Interface-induced hysteretic volume phase transition of microgels: simulation and experiment
Jannis Kolker, Johannes Harrer, Simone Ciarella, Marcel Rey, Maret Ickler, Liesbeth M. C. Janssen, Nicolas Vogel, Hartmut Löwen
IInterface-induced hysteretic volume phase transition ofmicrogels: simulation and experiment
Jannis Kolker, ∗ a Johannes Harrer, ‡ b Simone Ciarella, c Marcel Rey, b Maret Ickler, b LiesbethM. C. Janssen, c Nicolas Vogel, b and Hartmut Löwen a Thermo-responsive microgel particles can exhibit a drastic volume shrinkage upon increasing the sol-vent temperature. Recently we found that the spreading of poly(N-isopropylacrylamide)(PNiPAm)microgels at a liquid interface under the influence of surface tension hinders the temperature-inducedvolume phase transition. In addition, we observed a hysteresis behavior upon temperature cycling,i.e. a different evolution in microgel size and shape depending on whether the microgel was initiallyadsorbed to the interface in expanded or collapsed state. Here, we model the volume phase transi-tion of such microgels at an air/water interface by monomer-resolved Brownian dynamics simulationsand compare the observed behavior with experiments. We reproduce the experimentally observedhysteresis in the microgel dimensions upon temperature variation. Our simulations did not observeany hysteresis for microgels dispersed in the bulk liquid, suggesting that it results from the distinctinterfacial morphology of the microgel adsorbed at the liquid interface. An initially collapsed micro-gel brought to the interface and subjected to subsequent swelling and collapsing (resp. cooling andheating) will end up in a larger size than it had in the original collapsed state. Further temperaturecycling, however, only shows a much reduced hysteresis, in agreement with our experimental obser-vations. We attribute the hysteretic behavior to a kinetically trapped initial collapsed configuration,which relaxes upon expanding in the swollen state. We find a similar behavior for linear PNiPAmchains adsorbed to an interface. Our combined experimental - simulation investigation provides newinsights into the volume phase transition of PNiPAm materials adsorbed to liquid interfaces. a Institut fur Theoretische Physik II, Heinrich-Heine-Universität Düsseldorf, Univer-sitätsstrasse 1, D-40225 Düsseldorf, Germany. b Institute of Particle Technology, Friedrich-Alexander University Erlangen-Nürnberg,Cauerstrasse 4, 91058 Erlangen, Germany. c Theory of Polymers and Soft Matter, Department of Applied Physics, Eindhoven Uni-versity of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands.
Soft microgels consisting of a swollen polymer network possessfascinating properties. Their soft nature allows them to undergosignificant deformations, for example when adsorbed to an inter-face . In addition, stimuli-responsive properties can be encodedin their molecular structure, allowing microgels to undergo sharptransitions between swollen and collapsed states upon an exter-nal stimulus . This dynamic behavior has important conse-quences. From an applied side, stimuli-responsive microgels serveas dynamic and reversible emulsion stabilizers or vehicles fordrug-delivery, due to their ability to take up and release moleculeson demand . From a fundamental side, soft microgels serve asmodel systems for classical many-body systems to model atom-istic physical phenomena such as melting of crystal lattices ,solid-solid phase transitions , or complex self-assembly .Among the most prominent microgel systems are crosslinkedpoly(N-isopropylacrylamide) (PNiPAm)-based microgels, whichexhibit a drastic volume shrinkage in aqueous dispersion above ◦ C . During this volume phase transition, nonpolar groupsof the PNiPAm microgel aggregate and bound water is expelledfrom the macromolecule to increase the entropy of the sys- a r X i v : . [ c ond - m a t . s o f t ] F e b em . This volume phase transition has been extensivelyexplored through experimental studies , computer simula-tions and statistical theory .However, in many actual applications such as emulsion sta-bilization or surface patterning microgel particles areexposed and attracted to a liquid interface, where the spatialisotropy of the microgel is broken and a more complex behav-ior results. Therefore it is important to understand their struc-tural and dynamical properties when confined to a liquid inter-face. In fact, it was shown that at liquid interfaces, microgels de-form into a fried-egg-like structure where the core pos-sesses a disk-like shape and the surrounding dangling polymerchains form a quasi two-dimensional layer close to the interface,termed the corona. Correspondingly, the volume phase transitionis strongly modified due to the presence of the interface. Recentexperiments have analyzed the volume phase transition atthe interface and showed that the volume phase transition in thelateral direction is strongly hindered by the presence of the liq-uid interface. In addition, a hysteresis behaviour upon swellingand deswelling of microgels dependent on whether the microgelswere deposited in the swollen or collapsed state was found .Though monomer-resolved computer simulations have consid-ered the microgel structure at an interface , the hystereticbehaviour was not yet reproduced in computer simulations.Here we present extensive Brownian dynamics simulations toconfirm and investigate the hysteresis on a monomer-resolvedscale. The microgel particle is modelled by a coarse-grained poly-mer network with effective beads . The interface is describedby an external effective potential attractive for the beads. Withinour simulations we are able to reproduce and quantify the hys-teresis behavior and the corresponding structural change of themicrogel particle. In our protocol the microgel particle is firstequilibrated in the bulk either in the collapsed (high temperature)or in the swollen (low temperature) state and then brought to theinterface and "equilibrated" there again. Figure 1 schematicallyoutlines the temperature cycling experiments used for this inves-tigation. A swollen particle adsorbed to the interface respondsreversibly to further collapsing and swelling (resp. heating andcooling) cycles. An initially collapsed particle adsorbed to theinterface, however, experiences structural changes upon temper-ature cycling. Instead of reversible regaining its initial size, themicrogel remains significantly more expanded after undergoinga temperature cycle. We attribute this hysteretic behaviour to aninitial kinetically trapped collapsed state. This behavior is purelyinduced by the interface, as a similar temperature cycling of abulk microgel does not show any hysteresis. We investigate theseinterfacial volume phase transitions as a function of crosslinkerdensity. In both experiments and simulations, we find that lesscrosslinked microgels exhibit a larger hysteresis. In addition, weshow that even linear PNiPAm chains adsorbed to an interfaceundergo a hysteretic behavior upon temperature cycling, indicat-ing that the hysteresis does not only depend on the architectureof the microgel, but originates from the molecular nature of thepolymer itself. temperature initial collapsed state temperature collapsedstateswollenstate c oo li ng hea t i ng bulk interface reversible hysteretic volume surface area interfacial swollen state . c oo li ng . c oo li ng hea t i ng Fig. 1
Schematic visualization of the interface-induced hysteresis effect(right) compared to the bulk behavior (left). Bulk: reversibility betweenthe swollen and collapsed state within a cooling-heating cycle for a micro-gel particle. Interface: hysteresis in the first cooling-heating cycle of aninterface-adsorbed microgel particle. For subsequent cycles reversibilityoccurs.
We adopt the modelling proposed in Ref. describing amicrogel particle on a monomer-resolved level with no explicitsolvent. In detail, the microgel particle consists of two types ofbeads, monomers and crosslinkers, which define its internal ar-chitecture. A monomer is covalently linked to either a neigh-bouring other monomer or to a crosslinker by springs and themaximum number of these bonds is two. A crosslinking bead, onthe other hand, has four of such bonds. In terms of all other in-teractions (next nearest neighbour interactions etc.) monomersand crosslinkers do not differ. The covalent bonds are describedby a finite-extensible-nonlinear-elastic (FENE) potential (seeSupporting Information) with a characteristic energy scale ε , amaximal bond expansion ˜ R = . σ and an effective spring con-stant ˜ k f = ε / σ . The remaining bead-bead interactions aremodelled by a repulsive Week-Chandler-Andersen (WCA) poten-tial given explicitly in the Supporting Information which con-tains the size σ of the repulsive monomers as a length and therepulsion strength ε , as the same energy scale as for the FENEpotential. Henceforth we choose σ and ε as units of length andenergy.To incorporate the thermoresponsivity of the microgel parti-cle effectively, a further attractive bead-bead pair potential isadded given by V α ( r ) = − αε if r ≤ σ αε (cid:104) cos (cid:16) γ (cid:0) r σ (cid:1) + β (cid:17) − (cid:105) if σ < r ≤ R σ otherwise (1)with γ = π (cid:16) . − (cid:17) − and β = π − . γ . Importantly, theeffective attraction strength is controlled by the parameter α ,which mimics the quality of the solvent in an implicit manner. Thevalue α = . describes a strong attraction (relative to the beadrepulsions) imitating poor solvent conditions and thereforemimicking the collapsed state of the microgel. For good solventconditions the value α = is used such that there is no attrac-tion at all, reflecting the swollen state. In our simulations,we varied the effective attraction strength α between the two ex-treme cases in the range ≤ α ≤ . and refer to the α = case s the "fully swollen state" and the α = . case as the "fully col-lapsed state". As the solvent quality is controlled by temperaturein the experiments, α = corresponds to a low temperature, be-low the volume phase transition of the PNiPAm microgels, and α = . to a high temperature situation above the volume phasetransition. The internal architecture of the microgel particle isas in Ref. and depends on the crosslinker density. In oursimulations we represent each microgel particle by a total numberof N = beads, which includes a fraction of homogeneouslydistributed crosslinkers. The percentage of crosslinkers is an im-portant parameter in our simulations and will be systematicallyvaried between 0 and . .To mimic the effect of an air-water interface, we follow Ref. and add an external potential; the potential is defined suchthat the interface normal is along the z direction. Explicitly, foreach bead, the external potential is a combination of an effectiveLennard-Jones potential on the water side and a steep linear po-tential on the air side. We introduce a typical range σ ext for theeffective bead-interface interaction and a shifted z -coordinate ˜ z by ˜ z = z − σ ext ; the derivative of the potential (i.e. the force) isdefined to be continuous at a matching point ˜ z a < . In detail, V ext ( ˜ z ) = V LJ ( ˜ z ) ˜ z ≥ ˜ z a V LJ ( ˜ z a ) + ( ˜ z a − ˜ z ) dV LJ ( ˜ z ) d ˜ z (cid:12)(cid:12)(cid:12) ˜ z = ˜ z a ˜ z < ˜ z a (2)with the Lennard-Jones potential V LJ ( ˜ z ) = ε ext (cid:34)(cid:18) σ ext ˜ z (cid:19) − (cid:18) σ ext ˜ z (cid:19) (cid:35) , (3)where ε ext is an attraction energy strength. Therefore the inter-face tries to pin all beads to the position ˜ z = where the poten-tial is minimal. Physically this attraction towards the interfaceresults from surface tension reduction by reducing the bare air-water interfacial area when a bead is adsorbed at the interface.The large difference in the bead chemical potential between theair and the water phase is reflected by the steep increase of V ext ( ˜ z ) for ˜ z < ˜ z a . In the following we have chosen ε ext = . ε , to ensurea sufficiently strong adsorption strength towards the interface,and σ ext = . σ corresponding to a relatively deep and stiff mini-mum around ˜ z = . A harmonic expansion around the origin ˜ z = yields a large spring constant of ε / σ which is enforcing a beadmonolayer as observed by the very thin corona formed by polymerchains expanding at the liquid interface. Finally we have chosenthe matching position z a , i.e. the point where the Lennard-Jonesbehavior changes to a constant force, as ˜ z a = − . σ .We simulate the bead dynamics as Brownian dynamics with animplicit solvent, whereby the short-time bead self-diffusion coef-ficient D sets a characteristic Brownian time scale τ B = D / σ .The latter defines our unit of time in the following. A finite timestep of ∆ t = . τ B is used to integrate the equations of mo-tion with an Euler forward scheme. All of the Brownian dynamicssimulations are performed with the HOOMD-Blue package andare visualized by OVITO .In our modelling it is important to distinguish between a sol-vent bath temperature T ∗ – which sets the Brownian fluctuations – and the implicit temperature influence on the effective bead-bead attraction strength α . Since the temperature change is smallcompared to the absolute room temperature, we have fixed thesolvent bath temperature to k B T ∗ = ε throughout all of our sim-ulations but changed solely the effective attraction strength α , inagreement with typical protocols in the literature .Our simulation protocol is as follows: first we equilibrate themicrogel particle in the bulk (i.e. in the absence of the interfaceexternal potential (2)) and thus gain an equilibrated initial beadconfiguration. This is done separately for the "fully swollen state"with α = and for the "fully collapsed state" with α = . . Wethen instantaneously expose the initial bead configuration to thepotential (2) such that the z -coordinate of the center of mass ofthe particle is a distance of σ apart from the interface positionat ˜ z = . We then relax the system for a long waiting time t w of typically · τ B . The relaxed configuration is – in the pres-ence of the interface – subject to a sudden change in the attrac-tion parameter α from 0 to 1.55 or, respectively from 1.55 to0. Physically this means that the solvent quality temperature isabruptly changed from high to low (or vice versa). We then al-low the system to undergo relaxation for another waiting time;this relaxation process is referred to as "collapsing" (if α has beensuddenly increased), or as "swelling" (if α has been suddenly de-creased). Finally we reverse α to its initial value, thus establishingone "cycle". After a third waiting time the resulting configurationis compared to the first configuration at the same α relaxed atthe interface from its initial bulk state. If there is a significantdifference in the extent of the configuration, the system is calledhysteretic. The protocol is repeated several times leading to a"cycling process" which is composed of alternating "swelling" and"collapsing" processes. Appropriate averages are taken to ensurethat the behavior does not suffer from peculiarities of the initialconfiguration.The diagnostics to identify hysteresis is done via monitoringthe lateral radius of gyration, R lat ( t ) , as a function of time t . Thelateral (or projected) radius of gyration is defined as R lat ( t ) = N N ∑ i = ( x i ( t ) − X ( t )) + ( y i ( t ) − Y ( t )) (4)with (cid:126) r i ( t ) = ( x ( t ) , y ( t ) , z ( t )) as the location of the bead i , and (cid:126) R ( t ) = ( X ( t ) , Y ( t ) , Z ( t )) = N ∑ Ni = (cid:126) r i ( t ) is the instantaneous cen-ter of the microgel particle. Finally, we can also infer the lateralosmotic pressure from the fraction of beads in the interfacial re-gion. Here, by definition, a bead is in the interfacial region if its ˜ z -coordinate lies between − . σ and . σ . We also analyze theinternal core-corona structure in more detail by using a "hull pa-rameter" ∆ ( r ) , which brings us also into a position to extract acore radius R c and an outer corona radius. Details of the proce-dure are described in the Supporting Information. PNiPAm microgels crosslinked by N,N’-Methylenebis(acrylamide)with different crosslinking densities are synthesized by pre-cipitation polymerisation according to a previously publishedprotocol . The cycling experiments involving the differently rosslinked microgels are conducted in a similar fashion as de-veloped in an earlier publication . When starting from the col-lapsed state, 0.05 w% microgels dispersed in an ethanol/watermixture (1:1) with a temperature of ◦ C are spread at theair/water interface on a Langmuir-Blodgett trough (KSV Nima),which is preheated to the initial temperature of ◦ C. Similarly,when starting from the swollen state, the microgel dispersion isspread at room temperature. The trough is heated using a ther-mostat by increasing the temperature by ◦ C every 10 minutes.Conversely, the trough is passively cooled by switching off thethermostat. The interfacial arrangement as a function of temper-ature is then deposited onto a silicon wafer (0.5 x 10 cm ) lifted at0.1 mm/min through the interface, while the corresponding sur-face pressure is measured using the Wilhelmy plate method. Themicrogels are characterized ex-situ by image analysis of scanningelectron microscopy (Zeiss Gemini 500) images of the transferredinterfacial layer. From these images, the size of the microgel coreis quantified by a custom-written Matlab software . First, as a reference, we simulate the bulk behavior of a microgel.Figure 2 shows the time evolution of the radius of gyration dur-ing swelling and collapsing together with typical initial and finalsimulation snapshots before and after the swelling and collapsingprocesses. The initial configurations (marked with b) and d)) areequilibrated for a long relaxation time of typically τ B , andare therefore practically fully relaxed. After a sudden increase (ordecrease) of effective attraction α (mimicking an increase or de-crease the temperature, respectively) the radius saturates quickly(i.e. within a relaxation time of roughly τ B ) indicating equi-libration. After a long relaxation for a simulation time of about τ B two final configurations marked with c) and e) in Figure 2are reached. These two final configurations c) and e) are similarto the initial conditions d) and b) in Figure 2, i.e. both swollenand both collapsed configurations exhibit a similar radius of gy-ration. Importantly, this implies that swelling and subsequent col-lapsing are completely reversible in this bulk experiment on theaccessible time scale of the simulation, or in other terms, thereis no hysteresis. This non-hysteretic bulk behavior in size is inagreement with experiments . Once the reversible behavior ofthe volume phase transition is established, one can trivially add acycle of subsequent swelling and collapsing processes, which al-ways results in an equilibrated state. Likewise, one can perform amulti-cycle process by adding many subsequent swelling and col-lapsing processes. Again one finds reversible behavior such thatafter each cycling process the system is in the same equilibratedstate. The latter feature will change significantly at interfaceswhich we shall show next. Next, a microgel with an equilibrated bulk configuration is placedat the interface in silico (as described in Materials and Methods)
Fig. 2
Volume phase transition of a microgel in bulk. Radius of gyration R g (in terms of the bead size σ ) as a function of time t for a micro-gel particle in the bulk for both a collapsing process (yellow line) and aswelling process (green line) with the corresponding error bars in blackas obtained by simulation. The processes start from equilibrated con-figurations (shown as typical simulation snapshots b) and d)). Typicalsnapshots c) and e) after a long simulation time of τ B are also given.There is no notable hysteresis in the bulk. The data are obtained for acrosslinking density of . . and relaxed there for a long initial relaxation time of τ B .Subsequently, a swelling and collapsing process is initiated byan instantaneous change in the solvent quality, encoded in theeffective attraction α . The evolution of the lateral radius of gyra-tion following this change of α is presented in Figure 3 for botha swelling and a collapsing event. Similar to Figure 2, typicalinitial and final snapshots are given in Figure 3 both with a topview onto the microgel and a lateral view from the side. The sideview clearly shows the oblate core-corona morphology ofthe microgel, which is induced by lateral stretching caused by thereduction of the bare water-air interfacial tension. This is alsoaccompanied by the fact that the lateral radius of gyration is sig-nificantly larger than its bulked value shown in Figure 2. Theimportant message taken from Figure 3 is the presence of a hys-teresis, observed from the difference in the collapsed dimensionsfor the two phase transition scenarios. In particular, the collapsedstate e), reached after a relaxation time of τ from an initiallyswollen microgel at the interface, is much less contracted thanthe starting collapsed configuration b) from the initial interfacialadsorption of a bulk collapsed microgel. This difference is clearlyevidenced by the different lateral radii of gyration, and is alsoconsistent with previous experimental findings .The hysteretic behavior found for the volume phase transitionat the interface in the collapsing process is in contrast to theswelling process where the swollen end state c) practically ex-hibits the same lateral radius of gyration as the initial state d) ofa microgel adsorbed to the interface from a swollen bulk confor-mation. The clue to understand this difference lies in the natureof the starting state b). In the collapsed state at the interface,theequilibration is kinetically hindered by the strong attractive inter-actions between the beads. Therefore we hypothesize that theyare kinetically trapped since they cannot escape quickly to relaxall constraints. If a swelling process is induced by improving thesolvent quality, these constraints are released and the microgelrelaxes faster to its equilibrium state. Again we emphasize thatthe allotted relaxation time at the interface for the collapsed state ig. 3 Volume phase transition of a microgel at an interface. Lateral radius of gyration R lat (in terms of the bead size σ ) is shown as a function oftime t for a microgel at the liquid interface for both a collapsing process (yellow line) and a swelling process (green line) with the corresponding errorbars in black as obtained by simulation. The processes are started from configurations relaxed at the interface for a total time of τ B (shown astypical simulation snapshots b) and d)). Typical snapshots c) and e) after a long relaxation time of τ B are also given. There is a notable hysteresisfor the collapsing but not for the swelling process. The data are obtained for a crosslinking density of 4.5% is long as compared to the typical time for swelling and collaps-ing in the simulation but not enough to achieve full equilibra-tion. This "explosion" in relaxation time scales for interfaciallycollapsed states is in accordance with the experimental findingsin literature . We now go one step further and study a periodic sequence ofswelling and subsequent collapsing processes in a cycling way.In particular, this procedure allows investigation whether theswollen state arising by swelling from an initial collapsed stateis hysteretic upon further periodic decrease and increase of sol-vent quality (or temperature). Simulation data for such a cy-cling process are given in Figure 4a. These data show an almosthysteresis-free behavior after one cycle. This gives evidence thatthe hysteretic behavior is mainly attributed to the collapsed initialstate b). Once this adsorbed, collapsed microgel is swollen, theresulting interfacial state can almost reversibly be collapsed andswollen again without showing any further hysteresis in radius ofgyration.The hysteretic behavior is reflected in the two length scalescharacterizing the microgel particle, namely the lateral radius ofgyration R lat and the core radius R c as documented in Figure 4a.Indeed our analysis of the morphology by the hull parameter ∆ ( r ) (Fig. 7, Supporting Information) reveals a clear distinction be-tween core and corona for the different states in the cycling pro-cess and documents that our simulation scheme reproduces thefried-egg structure of a microgel at the interface . Thecore region is much more compressed due to the interfacial at-traction to the beads. Therefore the effect of kinetic arrest in thecore is much more amplified when the particle is brought to theinterface as compared to the bulk situation. This gives a clue tounderstand the underlying reason for the hysteretic behavior atthe interface.We compare the hysteresis observed in simulation with experi-mental data, obtained for an interfacial assembly of microgels onthe air/water interface of a Langmuir trough exposed to changesin the temperature of the water subphase. the diameter of the mi-crogel cores are quantified by image analysis of scanning electronmicroscopy images after transfer to a solid substrate. Interest- ingly, the experimental data shows a similar hysteretic behavior,evidenced by a change in core diameter between the initial ad-sorption in collapsed state and the first temperature cycling asseen in Figure 4b. However, in the experiments the change of sol-vent quality (temperature) is not stepwise but smoothened due toexperimental constraints (compare the gray line in Figure 4b tothe sharp jump of the effective attraction α shown along with thesimulation data in Figure 4a). Fig. 4
Temperature cycling of a microgel at an interface. Cycling pro-cess of swelling and collapsing for a microgel particle initially collapsed atthe interface. a) simulation data and b) experimental data. A hysteresiswithin the first temperature cycle can be clearly observed for both simula-tion and experiment. The simulation data are obtained for a crosslinkingdensity of . and the experimental data for a crosslinking density of . The corresponding effective attraction parameter α and the solventtemperature are also shown as a function of time in a) and b) to illustratethe cyclic nature of the process. Two different length scales are shown inthe simulation data in a) which give the same qualitative trend, namelythe core size R c and the lateral radius of gyration R lat . We now investigate the hysteretic behavior as a function of thedegree of crosslinking. In Figure 5 the interfacial volume phasetransition upon temperature cycling is compared for three dif-ferent crosslinking densities in simulation and experiment. Therelative amount of hysteresis is quantified as − R , where R is the ratio between the two microgel radii before and after thefirst cycling process. For the simulations, the ratio R is taken as R lat ( t = τ B ) / R lat ( t = τ B ) , see Figure 5a, and for the exper- ments we take for R the ratio of the two radii at times minand min, see Figure 5 b. In the absence of hysteresis, the tworadii coincide such that R is one and the amount of hysteresisvanishes.Both experimental and simulation data show that the relativeamount of hysteresis decreases with increasing crosslinking den-sity (5c,d). We rationalize this behavior based on an intuitive ar-gument stemming from the kinetics of the initially trapped state.For a more connected polymer network the microgel is more resis-tant against the stretching effect of the interface and therefore it isstretched out less. This explains why the hysteresis is smaller forhigher crosslinking density. Conversely, if the degree of crosslink-ing is small, long dangling chains can be much more entangledand therefore contribute much more to the amount of hysteresis.Indeed in the extreme case of a single linear chain, additional sim-ulation and experimental data indicate that the relative amountof hysteresis is even higher than the values found for the low-est crosslinking density of one percent; for simulation data seeagain Figure 5c. The number of beads in the simulation of thelinear chain is identical to that contained in the microgel particle,namely 5500 beads. Fig. 5
Temperature cycling for different crosslinking densities. Cyclingprocess of swelling and collapsing for microgel particles with differentcrosslinking densities initially collapsed at the interface: a) simulationdata and b) experimental data. Hysteresis within one cycle can be clearlyobserved for both simulation and experiment. On the simulation side thelateral radius of gyration R lat is used for comparison. The correspondingeffective attraction parameter α and the solvent temperature are alsoshown as a function of time in a) and b) to illustrate the cyclic na-ture of the process. The relative amount of hysteresis as a function ofthe crosslinking density: c) simulation, d) experimental data. Simula-tion data for the special case of a linear polymer are added as well forcomparison (blue). We now consider the surface pressure of an interfacial microgellayer as a function of temperature, which can be measured in experiments. Care has be taken in the interpretation of the ef-fect of microgels on the surface pressure, since the air/water sur-face tension also changes with temperature. As the latter showsa monotonic decrease, we attribute any deviation from such alinear behavior to the microgel layer and therefore subtract thechange in surface pressure of water from the plotted data. There-fore, we refrain here from a full quantitative comparison betweenexperiment and simulation but only consider qualitative trends.In the experimental system, the decrease in surface tension inthe presence of microgels (or, the increase in surface pressure)relates to the surface density of PNiPAm chains adsorbed to theair/water interface. In simulations, the corresponding quantity isthe surface density of adsorbed beads, which is directly accessible.Figure 6a and c compares the evolution of the adsorbed microgelfraction determined in simulation with the surface pressure mea-sured in experiment as a function of the effective attraction α or the temperature, respectively. The data are obtained by start-ing with an interfacially adsorbed microgel with high effectiveattractions α resp. high temperature T (solid points). This ini-tially collapsed microgel is then slowly swollen at the interface bya decrease the attraction (resp. temperature), as shown by bluepoints/blue line. The swelling increases the area occupied by themicrogel (see above), and thus decreases the surface tension - orincreases the surface pressure in experiment (6c). Concomitantly,the swelling causes more beads to adsorb to the interface as thesystem passes through the volume phase transition (6a,b).Notably, the maximal number of adsorbed beads occurs closelyat α ≈ . where the bulk collapse transition happens andis thus correlated to the bulk volume transition. Interestingly, inthe simulation data, a slight decrease of the adsorbed beads is ab-sorbed when the attraction vanishes. This is due to the fact thatparts of the chains go back into the bulk due to entropic reasons.The swollen microgels are subsequently collapsed (or heated)again. The red data points/red line in Figure 6a-c show the asso-ciated evolution of the adsorbed fraction of microgel beads (simu-lations) and surface pressure (experiments), respectively. In bothexperiment and simulation, a hysteresis is observed, in agreementwith the data shown before. The fraction of adsorbed beads andthe surface pressure increases for this increase in α or tempera-ture, compared to the initial values at low temperature culminat-ing in a maximum at a temperature which roughly coincides withthe bulk volume transition. We interpret this maximum as a jointeffect of the interfacial attraction to the beads and the effectiveattraction between the beads. When the bulk volume transitiontemperature is reached from below, the bulk attraction wins anddrags beads from the interface into the bulk. We remark that uponcooling a similar maximum is found in the simulations but not inthe experiments; the reason for this slight discrepancy remainsunclear.Finally, as a reference, we also show data for a linear polymerchain in Figure 6c-f. The linear chain shows similar trends inthe surface pressure and the fraction of the adsorbed monomersas the microgel, but there is no maximum in the adsorbed beadfraction during the swelling process. Thus the curves coincidenicely. ig. 6 Surface pressure cycle for different crosslinking densities. Evolution of surface pressure during the volume phase transition at interfaces, fora microgel and a linear polymer chain. a) simulation data for a microgel ( . crosslinking density). b) a zoom-in for the microgel simulation datafor effective attractions α between 0 and 1. c) experimental data for a PNiPAm microgel ( . crosslinking density). d) simulation data for a linearpolymer chain. e) a zoom-in for the linear polymer chain for effective attractions α between 0 and 1. f) experimental data for a linear PNiPAm polymerchain. A hysteretic behaviour of the volume phase transition is observed for both microgel and linear polymer chain in simulation and experiment. We used monomer-resolved computer simulations and interfacialexperiments to investigate the volume phase transition of PNiPAmmicrogels in bulk and adsorbed to an interface. Our results un-derline that the presence of an interface significantly changes thevolume phase transition. Particularly, we found a significant hys-teretic behavior for microgels undergoing the phase transition atthe interface. A microgel adsorbed in the collapsed state to the in-terface does not return to its initial configuration when subjectedto a temperature cycle. Instead, it relaxes into a more stretchedconfiguration. An initially swollen microgel, however, undergoesreversible transitions between collapsed and swollen states upontemperature cycling. We therefore attribute the hysteresis effectof the collapsed microgel at the interface to a kinetically trappedinitial state which can be released by swelling the microgel. Wefind that the hysteresis is more pronounced for weaker degree ofcrosslinking and is even observed for linear PNiPAm chains.Our results demonstrate that it is possible to model complexpolymer-interface phenomena via a comparably simple modelthat balances internal and interfacial attractions. This mod-elling approach may therefore also be transferred to other stimuli-responsive polymer systems, curved interfaces, or more crowdedsystems formed by multiple, overlapping interfacial microgels.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
N.V., H.L. and L.M.C.J. acknowledge funding from the DeutscheForschungsgemeinschaft (DFG) under grant numbers VO 1824/8- 1 and LO 418/22-1, respectively. N.V. also acknowledges supportby the Interdisciplinary Center for Functional Particle Systems(FPS). J.K. thanks Jens Grauer for helpful discussions.
The bead-bead interaction is modeled by a Weeks-Chandler-Andersen potential V WCA ( r ) = ε (cid:104)(cid:0) σ r (cid:1) − (cid:0) σ r (cid:1) (cid:105) + ε if r ≤ σ otherwise (5)where r is the radial distance between two beads, σ representsthe bead size and ε is the strength of repulsion. The bead-connecting covalent bonds are described by a finite-extensible-nonlinear-elastic (FENE) potential V FENE ( r ) = − ˜ k F ˜ R ln (cid:18) − (cid:16) r ˜ R (cid:17) (cid:19) if r < ˜ R otherwise (6)with ˜ k F = ε / σ an effective spring constant and ˜ R = . σ themaximal bond expansion. We characterize the internal core-corona structure of the microgelby a geometric analysis of the bead configurations. The mainidea is as follows: we consider all bead positions projected tothe xy -plane which have a distance less than a prescribed r fromthe (stationary) microgel center (cid:126) R , i.e. all bead positions whichfulfill ( x i ( t ) − X ( t )) + ( y i ( t ) − Y ( t )) < r . In the corona region we )ii)iii) vi)v)iv) Fig. 7 i)-iii) Typical bead configurations for a crosslinking density of . at the interface (top view) for the initial collapsed state i), the swollenstate ii) and the collapsed state after one cycle iii). Different hulls forthe core and corona are indicated in different colors (see legend). Thecorresponding ∆ ( r ) -profiles are shown in iv), v) and vi) with error bars.The tangent on the ∆ ( r ) -profile is shown as a full red line, the intersectionpoint with the dashed lines indicates the core and corona size. expect that the set of bead positions exhibits a relatively roughboundary while in the core region the boundary is fairly smooth.To quantify this further we consider the convex hull around theset of bead positions which provides a contour length L convex ( r ) .We also define a concave hull around the same bead positionsand compare its contour length L concave ( r ) to L convex ( r ) . Clearly, L concave ( r ) ≥ L convex ( r ) . In contrast to the convex contour, the pre-cise definition of the concave hull is not unique. In detail, forthe calculation of the concave hull we employed the algorithmin Ref. with the so-called k -nearest-neighbour approach wherewe used k = .The relative difference in the two contour lengths defines a pa-rameter ∆ ( r ) ∆ ( r ) = L concave ( r ) − L convex ( r ) L concave ( r ) (7)The spatial dependence of ∆ ( r ) contains valuable informationabout the core size R c and the corona size R co . Inside the core ∆ is small while it increases for increasing r until it saturates witha value ∆ ∞ of the order of one, concomitant with a "rugged" con-cave hull. We can therefore extract the core and corona size ap-proximatively by studying the behavior of ∆ ( r ) near its inflection point at r = r ∗ defined by the condition d ∆ ( r ) / dr | r = r ∗ = . Con-sidering the tangent ˜ t ( r ) through the inflection point given by thelinear relation ˜ t ( r ) = ∆ ( r ∗ )+ d ∆ ( r ) / dr | r = r ∗ × ( r − r ∗ ) , we define thecore size R c by the intersection of the tangent with the r -axis, i.e.by the condition ˜ t ( R c ) = . The corona size is defined by the dis-tance where the tangent reaches the saturation value ∆ ∞ , i.e. bythe relation ˜ t ( R co ) = ∆ ∞ .Concrete data for a typical microgel snapshot along with theprofiles for ∆ ( r ) are presented in figure 7i)-iii) corresponding tothree different situations: i) initially collapsed state of a microgelparticle brought to the interface from the bulk, ii) swollen micro-gel particle at the interface after the first temperature change, iii)collapsed state after one full cycle. The associated concave andconvex hulls for the inner and outer part are indicated in differ-ent colors. Moreover, in figure 7iv)-vi) the corresponding profiles ∆ ( r ) are presented together with the tangent ˜ t ( r ) (full red line)and the determination of the core size R c and the corona size R co . Notes and references
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