion condensation-induced neutralization removes the constraint of their osmotic pressure and allows the short-range attraction to dominate. Note that our plot only extends up to B ¼ 6s as higher electrostatics were so dense that no equilibration was possible. Compared to the single chains, the cases of f ¼ 0.5 and had reached similar sizes to their respective counterparts, resulting in a much higher slope in the descent and justifying our statement. Although the general trend in R E ð B Þ is the same, with the chain extension exhibiting a broad maximum for f ¼ 1.0 and 0.5, the peak position is shifted from around B ¼ 2s toward B ¼ 1s. Furthermore, it is much larger (R E s for f ¼ 1.0) due to the presence of the counterions whose osmotic pressure, negligible in the single chain case, exerts an additional swelling effect (increasing R E ) before being decreased by the removal of condensed ions (shifting B ). In case of the lower charge fractions, the chain extension decreases monotonically as the expansion due to the gas-like ions is removed by electrostatically binding them to the chains. Characteristic Ratios The characteristic ratios r ¼hR 2 E i=hr2 G i and a ¼hR2 G i1=2 =hr H i (see, e.g., ref. [54] ) as a function of the Bjerrum length B, displayed in Figure 7 and 8, reflect this behavior of the nodenode separation. They do not start off at a common value for B ¼ 0 and there is no sharp increase in a for small Bjerrum lengths, although the subsequent monotonic decay can be observed. The values found are also commonly higher than in the case of single PEs. Most notably, the ratio between end-to-end distance and radius of gyration r for, for example, f ¼ 1.0 and 0.5 reaches values of up to r 12, which indicates a strongly elongated conformation, which is significantly larger than for the isolated chains, due to the expanding force of the counterions. Although the very high intermediate values of the rod-like ratio for B 91s might seem surprising at first, this merely indicates an increased mass (i.e., monomer) agglomeration close to the centerof-mass of the chain. Recalling that for the chosen parameters and chain lengths we find one pearl per network strand (plus the shared ones at the nodes) which was located in the middle of that chain due to the electrostatic pearl-pearl repulsion, our observation fits Figure 6. Dependence of the end-to-end distance R E on the Bjerrum length B for the C-series (see Table 1). Compare to the single chain results in (ref. [53], Figure 10). (Charge-fractiondependent coloring is from red for f ¼ 1.0 to purple for f ¼ ) 729

10 B. A. F. Mann, K. Kremer, O. Lenz, C. Holm Figure 7. Dependence of the characteristic ratio r ¼hR 2 E i=hr2 G i on the Bjerrum length B for the C-series. Compare to the singlechain results in (ref. [53], Figure 21). network chains crosslinked there, the analysis of the radius of gyration R G for each of them individually only detects the mass agglomeration at the chain s center which is four times higher than the (symmetric) increase at its ends. For single PEs similar findings cannot be expected since it requires the biggest pearl to be localized close to the middle of the polymer, which is precisely the opposite of what is usually found as they exhibit an orb-like size profile for the pearls, that is, having their diameter grow with the distance from the chain s center. It would also be futile to try to generalize this observation toward a pearl-recognition criterion, because it is an artifact of the network topology which causes the pearls at the nodes to be shared among all neighboring chains, such that the individual network strand with one centered pearl only sees slightly increased local monomer concentrations at its end, without knowing that the latter will eventually add up to additional pearls. Longer chains with more than one individual pearl along their backbone will decrease the peak in r further, until it can no longer be distinguished if the ratio refers to extended conformations as in the case of good solvents or describes a pearl-necklace structure. Chain Length Dependence Figure 8. Dependence of the characteristic ratio a ¼hR 2 G i1=2 =hr H i on the Bjerrum length B for the C-series. Compare to the singlechain results in (ref. [53], Figure 22). perfectly as r > 12 now indicates recognition of the locally increased monomer density close to the chain s center. Figure 9 visualizes this, emphasizing that since the pearls at the nodes are composed of contributions from all four The choice of a sufficiently long chain length plays an important role in the poor-solvent regime, where the dominant effects (like the pearl-necklace regime) require a certain minimum amount of monomers to become visible. Therefore, we also investigated the impact of N m on the previously discussed results. Looking at the data of the A- series, we do not find dramatic changes once a minimum chain length is exceeded. As Figure 10 illustrates, the equilibrium end-to-end distance for the chosen parameter set of f ¼ and B ¼ 1:5s, that is, within the pearlnecklace regime, simply increases linearly with N m, in agreement with simple scaling predictions [24] expecting the chain extension to behave as R E ¼ bn m ð B f 2 =bþ 1=2 t 1=2. The characteristic ratio r ¼hR 2 E i=hr2 Gi is almost constant for N m 287 at values exceeding the rod-like value r ¼ 12 which indicates the anisotropic mass distribution along the chain. The increase in a ¼hR 2 G i1=2 =hr H i is more pronounced, as the additional pearls allowed on the larger chains become (indirectly) noticeable. This is also the only effect on the snapshots upon visual inspection, with more and more pearls emerging on the network chains. Dependence on the Charge Fraction Figure 9. Visualization of a ratio r > 12 of our network chains occurring for one central pearl, since the globules at the nodes are composed of contributions from all four strands crosslinked there. If one varies the charge fraction instead, choosing very long chains with N m ¼ 479 monomers at the same B ¼ 1:5s, one can observe the transition from the collapsed regime via the pearl-necklace regime toward stretched conformations, as the plots in Figure 11 show. While the node-node separation R E ( f) grows one order of magnitude, the characteristic ratio r increases from the random walk-like r ¼ 6 up to around 730

11 Hydrogels in Poor Solvents:... Figure 10. The dependence of node/node separation R E (left) and the characteristic ratios r (center), and a (right) on the amount of monomers per chain for the A-series ( f ¼ and B ¼ 1:5s). Figure 11. The dependence of node/node separation R E (left) and the characteristic ratios r (center) and a (right) on the charge fraction f for the E-series (N m ¼ 479 and B ¼ 1:5s). r ¼ 12 of a rod-like, strongly elongated conformation. Similar behavior is observed for a which mirrors the trends found for the shorter chains in Figure 8. It is, therefore, safe to assume that the chains in the C- series were sufficiently long to exhibit the full range of structures, with longer strands only marginally improving the results (e.g., by having more pearls to average over, etc.) while at the same time dramatically increasing the required computational effort. Consequently, future studies might also initially focus on the parameter regime we covered here, as it was shown to provide access to the complex behavior of PE networks in poor solvents. How Pearls Might be Experimentally Confirmed For poor-solvent PEs there has been always the question of the experimental verification of the pearl-necklace conformations. In our earlier articles, [52,53] we have argued that the peak in the structure factor that is related to the pearls might be to small to be detectable via scattering techniques that are sensitive to the polymer backbone, since the almost unavoidable polydispersity in the PE chains smears out the peak. As we have shown in ref., [34] this is basically also true for hydrogel networks. However, for certain monodisperse PE systems like PE stars or rings, there might be a way to obtain a signal. We learned of some experiments by Combet et al. [61] who examined a solution of CsPSS with the help of small-angle X-ray scattering (SASX). Replacing the sodium counterions by cesium counterions, the intensity of the scattering signal is increased and allows to analyze spatial structures in the counterion density instead of in the monomer density. As Figure 12 shows, when the signal is normalized by the concentration c, all data falls onto a single master curve for sufficiently large q. Furthermore, it reveals that the counterions have a common second maximum (marked by the arrow) in their total scattering function which indicates the existence of a dominant (short-range!) length scale there, at around q 0.2 Å. A sketch of the a spatial structure that is consistent with the data is given on the right of Figure 12. Claiming that those counterions which are no longer unperturbed by the electrostatic attraction of the chains condense or get very close to the PEs, it is argued that they form a shell around the PE s curled shape induced by the hydrophobic interactions with the solvent. Then, the second maximum found is indicating both the regularity of such a structure and its typical spatial extension j ¼ 2p=q 31 A. Such an interpretation would be compatible with a pearl-necklace conformation, where the counterions would be localized on the surface of the pearls. Although hydrophobic aggregates are easily detected by SAXS, non-sulfonated sequences could also form small domains (quenched hydrophobicity) which would be detected in a similar 731

12 B. A. F. Mann, K. Kremer, O. Lenz, C. Holm Figure 12. Scattering intensity I(q) obtained from SAXS measurements of CsPSS PEs. [61] Normalized by the concentration c all points fall onto the same master curve for large enough q, exhibiting a common maximum for short spatial distances (indicated by the arrow). On the right it is sketched how Rawiso et al. [61] attribute this finding to the PE s geometrical structure, where the darker core represents the polymeric backbone, the lighter shell the condensed counterions; the circles stand for the spherical hydrophobic aggregates (pearls), the elongated parts for the cylindrical strings. manner; careful preparation of the setup PEs can however minimize such anomalies, leaving only globular chain segments to be associated with the findings. Consequently, this could lead to a direct experimental proof of the pearl-necklace structures on PE chains in poor solvents, and should, therefore, be observable in our simulations as well, if the arguments are expected to hold at all. Therefore, we performed simulations within the pearl-necklace regime, and analyzed their total structure factor NpNm 2 X Sð~qÞ ¼ N p N m i¼1 XN pn m e i~qð~r i ~r j Þ j¼1 (10) spherically averaged within the periodic simulation box to [62] j~q 0 j¼qþ X 1 2 Dq SðqÞ ¼@ Sð~q 0 j~q 0 j¼q 1 2 Dq X j~q 0 j¼qþ 1 2 Dq j~q 0 j¼q 1 2 Dq 1 1 A 1 (11) and averaged over all 27 intermediate snapshots. To be able to estimate the impact of finite size effects, we also analyzed the structure of a significantly larger system, which had a diamond-like topology with N nodes ¼ 64 nodes and N p ¼ 128 chains. The result, shown in Figure 13, is very Figure 13. Total structure factor of the counterions for typical pearl-necklace conformations from the C-series (with N m ¼ 200 and f ¼ ). The arrow marks the second maximum which corresponds to the pearls in the structure. encouraging as it clearly shows a second maximum at large q, even if the still finite size of the simulation system prevents us from reaching a high enough resolution to cover the entire experimental range and to clearly desribe the first maximum at small values of q. With a q ranging from 0.8 to 1.1s 1, the second maximum corresponds to spatial structures of a size between 7.8 and 5.7s. This is in nice agreement with the outcome of the cluster algorithm which estimated n P ¼ 80;...; 140 monomers to be in one pearl, putting their extension at around 3:9s;...; 4:8s (close packing assumed), and, therefore, leads to an average characteristic length scale including the condensed shell of counterions on their surface of 5:9s;...; 7:3s. We finally conclude in this section that the via SAXS measurements of the counterion structure factor is seems to be possible to infer the size of the pearl-necklace structures predicted in theories and seen in simulations for some time. From our data we can clearly confirm the assumed connection between the existence of counterion shells around the pearls, and the occurrence of a second maximum in the scattering function of the counterions. Conclusion and Outlook PEs in poor solvents are known to exhibit within a certain regime so-called pearl-necklace conformations. In this work, we have been investigating the behavior of PE networks under poor-solvent conditions and compared them to the behavior of single PE chains under poor-solvent conditions. After reviewing previous results, we found similar equilibrium swelling conformations in the charged hydrogels under poor-solvent conditions, that is, collapsed structures for very low charge fractions or very 732

13 Hydrogels in Poor Solvents:... high electrostatics, local globules of monomers around the nodes, and (later on) on the chains forming pearl necklaces for increasing f and B, strongly stretched conformations for large f and intermediate B, and sausage -like structures for not too low charge fractions and large Bjerrum lengths. We ordered the conformations in a structure diagram using the charge fraction and the Coulomb coupling strength as parameters. Particularly, in the case of vanishing coupling strength, the gel did not collapse into a dense globule, since the entropy of the confined counterions prevented them to collapse completely. The regime in which pearl necklaces were observed, was smaller than theories anticipated, which is consistent with the single chain results where even in dilute solution the delicate interplay between the counterion distribution and the chain conformation for strongly charged chains limited the range of applicability for scaling predictions. We found the point of maximum extension in the network strands to be larger, but shifted toward smaller Bjerrum lengths; the former was attributed to the additional osmotic pressure of the ions, the latter to the onset of condensation effects which effectively remove particles from the ideal gas-like distribution. The same reason also rendered the network chains to be more elongated, even surpassing rod-like structures for parameters in the pearl-necklace regime. This phenomenon was extensively discussed as the cross-over between the different structures, which was shown to be smooth and continuous. Experimentally accessible observables such as characteristics ratios and form factors were also determined; they may aid the analysis of experimental data. However, we again found that the signatures of, for example, pearl-necklace structures are not very pronounced, although we are dealing with highly idealized and monodispersed model networks. However, a way out maybe to look at the structure factor of the condensed counterions, as is possible in SAXS measurements. Here, a signature of pearls could be observed, as a comparison with preliminary experimental data from Rawiso and co-workers [61] showed. To make the model more realistic in comparison to experimental situations, we would have to introduce random crosslinks, a polydisperse polymer distribution, dangling ends, and possibly added salt. This might lead to even more severe difficulties in distinguishing the various conformations of the network strands. Acknowledgements: This article is dedicated to B. A. F. Mann, upon whose PhD work the article is largely based, but who unfortunately passed away before completion of this article. We acknowledge helpful discussions with G. Fytas, A. Grosberg, W. Knoll, and U. Jonas, and thank P. Kosovan for a critical reading. We are especially grateful to F. Boué, J. Combêt, and M. Rawiso for discussions about the pearl-necklace structures, and for allowing us to use their data in Figure 11 prior to publication. Funds for this research were gratefully provided by the Deutsche Forschungsgemeinschaft (DFG) as part of the SPP 1259 Intelligente Hydrogele and under grant Ho 1108/13-1, and the SFB 625. We thank the VMD team [63] for their excellent visualization software. Received: April 20, 2011; Revised: June 8, 2011; Published online: August 19, 2011; DOI: /mats Keywords: hydrogels; molecular dynamics simulations; polyelectrolyte networks; poor solvents [1] K. S. Kazanskii, S. A. Dubrovskii, Adv. Polym. Sci. 1992, 104, 97. [2] M. J. Comstock, Superabsorbent Polymers (Eds: F. L. Buchholz, N. A. Peppas), Volume, 573 in ACS Symposium Series, American Chemical Society, Washington, DC 1994, American Chemical Society. [3] G. M. Eichenbaum, P. F. Kiser, A. V. Dobrynin, S. A. Simon, D. Needham, Macromolecules 1999, 32, [4] N. A. Peppas, P. Bures, W. Leobandung, H. Ichikawa, Biopharmaceutics, 2000, 50, 27. [5] T. Radeva, Physical Chemistry of Polyelectrolytes, Volume, 99 in Surfactant Science Series, Marcel Dekker Inc. New York, [6] W. E. Rudzinski, A. M. Dave, U. H. Vaishnav, S. Kumbar, A. R. Kulkami, T. M. Aminabhavi, Des. Monomer Polym. 2002, 5, 39. [7] W. Kuhn, B. Hargitay, A. Katchalsky, H. Eisenberg, Nature 1950, 165, 514. [8] T. Tanaka, Phys. Rev. Lett. 1978, 40, 820. [9] Responsive Gels: Volume Transitions I & II (Ed: K. Dušek), Volumes 109/110, in Advances in Polymer Science, Springer-Verlag, Heidelberg [10] A. Katchalsky, I. Michaeli, J. Polym. Sci. 1955, 15, 69. [11] I. Michaeli, A. Katchalsky, J. Polym. Sci. 1957, 23, 683. [12] P. Hansson, Langmuir 1998, 14, [13] H. S. Ashbaugh, L. Piculell, B. Lindman, Langmuir 2000, 16, [14] J. Sjöström, L. Piculell, Colloids Surf., A 2001, , 429. [15] P. Hansson, S. Schneider, B. Lindman, Prog. Colloid Polym. Sci. 2000, 115, 342. [16] P. Hansson, S. Schneider, B. Lindman, J. Phys. Chem. B 2002, 106, [17] A. Fernandez-Nieves, A. Fernandez-Barbero, B. Vincent, F. J. de las Nieves, Macromolecules 2000, 33, [18] A. Fernandez-Nieves, A. Fernandez-Barbero, B. Vincent, F. J. de las Nieves, Prog. Colloid Polym. Sci. 2000, 115, 134. [19] A. R. Khokhlov, S. G. Starodubtzev, V. V. Vasilevskaya, Conformational Transitions in Polymer Gels: Theory and Experiment, in Responsive Gels: Volume Transitions I (Ed: K. Dušek), Volume 109 in Advances in Polymer Science, Springer-Verlag, Heidelberg 1993, p [20] C. Mengel, W. H. Meyer, G. Wegner, Macromol. Chem. Phys. 2001, 202, [21] P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY [22] J.-L. Barrat, J.-F. Joanny, P. Pincus, J. Phys. II 1992, 2, [23] K. B. Zeldovich, E. E. Dormidontova, A. R. Khokhlov, T. A. Vilgis, J. Phys. II, 1997, 7, 627. [24] T. A. Vilgis, A. Johner, J.-F. Joanny, Eur. Phys. J. E, 2000, 3,