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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 [

Theoretical studies of the macroscopic properties of polyelectrolyte gels have a long history; a summary can be found in the review article by Khokhlov

Comparably little attention has been paid to finite-size cross-linked polyelectrolytes [

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 [

The rest of the paper is organized as follows. In

A microgel particle is comprised of a regular network of polymers, which are internally connected by _{c}_{d}_{d}_{c}_{d}_{d}_{c}_{d}

Topological structure of a microgel particle with _{c}_{m}_{D}

An individual polymer is modeled as a linear chain of _{m}_{i}_{s}_{ij}_{i}_{j}_{c}^{12} − (_{c}^{6}) and _{c}_{c}^{1/6}_{c}

Charge-charge interactions between monomers are captured in an effective manner by the Debye–Hückel potential:
_{B} is the Bjerrum length, _{D} = (4_{B}^{−1/2} is the Debye length with the ion concentration _{DH} = 5.3_{D}_{B}

In order to perform isothermal simulations, we couple the microgel monomers with the Brownian multiparticle collision dynamics method (B-MPC) [_{B}T_{i} the monomer velocity, is rotated around a randomly oriented axis by a fixed angle,

We consider microgels comprised of polymers with _{m}_{c}_{d}_{d}_{c}_{d}_{c}

We employ _{B}T_{B}T_{s}^{3}_{B}T^{2}. The collision time is Δ^{4} collision steps, which corresponds to 10^{6} molecular dynamics simulation steps, after reaching a stationary state in every simulation.

As a reference, we consider the polymer conformational properties of a microgel under good solvent conditions and _{D}_{c}

The system with _{c}_{m}^{v}_{c}

Further insight into the microgel structure is gained by the spherically averaged static structure factor [^{−1/0.62} with increasing _{m}^{0.62}. Hence, the polymer conformations are determined by thermal fluctuations, intramolecular and intermolecular interactions and the cross-links.

Dependence of the average root mean square radius of gyration _{m}_{c}_{D}^{0.59} and the dashed lineto ^{0.62}.

Spherically averaged static structure factor _{c}_{m}^{−1/0.62}.

_{m}_{c}_{D}

With increasing _{D}_{g}_{g}_{g}_{B}T_{D}

Radial monomer distribution functions _{m}_{c}_{B}T_{D}_{g}

The gel compaction is also reflected in the static structure factor displayed in ^{−4.1} for _{B}T^{−4} for a system with a sharp interface. In contrast, for _{B}T^{−1/0.6} in the range of 0.6 < _{m}^{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 _{B}T

Spherically averaged static structure factors _{p}_{c}_{B}T^{−4.1} and the straight dashed line to ^{−1/0.6}.

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,

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, _{D}_{D}_{DH}_{D}_{B}_{m}

The dependence of the radius of gyration, _{g}_{D}_{B}_{m}_{c}

_{D}_{D}_{D}/l_{D}/l^{−1/0.67} and ^{−1.05}, respectively, in the range of 0.05 ≲

Spherically averaged static structure factors _{c}_{m/}_{B}/l_{D}/l^{−1.05} and the dashed line to ^{−1/0.67}.

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:
_{m}_{p}_{p}_{p}_{p}_{p}_{e}_{e}_{D}_{e}_{p}_{p}_{p}_{p}_{g}/L_{p}^{0.6},

Dependence of the ratio _{p}_{p}_{c}_{p}_{B}/l_{p}^{0.6} and the red line to _{p}

_{D}/l_{D}_{D}_{D}/l_{g}_{D}/l_{B}T_{D}

Radial monomer distribution functions for various Debye lengths, _{D}/l_{c}_{m}_{B}/l_{B}T

We find a rather high monomer concentration in the vicinity of the center-of-mass of the microgel at larger _{D}

To gain insight into possible inhomogeneous polymer conformations, we consider the radial dependence of the polymer radius of gyration, _{cm}), in a microgel. As shown in _{p}_{p}_{p}_{D}/l_{cm}/_{g}_{p}_{D}/l_{cm}/_{g}

Dependence of the polymer radius of gyration on its radial center-of-mass position, _{cm}, for _{p}_{p}_{c}_{m}_{cm} are defined by the condition

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

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”.

The authors declare no conflict of interest.