Coarse-Grained Simulations of Nanogel Composites: Electrostatic and

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Coarse-Grained Simulations of Nanogel Composites: Electrostatic and Steric Effects María del Mar Ramos-Tejada and Manuel Quesada-Peŕ ez*

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Departamento de Física, Escuela Politécnica Superior de Linares, Campus Científico-Tecnológico, 23700 Linares, Jaén, Spain ABSTRACT: Monte Carlo coarse-grained simulations of nanogels loaded with nanoparticles are presented, paying special attention to two phenomena revealed by experiments with real composites: the deswelling and charge inversion observed after the absorption of nanoparticles. Both can be justified here exclusively in terms of electrostatic and excluded volume interactions. According to our results, charge inversion in composites presents its own peculiarities. For instance, it only takes place if the charge of the nanoparticle exceeds a threshold value that increases with the nanoparticle size. Simulations also show that the distribution of nanoparticles inside the polyelectrolyte network varies with the nanoparticle charge. In addition, the location of nanoparticles inside the nanogel strongly depends on whether charge inversion occurs.

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biopolymers,19−28 viral detection,29 or genome encapsidation.30 In particular, coarse-grained computer simulations have also been considerably useful in the past decade for the analysis of certain nonspecific aspects of the swelling of gels and nanogels, such as size and charge effects.31−40 Computational techniques allow us to explicitly consider the fluctuations of their polymer chains and their distribution of charge, the nonexistence of a perfect spherical surface, or the high degree of cross-linking (and complex topology) of nanogels. This could partially explain why theoretical treatments are so rare for nanogels. A few previous works have also paid attention to the permeation of small ions into these porous NPs.38,40,41 In particular, the interplay between electrostatic and steric forces was explored, and some theoretical approaches were also presented and tested with the help of simulation data.40,41 However, the uptake of NPs with sizes and charges larger than those corresponding to typical small ions has not been simulated or theoretically addressed so far to the best of our knowledge. Certainly, size and charge asymmetries between conventional ions and NPs make theoretical approaches much more difficult. But simulations have also proved that strong electrostatic couplings can induce intriguing phenomena in hard colloids (such as charge inversion or electrostatic attractive forces between like-charged particles). According to the previous paragraphs, the main goal of this work is to shed light on the role that electrostatic and steric interactions play when NPs are loaded in micro- or nanogels. This task will be tackled with the help of coarse-grained simulations. The rest of the paper is organized as follows. First, the model and the computational method employed are outlined. Then, results are presented and discussed, paying

INTRODUCTION Microgels are colloidal particles made of cross-linked polymer (or polyelectrolyte) chains. When these polymeric aggregates are synthesized with diameters of a few tens of nanometers, they are also known as nanogels. In any case, both micro- and nanogels are fascinating colloids because they can considerably swell by absorption of large amounts of solvent in response to different external stimuli, such as changes in pH or temperature, which introduces a level of control in their application. Because of this responsiveness, micro- and nanogels have been seen as promising materials for a number of cutting-edge applications such as drug delivery, sensors, chemical separation, catalyst media, micromechanical and optical devices, or photonic crystals.1−7 In some of these applications, microgels are used as containers for other smaller nanoparticles (NPs) in the size range 1−10 nm, such as drugs,8 metal NPs,9−13 magnetite,14,15 quantum dots, and other inorganic NPs.6,16,17 In fact, promising materials were prepared from microgels loaded with iron oxide,18 gold,3 silver, 13 or quantum dots.9 Consequently, the uptake of NPs into microgels and their subsequent release has become an area of growing interest in recent years. In many instances, both microgels and NPs are charged objects surrounded by an ionic atmosphere known as the electric double layer. Thus electrostatic interactions are expected to play a key role in the uptake and release of NPs by micro- and nanogels, which should be fully understood to control such processes. However, electrostatic interactions between nano-objects are a long-standing issue in colloid science, whose precise understanding continues to challenge many researchers. Over the years, coarse-grained models of charged polymeric systems have been quite helpful in elucidating factors controlling their properties and phenomena in which they are involved, such as the adsorption and collapse of © XXXX American Chemical Society

Received: December 13, 2018 Revised: February 4, 2019

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DOI: 10.1021/acs.macromol.8b02657 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. Nanogel Properties nanogel

monomers per chain

charged monomers per chain

bare charge

NG15+1e NG15+3e NG30+1e NG30+3e

15 15 30 30

1 3 1 3

100 300 100 300

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MODEL AND SIMULATIONS The coarse-grained representation of reality employed in this work is known as the bead−spring model for polyelectrolytes. In this picture, monomer units of the nanogel, ions, and NPs are represented as spheres, whereas the solvent is considered as a dielectric continuum. In particular, this model has also been employed in the simulation of nanogels in the past decade.32,34,36,42−46 The four nanogels simulated here consist of 100 polyelectrolyte chains connected by 66 cross-linkers using the same topology: The most inner cross-linkers (40%) connect four polyelectrolyte chains, but the most external ones only connect three or even two chains. Dangling chains were not considered in any case. A similar nanogel was employed for the first time in the pioneering work by Claudio et al.32 These four nanogels differ in the number of monomers per chain and the number of charged monomers per chain. The name employed for each nanogel refers to these two quantities. For instance, NG15+3e stands for the nanogel with 15 monomers per chain and three positively charged monomers per chain (e denotes the elementary charge). Together with the monomers that constitute the nanogel, the simulation cell also contains monovalent counterions neutralizing its charge, multivalent NPs in a given concentration, and the monovalent counterions that compensate their charge. All of these species are also modeled as spheres, as mentioned before. The diameters of monomers and hydrated counterions are 0.65 and 0.7 nm. The reader should keep in mind that in the case of ions, this diameter includes their hydration shell. The NP diameter was just a few nanometers. In simulation sets where this parameter remained fixed, a diameter of 6 nm was chosen. This value can be representative for some gold and silver NPs, quantum dots, and even small magnetite NPs (whose typical sizes are in the 7−20 nm range).14 The short-range repulsion between monomers, ions, and NPs due to excluded volume effects is modeled by means of the Weeks−Chandler−Andersen potential 6

2d

6

2d

ubond(r ) =

3.18 3.87 1.61 2.33

k bond (r − r0)2 2

(3)

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RESULTS AND DISCUSSION Nanogel Properties. In this work, the radius of a nanogel (or a composite) is described through a length (RNG) that is proportional to the radius of gyration (RNG = 5/3 R g ) and can give us an idea of the geometrical radius of the nanogel considered in a first approximation as a sphere. In fact, a sphere of radius RNG centered at the CM of the polyelectrolyte network contains 90% of the monomers. Given that the surface of a nanogel is not very well defined, this sphere is also

(1)

ZiZje 2 4ε0εrr

Ψ (RNG)

76.8 154.3 84.2 179.9

0.20 0.09 0.11 0.12

where kbond is the elastic constant (0.4 N/m) and r0 is the equilibrium bond length (0.65 nm). Simulations were carried out in a cubic box of length L and periodic boundary conditions. L must be large enough to contain the nanogel particle and a significant portion of the electric double layer around it. More specifically, L must be greater than 2(RNG0 + 4lD), where RNG0 is an initial estimate of the radius of the nanogel and lD is the Debye screening length. Monte Carlo (MC) simulations were implemented by means of the standard Metropolis algorithm. Three kinds of MC moves were performed. Most of the beads (monomers, crosslinkers, and small ions) executed single-particle moves. However, NPs were translated together with their nearest counterions (as if they formed a cluster due to a strong electrostatic coupling). The maximum displacement corresponding to each bead and cluster was individually adjusted so that its respective acceptance ratio was close to 50%. In addition, contractions and expansions of the whole polymer network and the particles inside were also attempted during equilibration to accelerate this process. In these contractions/ expansions, the position vector of each particle was multiplied by a scaling factor. Long-range Coulomb forces were handled through Ewald sums, which were implemented with the recommendations reported elsewhere.47 Although there are efficient simulation packages for coarse-grained models of soft matter, simulations were performed using a homemade computer code (in C) employed in previous works. At least 1 × 108 and 2 × 108 configurations were used for equilibration and statistics, respectively. The evolution of the radius of gyration of the nanogel, Rg, was monitored averaging this quantity in subsets of 5 × 105 configurations to check that an equilibrium value was reached after equilibration and to compute the standard deviation of this quantity, which gives us an idea of its uncertainty. Simulations also provide the concentration of particles at a distance r from the center of mass (CM).

where r is the center-to-center distance between a given pair of particles, εLJ = 4.11 × 10−21 J, d = (di + dj)/2, and di denotes the diameter of species i. All of the charged beads interact through the Coulomb potential uelec(r ) =

net charge

± ± ± ±

where Zie is the charge of species i, ε0 is the permittivity of vacuum, and εr is the relative permittivity of the medium (water at 293 K in this case). Consecutive monomers of a chain are connected by harmonic bonds, whose interaction potential is

special attention to the phenomenon of charge inversion in the composites formed when nanogels are loaded with NPs. Finally, some conclusions are highlighted.

l o ij d12 d6 1 yz o o o o 4εLJjjjj 12 − 6 + zzzz r ≤ 4{ r uLJ(r ) = m kr o o o o o o 0 r> n

radius (nm) 13.68 17.55 22.90 27.66

(2) B


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Macromolecules employed in this work as a border of the network to count the number of NPs and ions inside. The RNG values of the nanogels of this survey are included in Table 1. As can be seen, the size of the nanogel increases with the number of monomers per chain and the number of charged groups per chain, which is logical. Another important single-particle property of a nanogel is its net charge, defined here as the charge enclosed by a sphere of radius RNG (centered at the CM of the polyelectrolyte network). The net charge is also included in Table 1. As can be easily concluded, this quantity is smaller than the bare charge of the chains forming the network because the net charge also includes the charge of small ions and NPs inside. For classical hard colloids, whose surface is well-defined, it is also interesting to know the electrostatic potential at the surface or at the slipping plane (the ζ potential). However, electrokinetic concepts such as the slipping plane or the ζ potential are not so clearly defined for swollen nanogels because the surface of the nanogel is not unambiguously determined, as pointed out above. But if the surface of these soft particles is assumed to be located at a distance RNG from their CM, then ψ(RNG) can be interpreted as a surface electrostatic potential. Thus this quantity can provide valuable information to justify (and even predict) the electrokinetic behavior of nanogels and their composites. For that reason, Table 1 also includes the dimensionless electrostatic potential at RNG, Ψ(RNG) ≡ eψ/(RNG)/kBT, where kB is Boltzmann’s constant and T is the absolute temperature. In general, these values are >1. The dimensionless electrostatic potential inside the nanogel (not shown) is even larger, which means that the electrostatic potential energy of counterions inside the nanogel is (in magnitude) much greater than the thermal energy. In other words, such counterions are bound to the nanogel (or confined inside). It is quite instructive to find out if a simple Poisson− Boltzmann cell model (PBCM) put forward elsewhere37 can predict the net charges and the Ψ(RNG) values obtained in simulations. Table 2 displays these predictions and their

Figure 1. Radius of the nanogel/nanoparticle composite as a function of the mean concentration of nanoparticles added to the simulation cell for the four nanogels of this study.

nanogel NG30+3e (the one with the greatest charge and the longest chain), slight for NG30+1e and NG15+3e, and almost imperceptible for NG15+1e. In any case, the greatest change in size is observed when the NP concentration varies from 0 to 0.001 mM (the minimum concentration studied in this work). Similar behaviors in size with the NP content have been observed for ZnS, silver, magnetite, and ZnO in poly(Nvinylcaprolactam-co-acetoacetoxyethyl methacrylate)-based microgels6,13,14,17 and gold NPs in poly(N-isopropylacrylamideco-N-[3-(dimethylamino)propyl]methacrylamide)-based microgels10 and poly(N-isopropylacrylamide-co-methacrylic acid)-based microgels.11 The decrease in the size of the composite at low NP concentrations was attributed by some authors to the specific interactions of magnetite and ZnO NPs with the polymer network.14,17 However, it is somehow striking that these specific effects are observed in such a variety of NPs and microgels. In any case, we should keep in mind that the model employed here only includes excluded volume and electrostatic interactions, which are present in all charged systems. Although specific forces might also be present in real systems, this clearly reveals that the above-mentioned nonspecific forces can explain on their own the behavior reported. The mechanisms involved are described as follows. In the absence of NPs and multivalent ions, nanogels contain monovalent counterions that neutralize its bare charge to some extent. When oppositely charged NPs are added to the solution, they enter the nanogels and eventually expel a fraction of these monovalent conterions. Because the valence of the NPs is greater (and even much greater) than the valence of the counterions, the number of NPs inside the nanogel is considerably smaller than the initial number of monovalent counterions, which causes a reduction in the inner osmotic pressure. The deswelling of the nanogel at low concentrations of NPs can be attributed to this reduction of osmotic pressure, at least in part. In addition, NPs could play the role of crosslinking agents bridging polyelectrolyte chains together (due to the electrostatic attraction between them and the NPs). Finally, if the number of NPs inside the nanogel goes on increasing after having replaced the monovalent counterions, then the inner osmotic pressure also increases and the composite swells. In addition, the distance between the inclusions becomes smaller when the NP content grows.

Table 2. PBCM Predictions for Net Charge and Ψ(RNG) and Their Relative Deviation nanogel

net charge

relative deviation (%)

Ψ (RNG)

relative deviation (%)

NG15+1e NG15+3e NG30+1e NG30+3e

85.7 168.6 89.1 192.8

11.59 9.26 5.82 6.69

3.20 4.04 1.67 2.45

0.62 4.39 3.72 5.15

relative deviations from simulation results. As can be inferred, the relative deviation in Ψ(RNG) ranges from 0.6 to 5%, but the discrepancies are a bit greater for the net charge. Thus the PBCM predictions are quite good for the dimensionless electrostatic potential and acceptable for the net charge. Effect of the Nanoparticle Content. At this point, let us focus our attention on the composites formed by these nanogels after the absorption of NPs. Figure 1 shows the radius of the composite as a function of the mean concentration of NPs added to the simulation cell for the four nanogels studied in this work. The diameter and charge of these NPs are 6 nm and −20e, respectively. As can be seen, the radius decreases at low NP concentration. Then, this property passes through a minimum and finally increases. The rise is moderate for C


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Macromolecules Therefore, the repulsion forces between charged NPs could also contribute to the swelling of the composite. In relation to the size of the composite, the work by Sing et al.48 should be also mentioned. From an integral equation formalism, these authors recently demonstrated that a gel undergoes a first-order collapse in the presence of low multivalent electrolyte concentrations as a result of strong charge correlations. In addition, they also reported a reentrant swelling if the salt concentration was increased enough. Their study is focused on gels in the presence of trivalent ions, but their results resemble what different researchers have observed with microgels in the presence of highly charged NPs. Figure 2 displays the number of NPs absorbed into the four nanogels investigated here as a function of the mean NP

Figure 3. Net charge (in elementary units) of the composites studied in this work as a function of the nanoparticle concentration.

(or inversion) after the absorption of NPs. This change of sign is also observed in the normalized electrostatic potential at RNG, Ψ(RNG), also plotted in Figure 4. In fact, this figure

Figure 2. Number of nanoparticles absorbed in the four nanogels investigated here as a function of the mean NP concentration in the simulation cell.

concentration in the simulation cell. We should keep in mind that a NP is considered to be absorbed if the distance between its center and the center of mass of the nanogel is smaller than RNG. As can be easily concluded from Figure 2, the number of particles absorbed is not very sensitive to the NP concentration (at least in the range of concentrations simulated here). In the case of nanogels NG15+3e, NG30+1e, and NG30+3e, this number seems to increase slightly. For NG15+3e, the number of NPs inside the nanogel does not grow with the NP content. This insensitivity might also explain why the radius of the composite slightly grows when the NP concentration increases from 0.001 to 0.05 mM for the composites of NG15+3e, NG30+1e, and NG30+3e and it does not increase for NG15+3e (see Figure 1 excluding the data at zero concentration). The influence of the number of monomers per chain seems to be relatively small as well. According to Figure 2, the property that mostly controls the number of NPs loaded in the nanogel is its bare charge. When the bare charge of the nanogel is multiplied by 3, the number of NPs in the composite is roughly multiplied 3 as well. In other words, the absorption of NPs is electrostatically driven. As mentioned before, one of the most important properties of a colloidal particle is its charge. Figure 3 shows the net charge of the composite (bare charge of the nanogel plus the charge of the NPs and ions inside) as a function of the NP content again. The most outstanding feature of this figure is the fact that the composites are negatively charged. In other words, the net charge of the nanogels has undergone a reversal

Figure 4. Normalized electrostatic potential at RNG of the composites studied in this work as a function of the nanoparticle concentration.

suggests the existence of a reversal in the ζ-potential and electrophoretic mobility of such composites. Such reversals have been reported for gold NPs,10−12 ZnS NPs,6 and magnetite.14 In relation to the magnitude of the dimensionless electrostatic potential, it should be mentioned that this quantity is on the order of 1 and even smaller for most of the composites simulated here. The corresponding bare nanogels exhibit electrostatic potentials with much greater magnitudes, as mentioned above. Charge reversal is a counterintuitive phenomenon that has attracted the attention of many researchers of hard colloids and soft matter.49−54 We might naively imagine that the maximum number of NPs inside the nanogel should be such as to completely neutralize the bare charge of the polyelectrolyte network. But why do nanogels absorb NPs beyond that point? Different mechanisms have been proposed to justify the existence of charge inversion in other systems. For instance, highly charged polyelectrolytes (such as DNA) can adsorb onto an oppositely charged colloidal particle beyond the point of neutralization. After adsorption, the counterions condensed on the polyelectrolyte are released, which increases the entropy D


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Macromolecules due to their translational degrees of freedom.49,55 This entropy gain might contribute to charge reversal. Multivalent counterions can also overcompensate the charge of hard colloids as the result of strong positional correlations between them.50,51,56 The highly ordered structure formed by the counterions on the particle surface is similar to a 2D Wigner crystal (WC), and it is known as the strongly correlated liquid (SCL) by some authors.51 In addition, charge reversal has also been attributed to specific (but not always specified) ion−surface interactions.53,54 In our simulations, such specific interactions have not been considered, so the charge inversion observed in Figure 3 for nanogel composites must be exclusively explained in terms of electrostatic and excluded-volume forces, and only two of the mechanisms mentioned above (counterion release and SCL) might be operative. As mentioned previously, many monovalent counterions are usually trapped inside charged nanogels as a result of a large electrostatic potential energy (due to the so-called Donnan potential). When NPs with opposite charge enter nanogels, many of these counterions are released. Thus the charge inversion reported here could have an entropic contribution. In any case, it should be stressed that charge reversal presents its own peculiarities in the case of nanogels. For instance, it is well known that trivalent ions can induce charge inversion in hard colloids. However, this phenomenon was not observed in previous simulations of nanogels in the presence of trivalent cations.38 Thus there must be a minimum valence (or particle charge) from which charge reversal is reported. This is just what we analyze in the next section. It should also be mentioned that the PBCM previously employed to estimate Ψ(RNG) values and net charges of nanogels fails to predict charge reversals if ad hoc specific interactions are not considered.49,55 For that reason, its predictions were not compared with data shown in Figures 3 and 4. Effect of Nanoparticle Charge. In this section, we study composites of the four nanogels with NPs whose diameter remains fixed (6 nm) but whose charge varies from −2 to −40 elementary charges. In practice, this could be the case of a NP with weak acid groups on its surface, whose charge varies with pH. In particular, it is quite appealing to examine the behavior of the net charge of the composite when the NP charge changes, which is plotted in Figure 5 for different cases. This figure clearly reveals that charge inversion takes place only if the NP charge is greater (in absolute value) than 15e. This threshold seems to be the same for the four nanogel composites simulated here. In any case, we should bear in mind that the aspect of this figure and therefore the threshold for charge inversion might depend to some extent on the imaginary surface employed as the border of the composite to compute the net charge. It is interesting to analyze where the NPs are located (depending on their charge). Figure 6 shows the spherically averaged local NP number density, ρNP(r), as a function of the distance r from the CM of the polyelectrolyte network for NG15+1e and different NP charges. From this figure, it can be easily concluded that NPs with charges (in absolute value) smaller than (or equal to) 15e are preferentially located at the center of the nanogel, just where the electrostatic potential of the nanogel reaches very high values (not shown). However, NPs with greater charges are structured in shells at different distances from the nanogel CM. In other words, the

Figure 5. Net charge of the composites obtained from the four nanogels as a function of the charge of the absorbed nanoparticles.

Figure 6. Spherically averaged local nanoparticle number density, ρNP(r), as a function of the distance r from the CM of the polyelectrolyte network for the composites of NG15+1e and different nanoparticle charges.

distribution of NPs inside the polyelectrolyte network strongly depends on the existence of charge inversion in the composite. It is also quite instructive to analyze how NPs inside the nanogel are distributed around a given NP taken as a reference. This information is provided by the nanoparticle−nanoparticle radial distribution function (rdf), gNP−NP(r). Figure 7 displays this rdf for NPs of 6 nm inside the nanogel NG15+3e with charges ranging from −2e to −40e. As can be seen, the rdf obtained for −2e and −5e rises steeply at r ≈ dNP (where dNP denotes the nanoparticle diameter), yielding an asymmetric peak at this position. This is the typical behavior of hard spheres at high concentrations. In other words, the origin of this NP ordering is steric (due to excluded volume interactions). NPs with charges greater than −15e are also spatially ordered, but their ordering is characterized by more symmetric peaks at distances greater than dNP. In addition, the distance of minimum approach is also greater than dNP. This ordering can E


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Figure 7. NP−NP radial distribution function, gNP−NP(r), as a function of the distance r from the CM of a given nanoparticle taken as reference for the composites of NG15+3e and different nanoparticle charges.

be attributed to intense electrostatic forces between NPs when their charge grows, and this suggests that the origin of charge inversion for highly charged NPs is related to strong electrostatic correlations between them. The system with −15e can be considered a transition between these different behaviors. Its rdf steeply increases at r ≈ dNP, but the maximum is shifted to greater distances. As mentioned previously, the monovalent counterions inside the nanogel are partially replaced by the NPs. It is worth mentioning that the extent of this replacement depends on the NP charge; therefore, it is also related to charge inversion. Our simulations reveal that monovalent counterions are dominant at low NP charges but they are almost inexistent in the case of highly charged NPs (results not shown graphically). In this case, NPs are mostly responsible for charge inversion. It should also be pointed out that highly charged NPs carry small cations with them into the nanogel. To finish this section, let us turn our attention to the electrostatic potential. Figure 8 displays the normalized electrostatic potential at RNG of the composites studied in this work as a function of the NP charge. This figure shows that the surface electrostatic potential undergoes an inversion if the magnitude of the charge is >10e. This inversion point is slightly smaller than the one found for the reversal of the net charge. This minor difference in the inversion points can be understood recalling that the net charge and the surface electrostatic potential are not proportional quantities. According to Gauss’ law, the spherically averaged electric field at RNG is proportional to the net charge, but the electrostatic potential is computed by integrating from infinite to RNG the electric field. Effect of Nanoparticle Size on the Inversion Points. To begin with this section, it should be stressed that the inversion points previously found for the net charge and the surface electrostatic potential were obtained for NPs of 6 nm. Thus it would be interesting to find out if such inversion points change with the NP size. The answer to this question can be found in Figures 9 and 10, in which the net charge and Ψ(RNG) are plotted as a function of the NP charge for the composites of NG15+1e and NPs of three different diameters (2, 6, and 10 nm). Both figures reveal that the inversion points shift toward NP charges of greater magnitudes when the NP

Figure 8. Normalized electrostatic potential at RNG of the composites studied in this work as a function of the nanoparticle charge.

Figure 9. Net charge of the composites obtained from NG15+1e as a function of the charge of the absorbed nanoparticles for three NP diameters (2, 6, and 10 nm).

Figure 10. Normalized electrostatic potential at RNG for the composites obtained from NG15+1e as a function of the charge of the absorbed nanoparticles for three NP diameters (2, 6, and 10 nm).

F


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Innovación 2013-2016’, Project FIS2016-80087-C2-2-P and (ii) European Regional Development Fund (ERDF).

size grows. This shift can be the consequence of the competition between the excluded volume and electrostatic interactions. The distance of minimum approach between hard NPs is of the order of their diameter. Thus an increase in their size can weaken the electrostatic interaction between NPs, which is responsible for charge inversion. But this reduction in the electrostatic interaction can be compensated for by increasing the NP charge. Comparing Figures 9 and 10, one can also conclude that the effect of the NP size on the inversion point seems greater for the net charge. In fact, the inversion point of the net charge of the composite seems to be shifted to NP charges considerably greater (in magnitude) than 40e for NPs of 10 nm (see Figure 9). However, the inversion point of Ψ(RNG) only moves from −8e to −16e when the NP size increases from 2 to 10 nm.

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CONCLUSIONS The simulations presented here capture two phenomena reported in experiments of micro- and nanogel composites: (i) the reduction in size even at low NP contents and (ii) reversals in the electrophoretic mobility of composites related to charge inversion. Given that the model employed only involves nonspecific forces, a wide variety of composites should exhibit such phenomena. In fact, they are easily found in scientific literature. Our simulations also reveal that charge inversion only takes place if the NP charge is large enough. The threshold value of NP charge required for inversion increases when the NP diameter grows. Our results also demonstrate that the distribution of NPs inside the nanogel strongly depends on whether charge inversion occurs. The simulations presented here only consider the presence of one asymmetric electrolyte composed of multivalent NPs and their associated monovalent counterions. It would be interesting to figure out what would happen to the absorption of NPs and charge inversion in the presence of an additional monovalent electrolyte. Our results suggest that the charge inversion reported here is the result of strong electrostatic correlations between the NPs absorbed into the nanogel. Given that the screening of the electrostatic forces would increase with an additional electrolyte, charge inversion is expected to weaken. In fact, previous simulations revealed that charge inversion in hard colloids weakens when a monovalent salt is added.57 However, some theories based on the strongly correlated liquid claim that monovalent salts enhance charge inversion.51 Thus it would be noteworthy to shed light on this matter. Apart from these theoretical questions, many experimentalists can wonder how NPs and small ions compete in the absorption into nanogels.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Manuel Quesada-Pérez: 0000-0003-0519-7845 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the financial support from the following institutions: (i) ‘Ministerio de Economiá y Competitividad, ́ Plan Estatal de Investigación Cientifica y Técnica y de G


Article

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