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Comparison of a Microgel Simulation to Poisson-Boltzmann Theory

We have investigated a single charged microgel in aqueous solution with a combined simulational model and Poisson-Boltzmann theory. In the simulations we use a coarse-grained charged bead-spring model in a dielectric continuum, with explicit counterions and full electrostatic interactions under periodic and non-periodic boundary conditions. The Poisson-Boltzmann model is that of a single charged colloid confined to a spherical cell where the counterions are allowed to enter the uniformly charged sphere. We compare the simulational results to those of the Poisson-Boltzmann solution and find good agreement, i.e., for the number of confined counterions within the gel. We then proceed to investigate the origin of the differences between the results these two models give, and performed a variety of simulations which were designed to test for the influence of charge correlations, excluded volume interactions, and thermal fluctuations in the strands of the gel. Our results support the applicability of the Poisson-Boltzman cell model to study ionic properties of small microgels under dilute conditions.


I. INTRODUCTION
Hydrogel research has constantly attracted a lot of attention, due to its capability of absorbing large quantities of water, resulting in the expansion of the material up to 1000 times its dry volume. Recent review articles on hydrogel research range from general reviews 1 to application specific reviews such as sensors 2 , surface applications 3 , drug delivery applications 4 and other biomedical applications 5,6,7 .
A hydrogel is a crosslinked network of water soluble polymers. The hydrophilicity of the free uncrosslinked chain allows it to attract water molecules and be soluble in water up to infinite dilution. The crosslinking, however, restricts the dilution to a maximal swelling of the gel. The combination of these two properties allows the hydrogel to attract water molecules and keep them within the network, resulting in an overall swelling of the entire gel. Hydrogels can be made of neutral or charged polymers called polyelectrolytes (the backbone of polyelectrolytes, however, need not be hydrophilic). When polyelectrolyte gels are dissolved in water, charges are formed in the polyelectrolyte when the counter ions dissociate from the polymer. These counter ions are free to move within the solvent, but stay, in the absence of salt, within the vicinity of the polyelectrolyte to keep electrostatic neutrality of the gel. The increase in entropy due to these counter ions overcomes the extra enthalpy to bring in even more water molecules into the network. The overall effect is that charged hydrogels have an even greater swelling ratio than their uncharged counterparts. A hydrogel matrix can be made to a few centimeters in diameter, depending on the synthetic procedure used. Microgels, on the other hand, are hydrogels that are within the micrometer range.
A number of analytical theories on hydrogels and polyelectrolytes have been presented 8,9,10,11,12 . Corresponding simulations have also been done 13,14,15 , confirming the basic picture that the entropy of the counter ions exerts an extra osmotic pressure which in turn increases the swelling capacity of the hydrogels. Thus determining the location of these counter ions within the network, and specifically the investigation of ion condensation around the polymer chains, is an important aspect of hydrogel research. To our knowledge, the simulations that have been done, however, all consider quasi infinite systems via periodic boundary conditions that connect the chains of one side of the simulation box to its opposite side. This set up suffers from the limitation that the counter ions are, in a sense, always inside the polymer matrix. Thus they cannot examine the actual charge fraction of counter ions inside the network which could, in turn, change the amount of osmotic pressure exerted by the counter ions, and thus change the swelling properties.
In an attempt to attack this problem, we simulate a model microgel, which is a finite sized object made up of crosslinked polymers in a periodic box. The simulation details are described in section II A. The system is a spherical polymer network in a simulation box, as shown in figure 1.
This model simulation allows us to calculate the counter ion condensation not only within the vicinity of each chain, but also the fraction inside the sphere, and study the consequences for the chain conformations. This model is also similar to an even more reduced model of a single charged sphere with counter ions that can be situated inside or oustide this charged sphere. This reduced model can be calculated using Poisson-Boltzmann (PB) theory.
The simulation results can therefore be compared to a PB calculation, and thus provide evidence for the applicability of the Poisson-Boltzman cell model to study ionic properties of microgels.
There exists, however, quite a number of differences between the Poisson-Boltzmann method and simulation set-up, which can potentially confuse the interpretation of a direct comparison between the two. This paper examines the effects of these differences by presenting model simulations aimed at systematically removing, or adding, these factors in a step-wise fashion, with the final aim of studying the usefulness of the PB theory for a description of the swelling properties of microgels.
In section II, we describe the details of the microgel (MG) simulation (II A), brief description of PB theory (II B), and lastly the differences between PB calculation and the microgel simulation (II C). This list of differences (or assumptions) provides the rationale for the rest of the model simulations that we performed, aimed at connecting, in a stepwise manner, the PB model to the MG simulation. The methodologies and results of these simulations described in sections III through V. And in section VI, we give our counclusions.
In all simulations, the following reduced units were used: energy is placed in terms of k B T which is given a value of 1.0, values of lengths are in σ, charge is in units of e, and time is in τ . All particles are given a mass of 1.0. Table I where m i is the mass of particle i, V tot is the total potential acting on the particle, Γ is the friction coefficient that is set to 1.0, and W i (t) is a stochastic force that is uncorrelated in time and across particles. The friction and stochastic forces are linked by the fluctuationdissipation theorem where k B T = 1.0.

A. Microgel Simulation Details
The parameters for the microgel simulation are listed in Tables I and III, labelled as P-MG, where P indicates that periodic boundary conditions were used. Figure 1a shows how the starting structure of the microgel was constructed, whereas figure 1b gives a snapshot of an equilibrated microgel configuration. The microgel was simulated using a general bead-spring model 17,18 . The microgel was composed of 46 polymers, each composed of 10 monomers (beads), giving a total of 460 monomers. These polymers were connected by 29 crosslinks, each represented by one bead, giving a total of 489 beads (monomers plus crosslinks) in the microgel. Some crosslinks, especially those inside the microgel, were connected to the ends of four different polymers (serving as tetrafunctional nodes), while the rest were conneced to the ends of three different polymers (trifunctional nodes). Since all of the polymer ends were connected to one crosslink, the microgel had no dangling chains.
Upon examining figure 1a, one could see that the starting structure resembles that of a diamond lattice, with the edges cut off to approach a spherical shape. Once the particles are allowed to move, the average shape further approximates that of a sphere. The whole set-up was chosen to have an average equilibrated shperical shape, in order to make a comparison with the PB calculation of colloid using a cell model.
Half of the monomers (that is, 230) were given a charge of +1.0, while the others remained uncharged. The charged and uncharged monomers were placed along the polymer in an alternating sequence. Out of the 29 total crosslinks, 20 were also given a charge of +1.0, resulting to a total of 250 of the 489 total particles in the microgel having a charge of +1.0.
Thus 250 free counter ions, each with a charge of -1.0, were added to the simulation, giving a total 739 objects and an overall neutral charge.
The non-bonded interactions of the particles were described by a Weeks-Chandler-Anderson potential 19 of the form where σ 0 = 1.0, ǫ LJ = 1.0, r cut = 2 1/6 , and c shif t = 0.25, thus leaving only the repulsive part of the interaction. These are the standard parameters for bead-spring model of polymers in a good solvent 19 . The bonds between the particles were modelled using the FENE (Finite Extension Nonlinear Elastic) potential 17 where k F = 10.0 k B T σ 2 and r F = 1.5 14 . These parameters allow the bonds to equlibrate to r ≃ 1.0.
The potential energy between charges q i and q j separated by a distance r ij were described by the unscreened Coulomb potential where l B , the Bjerrum length, is defined as where e 0 is the elementary charge, ǫ 0 is the vacuum permittivity, and ǫ r is the relative dielectric constant of the solvent. The Bjerrum length is defined as the distance at which the Coulomb energy between two unit charges is equal to the interaction energy of k B T . This parameter provides a convenient way of tuning the strength of electrostatic interactions in a simulation. As a point of reference, the Bjerrum length of water at room temperature is 7.1Å. In this paper, a Bjerrum length of l B = 1 and k B T = 1 was used in all simulations (Table I) The MG simulation was performed in a cubic box with a length of 84 and periodic boundary conditions. This size was determined to be much larger than the Debye length of the system. The Debye length is the screening distance of a charged particle in a solution, beyond which its interaction with other ions is effectively screened. The Debye length is given as where n 0 is the molar conentration of ions in the reservoir, that is, the concentration of the counter ions remaining outside the sphere. As will be seen in the results, the microgel equilibrates to a radius of 15, with an average of 46% of the counter ions remaining outside the micorgel. This results to a Debye length of 14.1. The closest distance between two periodic images of the microgel is 54, which is roughly 3.8 times this Debye length. This method of calculating the molar concentration n 0 already yeilds the upper limit of the Debyle length. Choosing any other method for defining the n 0 -for example, n 0 as the total number of counter ions over the total box volume-would yeild a higher value for n 0 , and thus a shorter Debye length. This set-up therefore guarantees enough screening to prevent the microgel from being strongly electrostatically affected by its periodic image. Thus the microgel is in the dilute limit where each micgrogel behaves independently of other microgels.
This allows us to compare the results with the PB calculations of a colloid in a cell model, which also assumes that the interaction of one colloid with other colloids is negligible.
The Poisson-Boltzmann (PB) theory is a mean-field theory that describes the electrostatic interactions between charged particles in solution. The Poisson-Boltzmann equation is where φ(r) represents the electrostatic potential, and z i is the valency of the ion species i.
The charge density ρ(r), assumed to be a Bolztmann distribution of charges, is given by This theory assumes that the only interactions taken into consideration are Coulombic interactions between point charges, thus neglecting any finite size effects of the charges. The other assumptions are that the ion entropy is that of an ideal gas.
In a cell model 22 Table II. The results will be discussed in the suceeding sections. One other PB calculation was done, labelled as PB-2, with slightly different parameters listed in III. The reason for this will be explained in the discussion of the MG simulation in section V.

C. Rationale of the Simulation Strategy
The charged sphere in a cell model is already quite close to the microgel simulation.
However, there still are quite a number of assumptions made between the PB model and the MG simulation. These are outlined as follows: 1. Discrete Particles. Whereas the total charge of the sphere in the PB model is uniformly distributed throughout the charged sphere, discrete charges are used in the MG simulation.
2. Excluded Volume. The microgel particles, whether charged or uncharged, in the MG simulation have excluded volume, as parameterized by σ LJ , which in turn dertermines the local binding energy. This is non-existent in the charged sphere of the PB solver.
3. Arranged Particles. The charged particles in the MG simulation are situated along the chain, and are therefore not totally free to move relative to each other, differing from the even distribution of charges in the PB model. This difference also compares the electrostatic algorithms employed in the two systems.
When a simulation is done in a cell model, a direct sum of the Coulombic potential between all pairs of charged particles is calculated, whereas an Ewald sum approachin this paper, the P3M algorithm-is done for the MG simulation in the periodic box.
Due to these assumptions, one would easily suspect that there would also be differences between the results of the PB calculation and the microgel simulation. However, it would be impossible to directly link the differences in results to each of these assumptions, since all of these assumptions are simultaneously present and could easily confuse the analysis.
We therefore performed a series of model simulations that systematically remove from the original model one or more of the assumptions listed, thereby going from the PB picture to the MG simulation in a step-wise fashion. Using this procedure, we are able to attribute specific effects to the corresponding assumptions. These simulations are called Lattice Sphere  Table I so as to keep the counter ions inside the sphere. During the simulation, the counter ions are free to move throughout the whole sphere (note that R = 52 is the radius of the volume available for counter ion motion). A repulsive Lennard-Jones potential was used between counter ions and the charged particles in the lattice so as to avoid that two oppositely charged particles to interact too strongly via Coulombic forces. No Lennard-Jones potential was used between counter ions, since these were treated as point particles, following the assumption of PB theory (section II B). This set-up is very similar to the PB model, except that two assumptions were already were lifted: discrete charged particles (assumption 1) distributed as evenly as possible within a sphere, and the added excluded volume (assumption 2) of these particles instead of an even charge distribution. We used two ways of determining these smaller σ LJ values. The first way was to maintain the energy of closest approach, that is, the Coulombic energy between the counter ion (with a charge of -1.0) and the charged particle (whether charged +1.0, +0.516, or +0.279) in the three different systems were always be the same. From calculations C-LS-1 to C-LS-2 to C-LS-3, the total number of discrete charged particles in the inner sphere was increased from 250 to 484 to 894, with a corresponding decrease in partial charge and σ LJ from 1.0 to 0.517 to 0.279. Figure 3 shows the integrated counter ion probability of the simulations listed in Table II. The second to the last line in Table II refer to the results of the percentage counter ion in figure 3 where r is equal to the sphere radius r 0 . The dots in the inset of figure 3 at r 0 = 18σ indicate these percentages. Figure 4 shows the density of the counter ions as a function of the distance from the center of the inner sphere.
In general, the integrated counter ion probability (Fig. 3) and the counter ion density This does not mean, however, that excluded volume will not have an effect in polymeric systems. The next section discusses the more realistic arrangement of charged particles within a polymer chain.

A. C-ST Methodology
After having seen the effects of assumptions 1 and 2, we then proceeded to simulate another system with assumption 3 removed, that of charged particles in the system being placed along the chains in the microgel. In this case, we took the coordinates of an expanded microgel with the same parameters as described in II A and fixed their coordinates so as not to allow thermal fluctuations. A snapshot of this system is shown in Figure 2c. This In the case of the C-LS-1 configurations ( fig. 5a), there is enough space in between two charged particles for a counter ions to move around the charged particle. The closest distance between the two dotted circles (given by x) is greater than 1.0, which is the Bjerrum length in the simulation. Thus even in this case, their Coulombic repulsion would still be much less than their thermal energy.
That is not the case for the static microgel (C-ST-1, fig. 5b) which does not include the uncharged particles in the microgel. The closest distance between paths (dotted circles) of two counter ions of adjacent charged particles is less than the Bjerrum length. Thus, unlike the lattice sphere cases, the space right in between two charged particles cannot be Although excluded volume had no effect on the configuration of the lattice sphere simulations as described in section III B, the specific arrangement of the excluded volume, in this case, arranged in a polymer chain, can slightly lessen the available space for counter ion condensation and thus lower the fraction of counter ions inside the microgel.

V. MICROGEL IN A SPHERICAL CELL (C-MG) AND IN A PERIODIC BOX
(P-MG)

A. C-MG and P-MG Methodology
After removing assumptions 1 and 2 to the lattice sphere (C-LS) and assumption 3 to the static micro gel (C-ST), we then removed the next assumption 4, which looks at the effect of thermal fluctuations of the charged particles in the polymer. This is basically a microgel simulation, where the particles of the microgel are allowed to move, inside a spherical cell.
Only the central particle of the microgel was kept fixed so as to keep the microgel in the center of the sphere. The parameters are basically the same as those of the microgel, and are listed in Table II. Aside from thermal fluctuations, the other difference of between this system and static microgel in a sphere (section IV) is the decrease in microgel radius, thus decreasing the free volume in between the chains. The parameters are listed in Table II.
This simulation is labeled C-MG. As with the C-LS and C-ST simulations, the C-MG simulation was also done in a spherical cell model, with the electrostatics calculated as a direct sum.
To determine the effect of the last assumtion, that of periodic boundary conditions and differences in electrostatic algorithm, we compared the results of C-MG to P-MG. The methodology and parameters for the P-MG simulation has already been discussed in section II A.
Due to the thermal fluctuation of the entire microgel for both C-MG and P-MG, the radius of the microgel was also fluctuating, but generally maintaing its overall spherical shape. In order to determine the average radius of the microgel, we used the following equation 25 that relates the square radius of gyration of a rigid sphere to the radius of the sphere where R 2 g is square radius of gyration of the sphere with radius r.

B. Discussion: Thermal Fluctuations and Periodic Boundary Conditions
The parameters and results (second to the last line) of the C-MG and P-MG are listed in Table III. The integrated counter ion probability is shown in figure 3  Both C-MG and P-MG have practically the same results-configuration as measured by R 2 g , fraction of counter ion inside the microgel, integrated counter ion probability (firgure 3) and counter ion densities ( figure 4). Thus the spherical cell model simulation, whose electrostatics is calculated as a direct sum, is equivalent to the corresponding simulation under periodic conditions, as long as the sytsems are set up correctly. This means that the system size is done ensuring enough electrostatic screening (as calculated by the Debye length) in order to satisfy the "isolated system" assumption of the cell model, and that the P3M electrostatics is tuned correctly.

VI. CONCLUSIONS
We have simulated a model polyelectrolyte microgel setup with the corresponding counter ions in a periodic box, using the results mainly to calculate the fraction of counter ions inside the microgel. We compared these results to a Poisson-Boltzmann numerical solution 23,24 in order to see how the mean field solution differs from the results of the simulation. We outlined the differences between these two models, listed as assumptions in section II C, and proceeded to perform a series of simulations aimed at removing these assumptions in a step-wise fashion in order to isolate the effects of each assumption.
The very first assumption of going from a charged sphere with uniform distribution in the PB solver to having discrete charges showed a slight decrease in the fraction of counter ions inside the microgel. The fact that the sucessive "smearing out" of the discrete charges, thus approacing the uniform charge distribution in the PB solver, resluted in a sucessive approach of the LS results to that of the PB calculation shows that the discretization of charges causes the slight decrease of the fraction of counter ions inside the microgel. A further decrease is shown when going from the lattice sphere with evenly spaced charge distribution within the sphere to the static microgel configuration, where the charges are not as evenly distributed since they were lined up within the polymer. Aside from the discretization of charges, the density of charges of the microgel particles also plays a role in fraction of counter ions in the microgel, the two being directly proportional.
Even with the many assumptions in between the MG simulations and PB theory, we have seen that the results of the two methods are quite close to one another. We can therefore safely assume that if PB models with configurations that follow the more complex configurations found in molecular simulations of real systems are constructed, the results in terms of counter ion condensation from both approaches would be comparatively close, as long as we stick the the basic assumptions used in this study such as movable charges and counter ions, good solvent, and relatively isolated systems. This allows one to use the PB cell model to predict structural or thermodynamic properties of microgels in solution.   following parameters are common for all these calculations: the radius of the inner sphere r 0 = 18σ whose total charge is q sp = +250.0, the total number of counter ions is n CI = 250, the charge for each coutner ion is q CI = −1.0, each counter ion is treated as a point particle with no σ LJ interaction between them, all use the cell model and the electrostatic interactions are calculated via direct sum, the outer radius of the cell is R = 52σ. The entries left blank in this table are parameters that are not applicable to the specific computation.