Charge affinity and solvent effects in numerical simulations of ionic microgels
G. Del Monte, F. Camerin, A. Ninarello, N. Gnan, L. Rovigatti, E. Zaccarelli
CCharge affinity and solvent effects in numericalsimulations of ionic microgels
Giovanni Del Monte , , , ∗ , Fabrizio Camerin , , ∗ , AndreaNinarello , , Nicoletta Gnan , , Lorenzo Rovigatti , ,Emanuela Zaccarelli , , ∗ CNR Institute of Complex Systems, Uos Sapienza, piazzale Aldo Moro 2,00185, Roma, Italy Department of Physics, Sapienza University of Rome, piazzale Aldo Moro 2,00185 Roma, Italy Center for Life NanoScience, Istituto Italiano di Tecnologia, viale ReginaElena 291, 00161 Rome, Italy Department of Basic and Applied Sciences for Engineering, via AntonioScarpa 14, 00161 Roma, ItalyE-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
January 28, 2021
Abstract.
Ionic microgel particles are intriguing systems in which the propertiesof thermo-responsive polymeric colloids are enriched by the presence of chargedgroups. In order to rationalize their properties and predict the behaviour ofmicrogel suspensions, it is necessary to develop a coarse-graining strategy thatstarts from the accurate modelling of single particles. Here, we provide anumerical advancement of a recently-introduced model for charged co-polymerizedmicrogels by improving the treatment of ionic groups in the polymer network.We investigate the thermoresponsive properties of the particles, in particulartheir swelling behaviour and structure, finding that, when charged groups areconsidered to be hydrophilic at all temperatures, highly charged microgels donot achieve a fully collapsed state, in favorable comparison to experiments. Inaddition, we explicitly include the solvent in the description and put forwarda mapping between the solvophobic potential in the absence of the solvent andthe monomer-solvent interactions in its presence, which is found to work veryaccurately for any charge fraction of the microgel. Our work paves the way forcomparing single-particle properties and swelling behaviour of ionic microgels toexperiments and to tackle the study of these charged soft particles at a liquid-liquid interface.
Keywords : ionic microgels, charge affinity, solvophobic attraction, volume phasetransition, form factors
Submitted to:
J. Phys.: Condens. Matter a r X i v : . [ c ond - m a t . s o f t ] J a n harge affinity and solvent effects in numerical simulations of ionic microgels
1. Introduction
Soft matter is a very active branch of condensedmatter physics, which comprises, among other systems,colloidal suspensions, whose constituent particles cangreatly vary in shape, softness and function. Softmatter encompasses not only synthetic particles, butalso constituents of many biological systems, such asproteins, viruses and even cells, whose size rangesbetween the nano and the micrometer scale. A peculiaraspect of soft matter systems is the great variety ofamorphous states they can form, including glasses [1, 2]and gels [3, 4, 5]. Indeed, a large amount of work in thisfield is devoted to the study of these non-ergodic stateswhich may form due to different kind of interactions,such as steric, hydrophobic or electrostatic ones, bothof attractive and repulsive nature.Sometimes, a single colloidal particle is alreadyquite a complex object whose behaviour at thecollective level is strongly connected to the microscopicfeatures of the particle itself. This situation is typicalof soft colloids, i.e. deformable particles with internaldegrees of freedom strongly influencing their mutualinteractions, which makes them already intrinsicallymulti-scale. For these systems a theoretical approach isquite challenging even at the single-particle level, thusit is convenient to rely on the development of suitablecoarse-grained models [6] that allow to greatly reducethe system complexity, still capturing the importantingredients to be retained for a correct description ofthe collective behavior. This strategy is very profitablefor the case of microgel particles [7] that, combiningtogether the properties of colloids and polymers, can beviewed as a prototype example of soft particles [8, 9].A microgel is a microscale gel whose internal polymericnetwork controls its peculiar properties. By varyingthe constituent monomers, microgels can be maderesponsive to temperature, pH or to external forces [7].For their intriguing properties, they are employed in awide variety of applications, ranging from biomedicalpurposes [10, 11] to paper restoration [12].In order to be able to predict the behaviourof dense microgel suspensions and the formation ofarrested states, it is important to properly takeinto account the internal degrees of freedom of theparticles, by modelling their effective interactionsin such a way that the resulting object can stillshrink, deform and interpenetrate [13, 14, 15, 16].Hence, an accurate modelling of a single microgelis a necessary pre-requisite for a correct descriptionof bulk suspensions. To validate numerical modelsat the single-particle level, there are a number ofdifferent experiments we can refer to. One of the moststraightforward is the measurement of the effectivesize of the microgels via dynamic light scatteringexperiments. Upon varying the controlling parameter of the dispersion, the so-called swelling curves canbe determined. For instance, microgels synthesizedby employing a thermoresponsive polymer, such asPoly(N-isopropyl-acrylamide) (PNIPAM), undergo aso-called Volume Phase Transition (VPT) [7] at atemperature T VPT ≈ ◦ C from a swollen to a collapsedstate.In addition, form factors can be measured bysmall-angle scattering experiments of dilute microgelsuspensions, either using neutrons [17], x-rays [18] oreven visible light for large enough microgels [19]. Thisobservable directly provides information on the innerstructure of the microgels and shows that microgelsprepared via precipitation polymerization [20] can bemodelled as effective fuzzy spheres [17], where a ratherhomogeneous core is surrounded by a fluffy corona,giving rise to what is usually called a core-coronastructure. A more complex situation arises when ionicgroups are added to the synthesis to make microgelsresponsive also to external electric fields [21, 22] andto pH variations [23]. A case study of such these co-polymerized microgels is the one made of PNIPAM andpolyacrylic acid (PAAc) [20, 24, 25, 26], that is pH-responsive due to the the weak acidic nature of AAcmonomers.An increasing amount of work in the last yearshas focused on modelling single-particle behaviourboth of neutral [27, 28, 29] and ionic microgels [30,31, 32, 33]. For the latter case, we have recentlyshown [33] that it is important to take into accountboth the disordered nature of the network, as opposedto the diamond lattice structure [29], and an explicittreatment of charges and counterions. Indeed, mean-field approaches completely neglect the complex,heterogeneous nature of the charge distribution withinthe microgel.In this work, we extend our previous effort bygoing one step further towards a realistic numericaltreatment of ionic co-polymerized microgels. InRef. [33], we modelled a single microgel particle suchthat all of its monomers, including charged ones,experienced a solvophobic attraction on increasingtemperature. Here, instead, charged monomersexperience Coulomb and steric interactions only. Thisis expected to be more realistic, since charged orpolar groups always remain hydrophilic irrespectivelyof temperature, thus having a distinct behaviourwith respect to all other monomers. We examinethe consequences of this difference on the microgelswelling behaviour as well as on its structure andcharge distributions across the VPT. In the secondpart of the manuscript, we consider the presence of anexplicit solvent, to examine its effects on the structuralproperties of the microgel. In this way, we aim to makeour model suitable for situations where solvent effects harge affinity and solvent effects in numerical simulations of ionic microgels
2. Methods
The coarse-grained microgels used in this work are preparedas in Refs. [27] starting from N patchy particles of diameter σ , which sets the unit of length, confined in a sphericalcavity. A fraction c = 0 .
05 of these particles has fourpatches on their surface to mimic the typical crosslinkerconnectivity in a chemical synthesis, while all the othershave two patches to represent monomers in a polymer chain.During the assembly, an additional force is employed onthe crosslinking particles to increase their concentration inthe core of the microgel in agreement with experimentalfeatures [18]. Once a fully-bonded configuration is reached(when the fraction of formed bonds is greater than 0 . V WCA ( r ) = (cid:40) (cid:15) (cid:104)(cid:0) σr (cid:1) − (cid:0) σr (cid:1) (cid:105) + (cid:15) if r ≤ σ (cid:15) sets the energy scale and r is the distancebetween the centers of two beads. Additionally, bondedparticles interact via the Finitely Extensible NonlinearElastic potential (FENE), V FENE ( r ) = − (cid:15)k F R ln (cid:34) − (cid:18) rR σ (cid:19) (cid:35) if r < R σ , (2)with k F = 15 which determines the stiffness of thebond and R = 1 . − e ∗ to a given fraction f of microgelmonomers, to mimic the acrylic acid dissociation in water,where e ∗ = √ πε ε r σ(cid:15) is the reduced unit charge (whichroughly corresponds to the elementary charge e , considering (cid:15) ≈ k B T at room temperature and σ as the polymer’s Kuhnlength), and ε and ε r are the vacuum and relative dielectricconstants. Accordingly, we insert in the simulation box anequal number of positively charged counterions with charge e ∗ to ensure the neutrality of the system. Interactionsamong charged beads are given by the reduced Coulombpotential V coul ( r ) = q i q j σe ∗ r (cid:15), (3)where q i and q j are the charges of counterions or chargedmonomers. We adopt the particle-particle-particle-mesh method [37] as a long-range solver for the Coulombinteractions. As discussed in a previous contribution [33],the size of the counterions is set to 0 . σ to facilitate theirdiffusion within the microgel network and to avoid spuriousexcluded volume effects. They interact with all otherspecies simply via the WCA potential.The swelling behaviour of a thermoresponsive microgel canbe reproduced in molecular dynamics simulations either bymeans of an implicit solvent, namely by adding a potentialthat mimics the effect of the temperature on the polymer, orby explicitly adding coarse-grained solvent particles withinthe box. In the first case, we employ a solvophobic potential V α ( r ) = − (cid:15)α if r ≤ / σ α(cid:15) (cid:110) cos (cid:104) γ (cid:0) rσ (cid:1) + β (cid:105) − (cid:111) if 2 / σ < r ≤ R σ r > R σ (4)with γ = π (cid:16) − / (cid:17) − and β = 2 π − γ [38].This potential introduces an effective attraction amongpolymer beads, modulated by the parameter α , whoseincrease corresponds to an increase in the temperatureof the dispersion. For α = 0 no attraction is present,which corresponds to fully swollen, i.e. low-temperature,conditions. For neutral microgels, the VPT is encounteredat α ≈ .
65, while a full collapse is usually reached for α (cid:38) . Model I all monomersexperience a total interaction potential where V α (Eq. 4)is added to the Kremer-Grest interactions, as previouslydone in Ref. [33]; in Model II only neutral monomers ex-perience this total interaction potential, while the chargedmonomers do not interact with V α , i.e. α = 0 for them inall cases. This second situation is equivalent to leaving un-altered the behaviour of charged groups of the microgel asthe solvent conditions change, so that they always retain agood affinity for the solvent ( α = 0). A similar treatment isalso adopted for the counterions, for which α = 0 for both Model I and
Model II .In the second part of this work, we explicitly consider thepresence of the solvent in driving the Volume Phase Tran-sition. Solvent particles are modelled within the Dissipa-tive Particle Dynamics (DPD) framework in order to avoidspurious effects which may arise from the use of a standardLennard-Jones potential due to the excessive excluded vol-ume of the solvent [39]. In the DPD scheme, two particles i and j experience a force (cid:126)F ij = (cid:126)F Cij + (cid:126)F Dij + (cid:126)F Rij , where: (cid:126)F
Cij = a ij w ( r ij )ˆ r ij (5) (cid:126)F Dij = − γw ( r ij )( (cid:126)v ij · (cid:126)r ij )ˆ r ij (6) (cid:126)F Rij = 2 γ k B Tm w ( r ij ) θ √ ∆ t ˆ r ij (7)where (cid:126)F Cij is a conservative repulsive force, with w ( r ij ) =1 − r ij /r c for r ij < r c and 0 elsewhere, (cid:126)F Dij and (cid:126)F
Rij are adissipative and a random contribution of the DPD, respec-tively; γ is a friction coefficient, θ is a Gaussian randomvariable with average 0 and unit variance, and ∆ t is the harge affinity and solvent effects in numerical simulations of ionic microgels integration time-step. We set r c = 1 . σ and γ = 2 . a i,j quantifies the repulsion betweentwo particles i and j , which effectively allows the tuning ofthe monomer-solvent (m,s) and solvent-solvent (s,s) inter-actions. Following our previous work [39], we fix a s,s = 25 . a m,s ≡ a between 5.0 and 16.0, that is therange where the collapse of a neutral microgel takes place.The reduced DPD density is set to ρ s r c = 3 . ρ s theactual number density of solvent beads). With this choiceof parameters, we previously showed that this model re-produces the swelling behaviour and structural propertiesof a neutral microgel particle, in quantitative agreementwith the implicit solvent model that was explicitly testedagainst experiments [18]. To compare the explicit solventmodel with the implicit one, we only consider Model II ,where charged monomers always retain a high affinity forthe solvent. We will show later that, in the explicit treat-ment, this translates to having charged monomers-solventinteractions (ch,s) always set to a ch,s = 0. All other inter-actions are identical to the implicit solvent model.Simulations are performed with LAMMPS [40]. The equa-tions of motion are integrated with the velocity-Verlet al-gorithm. All particles have unit mass m , the integrationtime-step is ∆ t = 0 . (cid:112) mσ /(cid:15) and the reduced tempera-ture T ∗ = k B T /(cid:15) is set to 1.0 by means of a Nos`e-Hooverthermostat for implicit solvent simulations or via the DPDthermostat for explicit solvent ones. In the former case thenumber of monomers in the microgels is fixed to N ≈ N ≈ P ( q ) = (cid:42) N N (cid:88) i,j =1 exp ( − i(cid:126)q · (cid:126)r ij ) (cid:43) , (8)where r ij is the distance between monomers i and j ,while the angular brackets indicate an average overdifferent configurations and over different orientations ofthe wavevector (cid:126)q (for each q we consider 300 distinctdirections randomly chosen on a sphere of radius q ).Also, we determine the radius of gyration R g of themicrogels as a measure of their swelling degree. This iscalculated as R g = (cid:42)(cid:34) N N (cid:88) i =1 ( (cid:126)r i − (cid:126)r CM ) (cid:35) (cid:43) , (9)where (cid:126)r CM is the position of the center of mass of themicrogel.For each swelling curve, representing R g as a function ofthe effective temperature α (implicit solvent) or a (explicitsolvent), we define an effective VPT temperature, either α VPT or a VPT , as the inflection point of a cubic splineinterpolating the simulation points.Finally, the radial density profile for all monomers is definedas ρ ( r ) = (cid:42) N N (cid:88) i =1 δ ( | (cid:126)r i − (cid:126)r CM | − r ) (cid:43) . (10) By restricting the sum in Eq. 10 to only charged monomersor to counterions, we also calculate ρ CH ( r ) and ρ CI ( r ), thatare the radial density profiles of charged microgel monomersand of counterions, respectively. In addition, we define thenet charge density profile as ρ Q ( r ) = − ρ CH ( r ) + ρ CI ( r ) . (11)
3. Results and Discussion
On the role of the affinity of chargedmonomers for the solvent
We start by discussing theinfluence of charges on the VPT for microgels with N ≈ Model I )to the case where it remains unchanged (
Model II ).Representative simulation snapshots of the two modelsfor the highest value of charge fraction investigatedin this work ( f = 0 .
2) are reported in Fig. 1. Herewe focus on different swelling stages of the microgelsupon increasing α . We notice immediately that a largeamount of inhomogeneities persists in Model II at large α , in contrast with the behavior of Model I where afull collapse is achieved. This can be better quantifiedby the swelling curves, reporting the variation of theradius of gyration R g versus the effective temperature α , that are shown in Fig. 2 for different values of thecharge fraction f . For both models we observe thatthe increase of f shifts the transition towards largereffective temperatures, but important differences ariseat large α , as displayed in the snapshots. In ModelI , where charged beads experience Coulomb as well assolvophobic interactions, the VPT occurs at all studied f , as shown in Fig. 2(a). Using the α -temperaturemapping established in Ref. [18] through a comparisonto experiments, the VPT temperature observed for f = 0 . T ≈ ◦ C.However, experiments on ionic microgels, for which theamount of charges was systematically varied [41, 26],have shown that even for values of f smaller than 0 . ◦ C.As hypothesized in our earlier work [33],
ModelI neglects the interplay between the hydrophiliccharacter of the co-polymer and its charge content.However, charged or polar groups, such as AAcgroups, are known to remain hydrophilic even at hightemperatures [42], which would increase the stabilityof the microgel in the swollen state with increasing f . We thus incorporate such a feature in ModelII by removing solvophobic interactions for chargedmicrogel beads. The resulting swelling curves, shownin Fig. 2(b), clearly demonstrate that for f = 0 . α that would harge affinity and solvent effects in numerical simulations of ionic microgels α Figure 1. Simulation snapshots.
Ionic microgels with f = 0 . N ≈ Model I and(bottom)
Model II for α = 0 , .
74 and 1 . α R g [ σ ] f = 0 f = 0.032 f = 0.2 Model I α Model II (a) (b)
Figure 2. Swelling curves.
Radius of gyration R g as afunction of the solvophobic parameter α and different values of f for microgels with N ≈ Model I )and (b) have always a good affinity for the solvent (
Model II ). correspond to temperatures above 50 ◦ C, in qualitativeagreement with experimental observations [41, 26, 43].
It is now important tocompare the two models from the structural pointof view, to check whether major differences arise.We start the analysis by looking at the form factorswhich, in our previous work on
Model I [33], wereshown to exhibit novel features with respect to neutralmicrogels. In particular, we found evidence thatfor α < α
VPT , the standard fuzzy-sphere-like modelwas not able to describe the numerical form factors,Instead, the emergence of two distinct power-lawbehaviours was found immediately after the first peak,at intermediate and high q values, respectively [33].This was attributed to the presence of charges in theinhomogeneous structure of the microgel, which givesrise to different features for core and corona regions,each being characterized by a different domain size.It is now crucial to verify whether such distinctivebehaviour also persists when the interactions amongcharged beads are modelled more realistically.Fig. 3 reports the form factors for Models I and II with f = 0 .
032 and f = 0 .
2, in comparison to the harge affinity and solvent effects in numerical simulations of ionic microgels -1 P ( q ) small α large α -1 -1 P ( q ) f = 0Model IModel II -1 -1 q [ σ -1 ] -1 -1 f = . f = . Figure 3. Form factors.
Form factors for charged microgels with N ≈ f = 0 .
032 and (bottom) f = 0 .
2, simulatedin implicit solvent for
Models I and II . The models are compared at the same α : for f = 0 . α = 0 , . , . , . , .
4; for f = 0 . α = 0 , . , . , . , . f = 0) is also displayed for comparison. Straight linesin the central panel of the bottom row highlight the two power-law behaviours of the form factors at intermediate (full line) andhigh (dashed line) q values, that are present for both models, extensively discussed in Ref. [33]. ρ (r) [ σ - ] f = 0monomers, Model Imonomers, Model II small α large α ρ (r) [ σ - ] ions, Model Ic-ions, Model Iions, Model IIc-ions, Model II r [ σ ] Figure 4. Density profiles.
Top panels show the monomers density profiles for an ionic microgel with f = 0 . N ≈ r obtained in implicit solvent for Models I and II . Bottom panelsreport the ions and counterions (c-ions) density profiles for f = 0 . α :0 , . , . , . , . f = 0) is also displayed for comparison. neutral case ( f = 0), at different values of α . For f = 0 . α ,the form factor is that of a collapsed microgel in all cases, as expected from the swelling curves in Fig. 2.For f = 0 . α , the behaviour ofthe two models is again very similar, with the formfactors of ionic microgels showing a first peak thatis systematically smaller with respect to that of the harge affinity and solvent effects in numerical simulations of ionic microgels r [ σ ] ρ (r) [ σ - ] ions, α =0.9Model Ic-ions, α =0.9Model Iions, α =1.4Model IIc-ions, α =1.4Model II r [ σ ] ρ (r) [ σ - ] monomers, α = 0.9, Model Imonomers, α = 1.4, Model II Figure 5. Comparison of
Models I and II at the same R g . Radial density profiles for an ionic microgel with f = 0 . N ≈ R g ≈
21, where α = 0 . α = 1 . ModelsI and II , respectively. The inset shows the corresponding ionsand counterions (c-ions) density profiles. neutral case. At intermediate α , we find that twopower-law-like behaviours are compatible with bothsets of data for charged microgels, while the neutralcase does not show such a feature. This finding, alreadyelaborated in Ref. [33], appears to be a distinctivefeature of our numerical model of ionic microgelsand is the result of the combination of a randomcharge distribution within a disordered, heterogeneousnetwork topology with the explicit treatment of ionsand counterions. Such a distinctive feature wastentatively attributed to the different degree of swellingof the corona and of the core, but still awaits a directexperimental confirmation. However, hints of a similartwo-step decay for P ( q ) were reported in Ref. [44] andwould certainly deserve further investigation in futureexperiments.On the other hand, major differences between the twocharged models arise for large values of α . Indeed, in Model I the microgel approaches and crosses the VPTleading to a fully collapsed state, while in
Model II itremains in a quasi-swollen configuration for all studied α . Consequently, for high α values, the form factordoes not resemble that of a homogeneous sphere, withonly a second peak becoming evident, as opposed tothe neutral case where many sharp peaks emerge. Wenotice that Model I fully coincides with the neutralcase for very large α , even for f = 0 . α . Thesedata further indicate that, for Model II , the microgeldoes not achieve a collapsed state, as also visible fromthe behaviour of the profiles of charged monomersand of counterions, respectively. These are reportedin the bottom panels of Fig. 4, showing that, forboth models, the counterions are always found to be very close to the charged monomers, in order toneutralize the overall charge of the microgel. However,all profiles remain much more extended for
ModelII as compared to
Model I , for all α . We stressthat the comparison is performed for microgels withdifferent affinity of the charged monomers for thesolvent at the same α , which corresponds to verydifferent swelling conditions, as evident from Fig. 2.Additional information can be extracted by comparingthe two cases for a similar value of R g , as reportedin Fig. 5. Also in this case, we find that Model II displays a more slowly decaying radial profile, albeithaving a very similar mass distribution with respect to
Model I , which is due to the presence of more stretchedexternal dangling chains. Similar results also apply toions and counterions profiles, that are shown in theinset of Fig. 5: even at the same R g , there is a surplusof charges at the surface in the case where the affinityof charged monomers for the solvent does not changewith the effective temperature ( Model II ). Overall,these findings confirm an enhanced stabilization ofthe swollen configuration operated by the chargedgroups of the microgel, hindering the tendency of theremaining (neutral) monomers to collapse.To complete the structural analysis of the twomodels, it is instructive to consider the net chargedensity profile inside the microgels, that is reportedin Fig. 6 for both f = 0 .
032 and f = 0 .
2. We confirmthat, for both models, the net charge of the core regionis roughly zero. However, it was shown in Ref. [33] thatin the collapsed configuration a charged double layerarises at the surface of microgels, signalling the onsetof a charge imbalance that grows with α . This feature,that is clearly visible in the behaviour of Model I athigh α for all values of f , is also present for Model II forthe low charge case ( f = 0 . f = 0 .
2, due to the fact that, up to the largest exploredvalues of α , the microgel does not fully collapse. Inthis way, it maintains a low concentration of chargedbeads, that is always roughly balanced by counterions,resulting in a rather uniform charge profile. Instead,the peaks at the surface appear when the microgelcollapses: this is indeed the case for both models atlow charge fraction and even for large f when chargedmonomers are assigned a solvophobic behaviour ( ModelI ). We conclude from this analysis that the hy-drophilicity of the charged monomers at all effectivetemperatures enhances the tendency of the microgel toremain swollen, even when most of the monomers ex-perience a very large solvophobic attraction. Thanksto the charge neutralization operated by counterions,the microgel remains very stable in a rather swollenconfiguration up to very large α , avoiding collapse for harge affinity and solvent effects in numerical simulations of ionic microgels -0.04-0.0200.020.04 ρ Q (r) [ e * σ - ] small α large α Model IModel II r [ σ ] f = . f = . Figure 6. Charge density profile.
Net charge density profile ρ Q ( r ) as defined in Eq. 11 for ionic microgels with N ≈ f = 0 .
032 , (bottom) f = 0 .
2, as a function of the distance from the microgel center of mass r , simulated in implicit solvent for Models I and II . The models are compared at the same α : for f = 0 . α = 0 , . , . , . , .
4; for f = 0 . α = 0 , . , . , . , . large enough values of f . This scenario agrees wellwith experimental observations, where the suppressionof the VPT [26, 43, 42] is found when the concentra-tion of charged hydrophilic groups in the polymer net-work is large enough. These considerations imply that Model I should not be used to describe microgels withhigh charge content. Indeed, its identical treatment ofthe solvophilic character of both neutral and chargedmonomers leads the particle to collapse at extremelyhigh α . Incidentally, we report that this was observedalso for unrealistic values of f up to 0 . Model II in the future to correctly incorporate chargeeffects in modelling microgels in a realistic fashion.
Solvent effects
We now go one step further in modelling ionicmicrogels, by explicitly adding the solvent to thesimulations. This is a necessary prerequisite to tacklephenomena that cannot be described with an implicitsolvent, e.g. situations in which hydrodynamics orsurface tension effects at a liquid-liquid interface [34]play a fundamental role. In this subsection, wecompare results for swelling behaviour and structuralproperties of the microgels for implicit and explicitsolvent simulations. In particular, we restrict ourdiscussion to
Model II , having established this to bemore in line with experimental observations. Sincesimulations with an explicit solvent require a muchhigher computational effort, we limit the followingdiscussion to microgels with N ≈ a - α )mapping We start by reporting the swelling curvesof charged microgels, stressing the point that theyhave been obtained by fixing the value of a ch,s , whichtunes the solvophilic properties of charged beads andcounterions. We find that setting a ch,s = 0, while a m,s ≡ a varies, the explicit model is essentiallyequivalent to the implicit one. This means thatit is possible to find a relation that links everyimplicit system with a certain value of the solvophobicattraction α to an explicit one with solvophobicparameter a that shows the same structure andswelling properties.In order to establish such a a - α mapping, weexplored two different routes. The first one, referredto as linear mapping in the following, is based on theassumption that the dependence of a on α is linear,as previously adopted for neutral microgels [39]. Inthis way, the mapping relation is obtained througha horizontal rescaling of the relative swelling curves R imp g ( α ) /R imp g ( α = 0) and R exp g ( a ) /R exp g ( a = 0) for theneutral implicit and explicit microgels onto each other.Specifically, given two points for each curve, ( a , a )and ( α , α ), the rescaled x -coordinate is calculatedusing the following relationship: a lin ( α ) = ( α − (cid:104) α (cid:105) ) ∆ a/ ∆ α + (cid:104) a (cid:105) (12)where (cid:104) x (cid:105) = 0 . x + x ) and ∆ x = x − x with x = a, α . The second mapping a num ( α ), referred to asnumerical mapping, has been obtained by numericallyinverting the equation R imp g ( α ) /R imp g ( α = 0) = R exp g ( a ) /R exp g ( a = 0) , (13) harge affinity and solvent effects in numerical simulations of ionic microgels R g / R g , m a x implicit (lin)implicit (num)explicit f = 0 a [ ε / σ ] [ ε / σ ] f = 0.032 f = 0.1 f = 0.2 α a linear mappingnumerical mapping (c) (d) (e)(a) (b) Figure 7. Implicit-explicit solvent mapping and swelling curves. (a) Mapping between α and a obtained by comparingneutral microgels with implicit ( Model II ) and explicit solvents: the linear mapping is expressed by Eq. 12 and the numericalmapping via Eq. 13; (b-e) Normalized radius of gyration R g /R g,max as a function of the swelling parameter a for microgels withdifferent charge content: (b) neutral, (c) f = 0 . f = 0 . f = 0 .
2, for explicit (full lines and filled diamonds) andimplicit solvent conditions (rescaled along the horizontal axis using the linear mapping a lin ( α ), dashed lines and empty squares, andusing the numerical mapping a num ( α ), full lines and filled squares). The present figure and the following ones refer to the samemicrogel topology with N ≈ after spline fitting the two swelling curves. We reportboth mapping relations in Fig. 7(a), finding that theyfall onto each other for almost the entire range ofinvestigated solvophobic parameters in the two models,confirming the overall correctness of the assumptionof linearity. However, we find some differences in theregion α > . a > f without any further adjustments. The normalized swellingcurves with varying charge fraction f , comparingimplicit and explicit solvent, are reported in Fig. 7(b-e). Data from implicit simulations are mapped viaboth methods described above. For the neutral case,the presence of the solvent does not affect the swellingbehaviour, as shown in Fig. 7(b), where no appreciabledifferences are found between linear and numericalmapping even at high α . Using the same relations forcomparing charged microgels in explicit and implicitsolvent, we find that, remarkably, the same swellingbehaviour works for all charge contents. The swellingcurves are virtually identical, which ensures that theinclusion of the solvent does not alter the microgelbehavior in temperature even in the presence ofcharges. Small deviations, as expected from Fig. 7(a),appear only at large α values, being more pronouncedfor high charge content. This confirms the robustnessof the DPD model which, as already discussed inRef. [39], does not induce spurious effects, e.g. dueto excluded volume, even in the collapsed state. Animportant result of this work is that, even in thepresence of an explicit solvent, the microgel at high f does not fully collapse at large α , being entirelyequivalent to implicit Model II and compatible withexperimental findings.
In this subsection, we willshow that the implicit and explicit solvent treatmentswith the newly established numerical mapping (Eq. 13)lead to identical structural features of the microgels.Small differences arise when using the linear mapping(Eq. 12) at high f and large values of α .We show in Fig. 8 the monomer (top panels), ionand counterion (middle panels) and charge (bottompanels) density profiles only for the f = 0 . f = 0 . a or α , when using the linearmapping: as we can observe from the rightmost panelsof Fig. 8, the linear mapping fails to associate implicitand explicit states in the most collapsed state, wherea visible difference arises between the profiles.The distribution of ions and counterions withinthe microgel is an observable that should be moresensitive to the presence of the solvent. However,quite remarkably, also in this case, we find excellentagreement between the two models, as shown in themiddle panels of Fig. 8. In particular, the emergenceof a clear double-peak structure in the ion distributionis found in both models for large α (implicit) and a (explicit), signalling an accumulation of ions atthe exterior surface of the microgels. This can beunderstood from the fact that ions, remaining alwayshydrophilic, never completely collapse onto the coreof the particle. Thus, the appearance of a peak atdistances corresponding to the outer region of themicrogel is the result of an attempt of ions to maximizetheir contact with solvent. This is preceded by aminimum, which indicates a region where ions are harge affinity and solvent effects in numerical simulations of ionic microgels ρ (r) [ σ - ] m ono m e r s i on s , c oun t e r i on s n e t c h a r g e implicit (lin)implicit (num)explicit swollen VPT collapsed ρ (r) [ σ - ] ions, imp (lin)ions, imp (num)ions, exp c-ions, imp (lin)c-ions, imp (num)c-ions, exp ρ Q (r) [ e * σ - ] r [ σ ] Figure 8. Density profiles.
Density profiles of monomers (top row), charged beads and counterions (middle row) and net charge(bottom row) for ionic microgels with f = 0 . r obtained in explicitand implicit solvent conditions. Curves from the explicit case refer to values of a = 5 , , . , ,
16, from the (left) swollen to the(right) collapsed state. Implicit and explicit solvent cases are compared at values of α approximately corresponding to each a valueaccording to both the linear ( α = 0 , . , . , . , .
1) and the numerical ( α = 0 , . , . , . , .
2) mapping. depleted within the particle.This feature is the echo of the minimum thatarises in the net charge density distributions, alreadyanticipated for the large microgel treated with theimplicit model in Fig. 6. Importantly, a minimumalso occurs in ρ Q ( r ) for smaller microgels, as shownin the bottom panels of Fig. 8, for the most collapsedconditions. Here a charged double layer is clearlypresent, with an excess of positive charges insidethe microgel corona due to the increased amount ofcounterions in this region. At the same time, anegative charge surplus is found at the surface ofthe microgel, since charged ions preferably remainin contact with solvent particles. The net chargedistribution is also identical for explicit and implicitsolvent when using the numerical mapping, with againvery small differences arising for the linear mapping atlarge α .It is important to notice that, although a doublelayer was also observed with the implicit solvent inRef. [33] (equivalent to Model I ), the two distributions(the one in Fig. 8 of the current manuscript and thatreported in Fig. 6 of Ref. [33]) have opposite signs.Indeed in Ref. [33] the superposition of electrostaticand solvophobic effects led to an accumulation of counterions at the microgel surface, with the onset of aseemingly Donnan equilibrium [45]. Notwithstandingthe different origin of the double layer, both modelsdemonstrate that an almost perfect neutrality isachieved within the core of the microgel, and it isonly at the surface that inhomogeneous distributionsappear. Besides, the reduced size of the microgelsstudied with the explicit solvent facilitates the onsetof peaks due to the increased surface-to-volume ratioof the microgels. A more precise assessment of sizeeffects and a careful comparison to experiments will bethe subject of future works.Finally, the explicit solvent model allows usto quantify the amount of solvent that is locatedinside the microgel as temperature increases. Thisis illustrated in Fig. 9, where the solvent densityprofile ρ s ( r ) is reported for different values of a andall investigated charge fractions. These plots confirmthe reduced tendency to collapse of charged microgelswhich retain a large amount of solvent within thenetwork structure. No inhomogeneities within themicrogel are in general observed. At large f and α some oscillations arise which may be due to reducedstatistics. Finally, this study confirms that evenat temperatures above the VPT there is quite a harge affinity and solvent effects in numerical simulations of ionic microgels r [ σ ] ρ s (r) [ σ - ] f = 0 f = 0.032 f = 0.1 f = 0.2small a large a Figure 9. Solvent density profiles for charged microgels of different f values, as a function of the distance from the microgelcenter of mass r . The different panels refer to a = 5 , , , ,
16 from (left) good to (right) bad solvent conditions. residual amount of solvent within the microgel, that issignificantly enhanced by increasing the charge. Thesefindings are in line with expectations [43, 46], that arethus confirmed by our simulations.
4. Conclusions
In this work we report an extensive numerical studyof single microgel particles, a prototype of soft colloidsthat is of great interest for the colloidal community,particularly for the formation of arrested states withtunable rheological properties [9], including glasses [47,48] and gels [49]. The use of different polymers withinthe microgel network allows to exploit responsivenessto different control parameters, such as temperatureand pH, giving rise also to unusual responses in thefragility of the system [47, 50, 16].In order to be able to model dense suspensionsof these soft particles, we can rely on two possiblestrategies. On one hand, we can exploit highly coarse-grained models, such as the Hertzian one, whichcompletely neglect the polymeric degrees of freedom ofthe particles and thus cannot reproduce the complexphenomenology observed in experiments in the gelor glassy regimes, such as shrinking, faceting andinterpenetration [51, 15]. On the other hand, we cantry to model a single microgel in a realistic way, aimingto reproduce its structural properties and, from this, tobuild effective interactions which retain the polymericfeatures of the single particle.Adopting the second strategy, the aim of thiswork is to improve the current numerical modellingof single ionic microgels with randomly distributedcharged groups, aiming to describe PNIPAM-co-PAAc microgels across the Volume Phase Transition.In particular, we assess two different ways tomodel the interactions of the charged monomersbelonging to the polymer network, either consideringor not a solvophobic attraction that mimics theirhydrophilic/hydrophobic interactions. We find that, aslong as the charged groups maintain the same affinity for the solvent, the tendency of the microgel to remainin swollen conditions is enhanced even at high effectivetemperatures. Thus, for a charge fraction of f = 0 . α , is replaced by the DPD repulsive in-teractions between monomers and solvent. The latteris varied through a change of the parameter a control-ling the repulsion between non-charged beads, whilethe interaction between charged monomers always re-tains a solvophilic nature.We have thus carried out a careful comparisonbetween explicit and implicit solvent treatments,finding quantitative agreement between the two.Interestingly, the relation among a and α establishedby the comparison of neutral microgels can be used harge affinity and solvent effects in numerical simulations of ionic microgels f (some deviations occur only at f = 0 . α, a values), where the same correspondencebetween implicit and explicit solvent states is retrieved.We showed that a linear mapping between the twocontrol parameters of the interactions in the implicitand explicit case is sufficient to obtain a very goodagreement between the two descriptions.From our analysis of the internal structure of themicrogels across the VPT, we found that counterionshave a rather similar distribution within the microgelcore, effectively neutralizing the internal charge atsmall distances, but being in excess close to the surface.This gives rise to a charged double layer for largevalues of a and α . Interestingly, such peaks in thecharge density distributions are swapped with respectto the case of Model I , where ions do not experience atendency to remain at the surface, since they are alsotreated as solvophobic. These detailed predictions willhave to be compared to experiments on ionic microgelsas a function of charge fraction, pH and T , in orderto establish the limit of validity of our model and tofurther improve it, towards a more realistic descriptionof experimental microgels.In perspective, this work paves the way to studyrealistic charged microgels in diffusing conditions,such as in electrophoresis and thermophoresis exper-iments [52], or at liquid-liquid interfaces and to calcu-late their effective interactions, similarly to what hasbeen done for neutral microgels [34, 35]. In this way, wewill be able to determine the conditions under whichelectrostatic effects play a dominant role over elasticones. Another important line of research will be the as-sessment of the role of the network topology: examplesof interesting cases whose properties could be investi-gated are microgels consisting of two interpenetratednetworks [50, 53] or ultra-low crosslinked [54, 55] andhollow [56, 57] microgels. Finally, we hope that ourtheoretical efforts will stimulate further experimentalactivity on charged microgels to verify the predictedbehaviour so that it will be possible to tackle the in-vestigation of dense suspensions in the near future. Acknowledgments
This research has been performed within the PhDprogram in ”Mathematical Models for Engineering,Electromagnetics and Nanosciences”. We acknowledgefinancial support from the European Research Council(ERC Consolidator Grant 681597, MIMIC). FC andEZ also acknowledge funding from Regione Lazio,through L.R. 13/08 (Progetto Gruppo di RicercaGELARTE, n.prot.85-2017-15290).
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