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Polymers 2014, 6(5), 1602-1617; doi:10.3390/polym6051602
Abstract: The structural properties of model microgel particles are investigated by molecular dynamics simulations applying a coarse-grained model. A microgel is comprised of a regular network of polymers internally connected by tetra-functional cross-links and with dangling ends at its surface. The self-avoiding polymers are modeled as bead-spring linear chains. Electrostatic interactions are taken into account by the Debye–Hückel potential. The microgels exhibit a quite uniform density under bad solvent conditions with a rather sharp surface. With increasing Debye length, structural inhomogeneities appear, their surface becomes fuzzy and, at very large Debye lengths, well defined again. Similarly, the polymer conformations change from a self-avoiding walk to a rod-like behavior. Thereby, the average polymer radius of gyration follows a scaling curve in terms of polymer length and persistence length, with an asymptotic rod-like behavior for swollen microgels and self-avoiding walk behavior for weakly swollen gel particles
Microgels are cross-linked polymers, typically polyelectrolytes, with a network structure. They are able to undergo reversible volume phase-transitions in response to environmental stimuli, such as pH, temperature, the ionic strength of the surrounding medium, the quality of solvent and the action of the external electromagnetic field [1,2,3,4]. This renders them potential candidates for a broad-range of applications in drug delivery, sensing, the fabrication of photonic crystals, template-based synthesis of inorganic nanoparticles and separation and purification technologies [5,6,7,8,9,10,11,12,13].
Theoretical studies of the macroscopic properties of polyelectrolyte gels have a long history; a summary can be found in the review article by Khokhlov et al. . Computer simulations have mainly been performed during the last decade. Such simulations typically consider defect-free microgels applying periodic boundary conditions, i.e., only the bulk properties of the gel are considered [15,16,17,18,19,20,21,22,23,24,25,26,27]. Monte Carlo [15,16,18,20,22,25,26,27] or molecular dynamics [17,19,21,23,24] simulations have been performed using coarse-grained polymer models with an implicit solvent, but explicit counterions. A major aspect of these studies is the swelling behavior of the polyelectrolyte networks. These simulations provide valuable insight into the origin of the driving force responsible for gel swelling. It is generally accepted that the entropy of the free counterions is responsible for swelling, whereas Coulomb repulsion between the charged polymer chains shows no explicit effect [16,21]. A detailed study of the latter aspect shows that this is due to the cancellation of various pressure contributions [21,23].
Comparably little attention has been paid to finite-size cross-linked polyelectrolytes [28,29,30]. A markedly different behavior of the counterion distribution has been observed. Now, no longer are all counterions confined inside the microgel, but rather, a large fraction is distributed outside, around the colloidal gel particle [28,31]. This is caused by the permeability of microgels. Another important aspect is the presence of the surface. We expect that, when the surface effect is non-negligible compared to the bulk effect, this effect will cause inhomogeneities in the structure of microgels.
To characterize the structural properties of microgels, we perform large-scale computer simulations, combining molecular dynamics simulations for the polymers with the Brownian multiparticle collision dynamics (B-MPC) approach . Electrostatic interactions are taken into account by the Debye–Hückel potential. Hence, we consider counterions only implicitly. The study is intended as a reference to discriminate between effects caused by explicit charges and counterions—such as counterion condensation [33,34]—and effects due to repulsive interactions in finite-sized microgel particles. The simulations reveal a significant dependence of the microgel structure on the monomer interactions. Under bad solvent conditions, we find highly compact particles with a rather constant radial monomer distribution. Switching to good solvent conditions implies a swelling of a microgel particle. An additional increase of the Debye length, i.e., an increasing electrostatic repulsion, leads to further swelling, and the monomer distribution becomes inhomogeneous. Thereby, the polymer conformational properties change from self-avoiding walk to rod-like behavior. At the same time, the surface properties of the microgels change from a rather sharp interface to a more fuzzy one for neutral, non-compact gels, back to a rather well-defined one at large Debye lengths.
The rest of the paper is organized as follows. In Section 2, the microgel model and the simulation approach are outlined. The results are presented in Section 3, and Section 4 summarizes our findings.
A microgel particle is comprised of a regular network of polymers, which are internally connected by Nc tetra-functional cross-links and with Nd dangling ends at the surface, due to its finite size, as illustrated in Figure 1. The ratio Nd/(Nc − Nd) indicates the relative importance of surface effects . The limit Nd/(Nc − Nd) → 0 corresponds to the case of a macrogel, where surface effects are negligible.
An individual polymer is modeled as a linear chain of Nm monomers. A nearly constant bond length, l, is maintained by the harmonic potential:
Charge-charge interactions between monomers are captured in an effective manner by the Debye–Hückel potential:
2.2. Brownian Multiparticle Collision Dynamics
In order to perform isothermal simulations, we couple the microgel monomers with the Brownian multiparticle collision dynamics method (B-MPC) [32,36,37]. This is a non-hydrodynamic version of the multiparticle collision dynamics approach for fluid systems [36,38]. In B-MPC, a monomer performs a random collision with the fluid after a time increment, Δt, which is denoted as collision time. Thereby, we assign an effective fluid particle to every monomer. The velocity of the fluid particle of mass M (equal to the mass of a monomer) is taken from a Maxwell–Boltzmann distribution of variance MkBT. In the collision, the relative velocity of every monomer, with respect to the center-of-mass velocity:
We consider microgels comprised of polymers with Nm = 20 and 40 monomers of Nc = 147 and 729 cross-links, respectively. With the number of dangling ends Nd = 64 and 204, the ratios Nd/(Nc − Nd) are 0.39 and 0.77. In the largest system, the total number of monomers is N = 50,169. The topological structure of a microgel for Nc = 729 cross-links is illustrated in Figure 1.
We employ l as the unit of length, kBT as the unit of energy and M as the unit of mass. Thus, the unit of time is . The Lennard–Jones parameters are σ = 0.8l, ε/kBT = 0.5, 1.0 and 1.5 for a poor solvent and 1.0 for a good solvent. For the bonds, we set ks = 103kBT/l2. The collision time is Δt = 0.1τ, and we perform 20 molecular dynamics simulation steps between collisions. To achieve a reasonable statistical accuracy, we performed, at least, 5.0 × 104 collision steps, which corresponds to 106 molecular dynamics simulation steps, after reaching a stationary state in every simulation.
3.1. Microgel in Good Solvent (lD = 0)
As a reference, we consider the polymer conformational properties of a microgel under good solvent conditions and lD = 0. Figure 2 displays the chain-length dependence of the polymer radius of gyration for Nc = 0, 147 and 729 . Here, is the square root of the average over all polymers of the mean square radius of gyration, where the radius of gyration of a polymer ( ) itself is defined by:
The system with Nc = 0 corresponds to a non-cross-linked network. As expected, the radius of gyration exhibits the power-law dependence ~ (Nm − 1)v with the number of bonds. Thereby, the critical exponent, v, for the non-cross-linked polymers closely follows the theoretical prediction v ≈ 0.59 . Similarly, the values for the systems with Nc = 147 and 729 follow a power law; however, with the somewhat larger exponent v ≈ 0.62. Hence, cross-linking leads to swelling of the polymer chains.
Further insight into the microgel structure is gained by the spherically averaged static structure factor :
3.2. Microgel in Poor Solvent (lD = 0)
Figure 4 shows radial monomer distribution functions P(r) with respect to the microgel vcenter-of-mass for Nm = 40, Nc = 729 and various attraction strengths ε under poor solvent conditions.In addition, P(r) with lD = 0 under good solvent conditions is shown. The distribution function is normalized, such that:
With increasing ε, the size of a microgel particle shrinks and its density increases. As expected, microgels in a good solvent with lD = 0 are more swollen. This behavior is in qualitative agreement with experimental results [40,41,42]. The inset of Figure 4 indicates that P(r) decreases fast for r > Rg and vanishes at r ≳ 1.5Rg, where Rg is the average microgel radius of gyration. Thereby, the fuzziness of the surface decreases with increasing ε and the interface becomes rather sharper for large ε. The fuzziness for ε/kBT = 0.5 is comparable to that of a microgel in a good solvent with lD = 0.
The gel compaction is also reflected in the static structure factor displayed in Figure 5. The shift of the strongly dipping part of S(q) at ql ≈ 0.1 with increasing ε is reminiscent of the shrinkage of the microgel. In the range of 0.3 < ql < 2.0, S(q) is proportional to q−4.1 for ε/kBT = 1.0 and 1.5. This behavior is in close agreement with Porod’s law S(q) ∝ q−4 for a system with a sharp interface. In contrast, for ε/kBT = 0.5, S(q) is proportional to q−1/0.6 in the range of 0.6 < ql < 2.0,. Thus, scales as ∝ (Nm − 1)0.6 with polymer length, and the interface is less sharp. The scaling exponent, 0.6, is smaller than 0.62 obtained for microgels in a good solvent. Hence, the polymers are somewhat more compact for ε/kBT = 0.5 as compared to the bare good solvent system. The attractive interaction brings the polymers closer to the scaling behavior of free polymers.
3.3. Microgel with Debye–Hückel Interaction
We now elucidate the conformational properties of microgels, where the monomers interact via the Debye–Hückel potential Equation (3). We focus on good solvent conditions, i.e., only the repulsive part of the Lennard–Jones potential is taken into account.
3.3.1. Microgel Radius of Gyration
The Debye–Hückel potential Equation (3) is a short-range potential, since it decays very fast with the distance between two particles. The actual range depends on the Debye length, lD. For lD ≪ σ < l, UDH decays so fast that it is essentially negligible. This corresponds to the limit where the charge-charge interactions in a system are screened and only excluded volume interactions remain. In the opposite limit, when the Debye length is larger than the microgel particle, there is a strong repulsion between monomers, and we expect the microgel to be swollen. This corresponds to unscreened charge-charge interactions within the microgel. Hence, by increasing lD, we expect a conformational change of the gel particle from a good solvent state to a fully swollen state and essentially stretched polymers. This expectation is reflected in Figure 6, where the microgel radius of gyration is shown for various Bjerrum and Debye lengths. The onset of the increase in particle size depends on the Bjerrum length; the larger the Bjerrum length, the earlier the increase. Moreover, the slope of the increasing curve increases somewhat with increasing lB. Interestingly, the swelling behavior only weakly depends on the number of cross-links. Here, we conclude that the microgel behavior should be very similar to the behavior of a macroscopic gel, i.e., there is little effect due to the finite size of the microgel. However, we observe a polymer-length dependence of the swelling behavior. The microgels with Nm = 40 swell significantly faster with increasing Debye length compared to the shorter polymers. Hence, we speculate that very long polymers will show a very steep increase in the form of a first order phase transition.
3.3.2. Microgel Structure Factor
Figure 7 displays structure factors of microgels for various Debye lengths. The shift of the initial decay (small q values) with increasing lD toward smaller q values reflects the swelling of the microgel. The surface is rather sharp for all lD, as indicated by the steep decay and the appearing successive oscillations. More importantly, on the length scale of the polymers, we see a crossover in the chain conformations from a self-avoiding to a rod-like polymer. For lD/l = 0.4 and lD/l = 6.3, S(q) is proportional to q−1/0.67 and q−1.05, respectively, in the range of 0.05 ≲ ql ≲ 0.3. We obtain a somewhat larger scaling exponent v = 0.67 compared to the good solvent value due to monomer repulsion.
3.3.3. Polymer Size Scaling
From simulations, we can extract the average polymer radii of gyration for the various Debye lengths. Since the Debye–Hückel potential is of short-range order, we assume that the polymer radius of gyration obeys the scaling relation:
3.3.4. Radial Monomer Distribution
Figure 9 shows radial monomer distribution functions for various Debye lengths and a microgel in a poor solvent. Already for small Debye lengths, characteristic peaks appear due to the underlying diamond lattice structure of the cross-linked microgel. The undulations are weak for lD/l ≲ 1, but they become rather sharp for large lD. Note that the density distribution for a microgel under good solvent conditions with lD = 0 is close to the distribution for lD/l = 0.4. We do not necessarily expect such a detailed structure for a real microgel, since the polymer lengths are rather polydisperse and the cross-link density is inhomogeneous. In the range of r/Rg > 1, the density profiles decrease for lD/l > 3 as fast as P(r) of a microgel in a poor solvent with ε/kBT = 1.5. This indicates that the microgel has a sharp boundary, and peripheral polymers are smaller than internal polymers. For small lD, P(r) gradually decreases in the vicinity of the surface. The fuzzy boundary implies that the radius of gyration of a polymer at the surface is larger than that in the interior.
We find a rather high monomer concentration in the vicinity of the center-of-mass of the microgel at larger lD. This is a consequence of the on average isotropic distribution of monomers and a cross-link at the location of the center of mass.
3.3.5. Radial Polymer Conformation
To gain insight into possible inhomogeneous polymer conformations, we consider the radial dependence of the polymer radius of gyration, (rcm), in a microgel. As shown in Figure 10, we find a qualitative difference for /Lp > 2.5 and /Lp < 1. For /Lp > 2.5, i.e., lD/l ≲ 0.45, exceeds the mean value at large rcm/Rg. Hence, polymers near the surface are swollen as compared to internal polymers. We attribute the inhomogeneities to anisotropic intramolecular interactions by short-range repulsion. Internal polymers experience an almost isotropic interaction, whereas at the surface, the symmetry is broken and repulsion is stronger from inside to outside, which leads to a swelling. For /Lp < 1, i.e., lD/l ≳ 2, is smaller than the average value at large rcm/Rg. Thus, the outside polymers are somewhat more compact than the internal polymers. This is attributed to the inhomogeneous radial monomer distribution, as displayed in Figure 9. The larger interaction range of the Debye-Hückel potential for these parameters combined with a larger number of neighbors leads to the stronger swelling of internal polymers than those at the surface. Overall, the effect is small, but is a consequence of the finite size of a microgel and is thus not present in bulk systems.
4. Summary and Conclusions
We have performed large-scale molecular dynamics simulations to unravel the structural properties of finite-size polyelectrolyte microgel particles and the conformations of their comprising linear polymers. Counterions are treated implicitly via the Debye-Hückel potential. We find radially inhomogeneous polymer conformations due to the finite size of the microgel. Thereby, polymers at the surface are swollen compared to internal polymers for microgels where polymers exhibit self-avoiding walk scaling behavior. Oppositely, peripheral polymers are more compact for microgels where polymers behave rod-like. The difference in polymer conformations is also reflected in the radial monomer distribution function. Due to the swelling of surface polymers, the surface of a microgel becomes fuzzy, and the density decays gradually to zero. Oppositely, microgels display a rather sharp surface with a fast decaying monomer density in the case of more compact polymers. Although differences in the radial dependence of the polymer radii of gyration shown in Figure 10 are small, it is a consequence of the finite size of a microgel.
In general, the radial variations of the polymer properties are surprisingly small. We expected more severe inhomogeneities. The smooth variations suggest that the microgel properties are rather similar to those of the bulk properties of macrogels, at least as along as the Debye-Hückel description applies.
We considered a model microgel with an underlying diamond-lattice structure of the cross-links. This certainly is not the case of typical synthetic microgels. Although we do not expect severe differences between microgels with a more random distribution of polymer lengths and cross-link density for the adopted Debye-Hückle description, a more generalized description is desirable for a quantitative comparison with experimental results. The structure will definitely matter for the transport of particles in the microgel.
The presented results are intended as a reference for simulation studies of microgels where monomers interact by the bare Coulomb potential and where counterions are taken into account explicitly. The comparison will shed light on the influence of explicit ions on the structure of a microgel. Specifically, the inhomogeneous charge distribution of the spherical particle implies an inhomogeneous distribution of counterions (in the simplest case, it can be considered as a radially dependent Debye length), and correspondingly, interesting effects appear, which reach beyond those presented in the current article. Such studies are currently under way.
The authors gratefully acknowledge the computing time granted on the supercomputer JUROPA at Jülich Supercomputing Centre (JSC). We also acknowledge financial support from the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich SFB 985 “Functional Microgels and Microgel Systems”.
Conflicts of Interest
The authors declare no conflict of interest.
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