Collapse-to-Swelling Transitions in pH- and Thermoresponsive

Oct 24, 2013 - We present a theory of a conformational collapse-to-swelling transition that occurs in aqueous dispersions of multiresponsive (pH- and ...
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Collapse-to-Swelling Transitions in pH- and Thermoresponsive Microgels in Aqueous Dispersions: The Thermodynamic Theory Alexey A. Polotsky,† Felix A. Plamper,‡ and Oleg V. Borisov†,§,⊥,* †

Institute of Macromolecular Compounds of the Russian Academy of Sciences, 31 Bolshoy pr., 199004 St.-Petersburg, Russia Physikalische Chemie II, RWTH Aachen University, 52056 Aachen, Germany § St.Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101, Kronverkskiy pr., 49, St.Petersburg, Russia ⊥ Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, UMR 5254 CNRS/UPPA, Pau, France ‡

ABSTRACT: We present a theory of a conformational collapse-to-swelling transition that occurs in aqueous dispersions of multiresponsive (pH- and thermoresponsive) microgels upon variation of ionic strength, temperature, or pH. Our theory is based on osmotic balance arguments and explicitly accounts for ionization equilibrium inside microgel partices. The theory predicts complex patterns in the dependence of the microgel particle dimensions on the control parameters: An increase in temperature leads to worsening of the solvent quality for the gel forming LCST-polymers and to concomitant decrease in the dimensions of the gel particles. This collapse of the gel particles provoked by an increase in temperature occurs either smoothly (at high or low ionic strength), or may exhibit a jump-wise character at intermediate ionic strength. The theory further predicts that the degree of swelling of microgel particles varies nonmonotonously and exhibits a maximum as a function of salt concentration at a pH close to the pK. This nonmonotonous variation of the particle dimensions occurs continuously at temperatures below or slightly above LCST (good or marginal poor solvent strength conditions, respectively), whereas at higher temperatures the jump-wise swelling of the gel particles is followed by either continuous or jump-wise collapse induced by progressive increase in the salt concentration. A decrease/increase in pH leads to deswelling of the weak polyacid/polybase gel particles, which occurs smoothly at temperatures below LCST, but may exhibit a discontinuity above LCST. These theoretical predictions can be used for design of smart stimuliresponsive microgels.

I. INTRODUCTION The ability of ionic polymers (polyelectrolytes) to change their conformations under the action of external triggers, such as ionic strength and pH, is exploited in many applications ranging from oil recovery and paper-making to nanotechnology and nanomedicine. A remarkable progress in understanding the properties of linear1,2 and branched3 polyelectrolytes has been achieved in the past few years. There are important distinctions in properties of strong (quenched) and weak (annealing) polyelectrolytes. In a strong polyelectrolyte the fraction α of charged monomer units is fixed chemically and does not depend on the environmental conditions. In a weak polyelectrolyte the fraction α of charged monomer units coincides with the degree of ionization and is governed by the local pH which may noticeably differ from the pH of the buffer.4 The latter is especially true for branched polyelectrolytes3 and ionic gels. Hereby, polyelectrolyte gels can be regarded as a special class of branched ionic polymers. Because of osmotic pressure of the confined counterions, they exhibit often tremendous swelling in solvents of high dielectric constant like water, while still having a defined macroscopic shape. The swelling is especially pronounced when it comes to highly charged gels. Generally, © XXXX American Chemical Society

circumstances are known where the macroscopic size of gels is detrimental. E.g., the characteristic switching time is proportional to the surface of the gel.5 Therefore, gels in the nanometer to micrometer scale can provide beneficial switching kinetics and rapid uptake/release of guest molecules. These gels are termed as microgels6−8 and will be regarded within this investigation. In presence of additional attractive (solvophobic) interactions, discontinuous volume transitions in the ionic hydrogels can appear. This means that large effects in the swelling can be observed when only small changes in the solvent strength are applied.9−11 The ion pair formation in less polar medium may provide additional driving force for the discontinuous gel collapse.12−15 These examples indicate that polyelectrolytes and the gels thereof are interesting species. A subtle balance between attracting and repellant interactions appears to offer this rich behavior. Therefore, the use of weak polyelectrolytes with Received: July 4, 2013 Revised: September 18, 2013

A

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occurring in pH-sensitive polyelectrolyte brushes upon variation of salt concentration. Furthermore, combined effects of added salt and tunable hydrophobicity of the monomer units on conformations of pH-sensitive star-branched polyelectrolytes were recently investigated.39,40 It was demonstrated, that salt-induced re-entrant swelling-to-collapse intramolecular conformational transitions in star-branched hydrophobic polyions occur continuously either at high or at low ionic strength but may exhibit features of a first-order phase transition (i.e., intramolecular microphase separation) at intermediate salt concentration. The aim of the present paper is to investigate volume transitions occurring in dilute aqueous dispersions of multiresponsive (pH- and thermoresponsive) microgel particles upon variation of external parameters, such as temperature, ionic strength and pH. The latter two affect the strength of Coulomb interactions between charged monomer units, whereas temperature controls the strength of short-range excluded volume interactions (solubility of the polymer in the absence of charges). In addition, hydrostatic pressure might be considered as additional parameter, which affects the strength of excluded volume interactions and thus solubility of the monomer units of the microgel in water.41−43 As we demonstrate below, a competition of short-range attractive and long-range repulsive interactions with ionization equilibrium inside the gel leads to complex patterns in the dependence of the microgel particles dimensions on these three control parameters even in the absence of any polymeror ion-specific interactions in the system. Hence, our conclusions are relevant for a wide class of thermosensitive weak polyelectrolyte microgels, and enables one to rationalize stimuli-responsive behavior of e.g. microgels of PNIPAM with pH-sensitive ionic comonomers or PDMAEMA in aqueous dispersions.

tunable charge and hydrophilicity might envision further stimuli-responsiveness for future applications. There are literature reports on the stimuli-responsive behavior of the microgels formed by poly(acrylic acid) (PAA) or different kinds of copolymers including monomer units of acrylic acid.16,17 The swelling behavior of these weak polyelectrolyte microgel particles can be efficiently controlled by pH: a decrease in pH causes a decrease in the degree of ionization of the microgel particles, drop in the intragel osmotic pressure of the counterions and concomitant collapse of the microgel. In contrast to pure PAA, poly(N,N-dimethylaminoethyl methacrylate) (PDMAEMA) and especially copolymers of Nisopropylacrylamide (NIPAM) and its derivatives exhibit additional thermosensitive features.18−21 That is, the polymer is soluble in water under ambient conditions but the solubility decreases upon an increase in temperature. Hence, aqueous solutions of such polymers exhibit lower critical solution temperature (LCST): Below the LCST the polymer is soluble at any concentration, while above LCST macroscopic phase separation into dilute and concentrated phases (precipitate) may occur.22 Regarding microgels, collapse of the microgel particles dispersed in aqueous media occurs upon an increase in temperature above LCST which can be termed in this case as volume phase transition temparature (VPTT).6 Remarkably, the microgel dispersions may exhibit colloidal stability even at temperature above VPTT. We have already revealed23,24 an interplay between protonation and thermoresponsive behavior in case of linear and branched PDMAEMA. Dependent on the pH, PDMAEMA turns water-insoluble upon heating. Furthermore, microgels of PDMAEMA show an excellent pH-responsiveness. Charged thermosensitive microgels of PDMAEMA have been prepared.25,26 Otherwise, cationic or anionic monomers can be copolymerized with monomers (like NIPAM) yielding thermosensitive polymers.18,19,27 For both weak and strong polyelectrolytes, the swelling of the gel particles can be manipulated by changing the strength of electrostatic interactions by, e.g., addition of salt ions.28 For weak polyelectrolytes pH can be considered as additional control parameter since its variation enables a tuning of the charge of the gel particles. In addition, issues of colloidal stability of the microgel dispersion might arise.29 Concluding, the combination of pH and thermosensitive properties of, e.g., PDMAEMA makes the stimuli-responsive behavior of the microgel more sophisticated, especially when it comes to the salt dependence. Indeed, such complex patterns as re-entrant swelling-to-collapse transitions provoked by increasing salt concentration were predicted theoretically30,31 and observed experimentally32−34 for pH-sensitive polyelectrolyte brushes and for star-branched polyelectrolytes.35,36 The nonmonotonous salt dependence of the degree of swelling of pHsensitive branched polyions (or polyelectrolyte brushes) was explained by dual effect of added salt on the differential intramolecular osmotic pressure: the initial increase in salt concentration provokes an increase in the osmotic pressure due increasing ionization of the monomer units, whereas further increase in salt concentration leads to a leveling off of the concentrations of small ions inside the star (or in the brush) and in the solution and to a concomitant decrease in the differential osmotic pressure. More recent studies based on more advanced theoretical approaches37,38 have enabled a detailed analysis on the internal structural rearrangements

II. MODEL AND FREE ENERGY Our approach is based on the analysis of the Gibbs free energy of a swollen microgel particle (calculated per subchain) which can be presented in a mean-field approximation as F = Fconf + Fex . vol + Fion

(1)

The first term in eq 1 accounts for conformational free energy of a stretched subchain comprising N monomer units and extended up to the size R Fconf /kBT =

3R2 2Nb2

(2)

where kB is the Boltzmann constant and T is the temperature. Here we assume that subchains are extended with respect to ideal (Gaussian) dimensions, exhibit Gaussian elasticity and are intrinsically flexible, that is, the statistical segment length is on the order of the monomer unit length b. The condition of the subchain extension, R ≥ bN1/2, may be violated in the microgel particles collapsed in poor solvent. The corresponding correction term which accounts for conformational entropy losses in contracted subchains could be introduced following the lines of ref 44. This correction, however, is negligible compared to other contributions to the free energy in the collapsed state and does not qualitatively affect the results of our analysis. The subchain dimension R is related to the polymer volume fraction c in the microgel as B

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Macromolecules c=

qNb3 2R3

=

QNb3 V

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larger than its characteristic size V1/3 reduced by the Bjerrum length lB = e2/ϵkBT (here ε is the dielectric constant of the medium). Since for the microgel particle V scales proportionally to Q and the dimensions of a subchain in the low salt limit can be estimated as ∼ α1/2Nb, the microgel particles comprising Q ≫ (α−1/2b/lB)3/2 subchains can be considered as almost electroneutral. Hence, the local electroneutrality condition applies with a good accuracy to sufficiently large microgel particles, with the typical particle size on the order of 100 nm or larger. It is essential, however, that volume fraction occupied by the gel particles in the solution is sufficiently small to ensure quasi-constant values of chemical potentials of all mobile ion species. The degree of ionization α of subchains in the microgel depends on the local pH inside the gel, which in turn depends on the pH in the buffer and on the excess local electrostatic potential in the gel. By combining the mass action law with the local electroneutrality condition the degree of ionization α of the gel can be expressed as3

(3)

where q is the number of subchains emanating from a junction point, Q is the number of subchains and V is the volume of the microgel particle. The free energy of nonelectrostatic (excluded volume) interactions can be presented in the virial approximation as Fex . vol /kBT = N (vc + wc 2)

(4)

where v and w are the second and the third virial coefficients, respectively. The second virial coefficient (the excluded volume parameter) v can be tuned by variation in the temperature. For most of the water-soluble polymers (including PNIPAM and PDMAEMA), v is a decreasing function of the temperature: vT≤LCST ≥ 0 and vT≥LCST ≤ 0 correspond to good and poor solvent conditions, respectively, whereas vT=LCST = 0 corresponds to the Θ-point. In the vicinity of LCST the second virial coefficient is expected to vary approximately linearly as a function of temperature, v ∼ (LCST − T). An increase in pressure at constant temperature is known to improve solubility of polymer in water,41 that is, v should be an increasing function of pressure. On the contrary, the third virial coefficient is virtually independent of temperature, w = const. It is known, that solubility of nonionic polymers, like e.g. PNIPAM in water is affected by salt concentration,45 that is, the second virial coefficient v may exhibit a dependence on the ionic strength. In our analysis we, however, disregard this dependence because of the primary effect of salt concentration on the strength of ionic interactions acting between charged monomer units of a pHsensitive microgel. We also disregard possible dependence of v on the pH of the solution, which may take place in the case of, e.g., hydrogen bond formation between nonionized monomer units. The ionic contribution to the free energy is specified in the framework of the local electroneutrality approximation, i.e., by assuming that the charge density of ionized monomer units inside the gel is locally compensated by excess density of mobile counterions. With the account of equilibrium exchange of mobile ions between microgel volume and the surrounding solution (where concentrations of all mobile ion species are assumed to be fixed), the ionic contribution to the Gibbs free energy is specified as3

α 1 − αb = 1 − α αb

⎛ αc ⎞ 2 αc 1+⎜ ⎟ − cs ⎝ cs ⎠

(6)

The degree of ionization αb of the respective nonpolymeric ionizable groups in bulk solution is related to pH of the buffer solution as αb = (1 + 10±(pH − pK ))−1

(7)

where pK = −log K, and K is the ionization constant of a single ionizable group and signs “+” and “−” refer to cationic and anionic microgels, respectively. It is convenient to characterize the extent of swelling of the microgel particles by the dimensionless parameter ⎛ c ⎞1/3 R = ⎜ θ ⎟ ≡ β1/3 ⎝c⎠ Rθ

where R θ = N1/2bw1/8q1/4 2−1/8

and cθ =

⎧ ⎡ ⎤⎫ ⎛ αc ⎞ 2 ⎪ ⎪ cs ⎢ Fion(c)/kBT = N ⎨ln(1 − α) − ⎢ 1 + ⎜ ⎟ − 1⎥⎥⎬ c ⎝ cs ⎠ ⎪ ⎣ ⎦⎪ ⎭ ⎩

(q/2)1/4 N1/2(2w)3/8

are respectively the subchain dimensions and the volume fraction of polymer in the microgel at the Θ-temperature (i.e., at v = 0) in the absence of ionic interactions. Assuming that in a dry state the volume fraction of polymer in the microgel equals unity, we can express the degree of swelling as R/Rdry = β1/3· N1/6(2w)1/8(q/2)−1/12. Below we characterize the temperature-controlled solvent strength using the dimensionless parameter τ/cθ, where τ = −v/ 2w. An increase in temperature leads to a decrease in v and to an increase in τ. Under poor solvent conditions, v ≤ 0, τ coincides with the volume fraction of polymer in the precipitate (or in a large polymer globule48). The average number N of monomer units in a subchain is inversely proportional to the degree of cross-linking and can be considered as a variable architectural parameter of the gel. An increase in N is manifested in a decrease in cθ. Below we shall analyze how the pH-/thermo-responsive properties of the microgels depend on the degree of cross-linking.

(5)

where cs is the overall concentration (volume fraction) of monovalent ion species (including H+ and OH− ions and added monovalent salt) in the solution. The local electroneutrality approximation is justified for dilute dispersions of ionic microgels with or without added electrolyte provided that the size of the gel particles is sufficiently large to ensure that the major fraction of mobile counterions is retained by the Coulomb attraction inside the microgel particles. The solution of the corresponding Poisson− Boltzmann equation for a branched polyion (e.g., a microgel particle) placed into an electroneutral Wigner−Seitz cell46,47 proves, that with the accuracy of logarithmic factors, the elecrtroneutrality condition is fulfilled if the bare charge of the microgel particle, αNQ, (measured in elementary charges e) is C

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Figure 1. Dependences of the reduced size of the microgel particle R/Rθ (a) and degree of ionization α (b) on reduced temperature τ/cθ for N = 20, q = 4, αb = 0.6 and different salt concentrations.

contraction of the gel occurs continuously upon a decrease in the solvent strength. For weak (pH-sensitive) polyelectrolyte gels, the degree of ionization of the subchains depends on the polymer concentration inside the gel and on the ionic strength in the surrounding solution, eq 6. Therefore, the temperatureinduced collapse transition in such microgels acquires novel features. Figure 1a illustrates how the character of the temperatureinduced collapse transition changes upon variations in salt concentration cs. The microgel gets contracted continuously as a function of decreasing solvent strength (increasing temperature) at low or high salt concentrations, but exhibits a jumpwise collapse below the Θ-point (above LCST) in the intermediate range of salt concentrations. This discontinuity is also reflected in the temperature dependences of the degree of ionization, presented in Figure 1b. At any salt concentration cs an increase in temperature (in τ) leads to compactization of the microgel particles and thus provokes a concomitant decrease in degree of ionization α. Indeed, α is a monotonously decreasing function of polymer concentration, eq 6, since repulsive electrostatic interactions of charged monomer units hinder ionization. Therefore, the degree of ionization in the collapsed state is much smaller than that in the swollen state. In Figure 2 the critical value of the bulk degree of ionization, αb,crit(cs), is presented as a function of salt concentration. At each given salt concentration cs the temperature-induced contraction of the microgel particles occurs continuously, if the bulk degree of ionization αb ≤ αb,crit(cs). On the contrary, at

The equilibrium degree of swelling of the microgel particle with respect to the Θ-dimensions, R/Rθ  β1/3 obeys the following equation, which results from minimization of the Gibbs free energy determined by eqs 1, 4, 5 and 6: β

8/3

⎡ ⎤ ⎛ αc θ ⎞ 2 cs 3⎢ τ ⎥ + β− β 1+⎜ ⎟ − 1⎥ = 1 cθ 2wcθ3 ⎢⎣ ⎝ csβ ⎠ ⎦

(8)

where the degree of ionization α is given by α 1 − αb = 1 − α αb

⎛ αc θ ⎞ 2 αc 1+⎜ ⎟ − θ csβ ⎝ csβ ⎠

(9)

The free energy of the microgel (per subchain) can be expressed as F 2wcθ NkBT 2

=

⎡ ⎤ ⎛ αc θ ⎞ 2 cs ⎢ 5 2/3 1 ⎥ β − − β + − 1 1 ⎜ ⎟ ⎥ 2 2β 2 wcθ 3 ⎢⎣ ⎝ csβ ⎠ ⎦

(10)

In the first order transition point the polymer density and the ionization degree in the coexisting collapsed and swollen states (denoted as state 1 and state 2) are determined from the condition F(β1 , α1) = F(β2 , α2)

(11)

solved together with eqs 8, 9. The critical line in (αb, cs) plane separating the ranges of continuous and discontinuous temperature induced collapse transitions is obtained from the condition dτ(β)/dβ = d2τ(β)/dβ 2 = 0

(12)

III. RESULTS A. Temperature-Induced Swelling-to-Collapse Transition. An increase in temperature leads to a decrease in solubility (enhancing hydrophobicity) of polymer chains forming the microgel and, as a result, to deswelling of the gel particles. This deswelling is, however, opposed by excess osmotic pressure of the mobile ions inside the gel. For a microgel with permanent charges (strong polyelectrolytes) in salt-free solution the competition of increasingly strong shortrange monomer−monomer attractions with the osmotic pressure of confined counterions results in a jump-wise collapse transition, whereas at sufficiently high ionic strength the

Figure 2. Critical value of the bulk degree of ionization αb,crit vs salt concentration calculated for different values of N, as indicated in the Figure. At αb ≥ αb,crit the temperature-induced collapse of the microgel occurs as a jump-wise first order phase transition. D

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Figure 3. Dependences of reduced size of the microgel particle R/Rθ (a) and degree of ionization α (b) on reduced salt concentration cs/cθ for N = 20, q = 4, αb = 0.6 and different temperatures. Solid points (circles) and dashed lines indicate the position of the first order phase transition in the cases τ/cθ = 2, 3, and 3.25.

αb ≥ αb,crit(cs) an increase in temperature triggers a jump-wise collapse of the microgel. The critical value of αb,crit(cs) depends nonmonotonously on salt concentration cs, that is, decreases at low salt concentrations, passes through a minimum and then increases at high salt concentrations. Hence, at fixed pH (fixed αb) an increase in temperature provokes progressive deswelling of the microgel particles either at small or at high salt concentration, whereas discontinuous temperature-induced volume transition can be expected in a certain range of intermediate salt concentrations. Remarkably, a decrease in N (equivalent to an increase in the degree of cross-linking) leads to shrinkage of the region of jump-wise collapse transition toward higher values of αb and to a decrease in the magnitude of the jump in the size of the gel particles. At given (buffered) value of pH, the range of salt concentrations where discontinuous collapse may be observed experimentally narrows upon a decrease in N and for strongly cross-linked gel temperature-induced deswelling is expected to occur smoothly at any salt concentration. B. Salt-Induced Swelling-to-Collapse Transition. The conformational response of the microgel particles to the variation in the ionic strength of the solution at fixed temperature and pH is more complex: The initial increase in the salt concentration promotes ionization and, consequently, provokes an increase in the osmotic pressure inside the gel. As a result, swelling of the microgel particles occurs. As soon at the degree of ionization of the microgel reaches (upon an increase in salt concentration) its maximal possible value α ≈ αb, (at given pH in the buffer) then a decrease in the differential osmotic pressure caused by added salt governs deswelling of the gel. As a result, the dependence of the size of the gel particles on salt concentration acquires nonmonotonous character. A similar effect has been predicted earlier for pH-sensitive polyelectrolyte brushes30,31 and stars36 where stretching of the brush-forming chains or of the star arms is governed by the excess osmotic pressure exerted by entrapped counterions. The dependence of the reduced size of the microgel particle on salt concentration cs at different temperatures is presented in Figure 3. As one can see from the Figure, under good solvent conditions (T ≤ LCST) as well as in the vicinity of the Θ-point (T ≈ LCST) the nonmonotonous but continuous variation in the dimensions of the gel (swelling-contraction) occurs upon an increase in salt concentration. On the contrary, under moderately poor solvent conditions (T ≥ LCST) one or two sharp conformational transitions accompanied by abrupt changes in the dimensions of the microgel particles occur upon progressive increase in salt concentration. As demon-

strated by Figure 3a, a continuous increase in the microgel particle size caused by addition of small amounts of salt can be interrupted by a jump-wise swelling transition. Upon further increase in salt concentration the gel shrinks. Depending on temperature, this contraction either occurs continuously or involves a jump-wise collapse transition. As one can see in Figure 3b, the degree of ionization α of the gel particles is a continuous and smoothly increasing function of salt concentration under good or nearly Θ-solvent conditions, T ≈ LCST, as well as under very poor solvent conditions. Under moderately poor solvent conditions the degree of ionization exhibits discontinuities which are coupled to abrupt swelling and collapse of the gel particles upon an increase in salt concentration. Moreover, in a narrow range of temperatures, the degree of ionization exhibits a weak maximum as a function of the ionic strength. These peculiarities in the dependence of the degree of ionization of the gel particles on salt concentration arise due to coupling between ionization and local polymer density, that is expressed by eq 6. The phase diagram of the microgel solution in (temperature, salt concentration) coordinates is presented in Figure 4. The

Figure 4. Phase diagram of the microgel solution in the temperature− salt concentration coordinates calculated for αb = 0.6 and different degrees of cross-linking (the values of N indicated at the lines). The lines are terminated by the critical points.

τtr(cs) lines in the Figure 4 correspond to the conditions of coexistence between collapsed and swollen states for the microgels with different degrees of cross-linking. Crossing of the coexistence lines upon variation in temperature (at constant ionic strength) or upon variation in the ionic strength (at constant temperature) implies jump-wise transitions between swollen and collapsed states of the microgel. Each of the firstorder transition lines ends in two critical points corresponding E

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Figure 5. Dependences of the reduced size of the microgel particles R/Rθ (a) and degree of ionization α (b) on the pH-controlled bulk degree of ionization αb for N = 20, cs/cθ = 0.1 and different temperatures, as indicated at the curves. The dotted line in panel b corresponds to α = αb. Solid points (circles) and dashed lines indicate the position of the pH-induced first order phase transition in the cases τ/cθ = 3 and 4.

pH-triggered transition from collapsed to swollen states occurs with a jump in the gel size, as the first order phase transition and it is accompanied by a concomitant jump-wise increase in the degree of ionization α.

to the lower and the upper critical salt concentrations. The temperature-induced contraction of the gel particles occurs continuously, without a jump in the size, if the salt concentration in solution is kept below the lower or above the upper critical salt concentration. The difference between the former and the latter diminishes upon an increase in the degree of cross-linking. Remarkably, for weakly cross-linked microgels the dependence of the transition temperature τtr on salt concentration cs is a nonmonotonous function, that is, exhibits a maximum. The presence of the maximum in the τtr(cs) dependence, Figure 4, for weakly cross-linked microgels, indicates that in a certain temperature range above LCST a progressive increase in the salt concentration provokes two successive jump-wise conformational transitions in microgel particles: The first one (which occurs at lower salt concentration) is the transition from the collapsed weakly ionized state to the swollen strongly ionized state, whereas the second one (which occurs at higher salt concentration) is the transition from the swollen to the collapsed state of the gel with concomitant drop in the degree of ionization α. (For strongly cross-linked microgels the latter transition is continuous). C. pH-Induced Collapse-to-Swelling Transition. The swelling of pH-sensitive microgel particles can be naturally governed by changing pH of the solution that affects the degree of ionization of the microgel particles. As follows from eq 6, the degree of ionization α of the microgel increases/decreases monotonously as a function of the bulk pH for anionic/cationic gel. Importantly, at any pH in the buffer, α remains smaller than αb which is directly related to the buffer pH by eq 7. Figure 5b demonstrates that under good solvent conditions (T ≤ LCST) the degree of ionization α in a swollen microgel closely follows increasing αb, which is reflected in continuous increase in the size of the gel particles, Figure 5a. The difference between α and αb becomes more pronounced upon an increase in temperature, particularly above LCST. The higher the temperature, the larger the polymer concentration in the microgel, the more significant is the deviation of α in the collapsed (at small αb) gel from αb. On the contrary, the difference between α and αb in the swollen gel is small at any temperature at large αb. The swelling of the microgel particles caused by an increase in αb (governed by the pH) occurs smoothly at temperatures below and around LCST (good, Θ, and marginal poor solvent conditions). Upon an increase in temperature the pH-triggered swelling transition becomes more sharp (occurs in a narrower pH range). At high temperature (sufficiently above LCST) the

IV. DISCUSSION AND CONCLUSIONS The presented above theory suggests that aqueous dispersions of multiresponsive (pH- and thermoresponsive) microgels exhibit a complex response to variations in temperature, ionic strength of the solution and pH. Our conclusions are based on the thermodynamic analysis of tunable balance between shortrange solvophilic/solvophobic interactions and excess osmotic pressure exerted by the ions partitioned between the microgel and the solution with the account of ionization equilibrium for the monomer units of the gel-forming subchains. Assuming that inside the microgel the charge of the ionized monomer units is approximately compensated by excess concentration of mobile counterions, we were able to derive analytical equation for the degree of swelling of the microgel as a function of external parameters (solvent strength, salt concentration, and pH in the buffer). Analysis of the temperature dependence of the degree of swelling indicates that an increase in temperature (decrease in the solvent strength) triggers the gel contraction. This contraction occurs continuously at low and high ionic strength whereas at intermediate salt concentration a jump-wise temperature-induced shrinkage of the microgel particles is expected. The range of salt concentration corresponding to the discontinuous temperature-induced collapse of the gel becomes wider upon a decrease in the degree of cross-linking (an increase in N). Furthermore, similarly to pH-sensitive polyelectrolyte brushes30,31 or star-branched polyelectrolytes,35,36 the microgel particles are expected to exhibit nonmonotonous dependence of their dimensions on salt concentration: an increase in the microgel size at low ionic strength is followed by a decrease at high ionic strength. The appearance of a maximum in the particle size dependence on salt concentration is explained by the fact that the ionization inside the microgel is suppressed at low ionic strength but grows upon moderate increase in salt concentration. The theory-predicted variation of the microgel particle size as a function of salt concentration is smooth under good, Θ, or marginal poor solvent conditions (below or around LCST), whereas under moderately poor solvent conditions (above LCST) successive swelling and collapse of the microgel may occur as a jump-wise first-order phase transition. F

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concentration results in progressive repartitioning of the chains between the two phases and concomitant smooth variation of the experimentally measurable dimensions of the stars or brushes. On the contrary, under the same conditions the crosslinked pH- and thermoresponsive microgel particles are expected to exhibit discontinuous volume transitions manifested in abrupt variation in the particles dimensions and in coexistence of collapsed and swollen microgel particles in solution in the vicinity of the transition point. Finally, we remark that an interplay of short-range monomer−monomer attraction with long-range Coulomb repulsion may result (at low ion concentration inside the gel) in formation of nanoscale collapsed hydrophobic domains along subchains,64 association of subchains into bundles,65,66 or other types of the domain structures.67,68 As has been discussed in ref 66, because of branched topology, multiple morphologies of microdomain structures may appear in partially collapsed hydrophobic polyions. The formation of locally collapsed hydrophobic domains may lead to smoothening of abrupt firstorder volume transitions in hydrogels. A more refined analysis of these effects based on Monte Carlo or molecular dynamics simulations is necessary. The predictions of our theory can be checked, e.g., by dynamic light scattering or viscosimetry in dilute aqueous dispersion of multiresponsive microgels. Indeed, the aqueous dispersions of the microgels often convey colloidal stability (or exhibit very slow coagulation) even in the collapsed state due to residual charges of the particles. In addition, the particle concentration can be low, while still allowing efficient light scattering.

The universality of predictions of our theory concerning complex behavior of the microgel upon variation in temperature or/and salt concentration and character of the stimuliinduced volume transitions is assured by the account of only main thermodynamic forces (contributions to the free energy) acting in the system. Additionally, ion binding to the chains, variation in local dielectric constant upon gel collapse, etc., may influence quantitatively the shape of the swelling curves. However, corresponding analysis requires application of more sophisticated numerical schemes for the free energy minimization60 and involves unavoidably empirical adjustable parameters. At the same time, phenomenology of any particular experimental system can be even richer due to polymer- or ionspecific effects possibly coming into play. For example, similarly to polyelectrolyte brushes and stars, reswelling of cationic hydrogels collapsed at moderate salt concentration may occur at very high ionic strength (more than one mol per liter) if I− or Br− are used as counterions.61,62 This behavior was explained theoretically using modified Donnan theory in terms of volume of the ions in ref 63, but it is not related to the pH-sensitive properties of the gel, i.e., could be expected for strong polyelectrolyte gels as well. The comprehensive analysis of such ion-specific effects is, however, beyond capabilities of the meanfield theoretical approaches. Furthermore, in the present study we focused only on the influence of monovalent salt on the swelling of the microgels. It is known that multivalent counterions provoke much stronger collapse of polyelectrolyte brushes or branched polyelectrolytes as compared to that induced by monovalent salt ions.3,49−52 Similar effects of the ion valence should be expected for microgels as well and can be qualitatively explained by preferential localization and lower osmotic pressure exerted by multivalent ions inside polyelectrolyte brushes or stars (or, equally, in microgels). At the same time, in the case of multivalent counterions quantitative analysis points to the importance of ion−ion or/and ion−polymer correlations, which leads to stronger collapse as compared to prediction of the theory based on the osmotic balance arguments.50,52 A quantitative analysis of the ion−ion correlations in strongly charged polyelectrolyte gels based on the liquid-state integral equation theory has been presented in the recent work.53 The contribution of ion−ion correlations to swelling of polyelectrolyte brushes54 or gels is, however, negligible in the case of monovalent ions considered here. At this point it is worth mentioning that a possibility of discontinuous collapse transitions provoked by decreasing solvent strength has been suggested earlier for star-branched polyelectrolytes and polyelectrolyte brushes on the basis of theories which exploited box-like models.54−57 More accurate analysis with the account of nonuniform distribution of polymer density and nonequal stretching of chains in polyelectrolyte stars or in brushes proved that under poor solvent conditions coexistence of swollen and collapsed conformations occurs in polyelectrolyte stars or brushes on the level of individual chains.58,59 More specifically, a decrease in the solvent strength or/and an increase in salt concentration provokes microphase segregation in a star (or in a brush). This segregation is manifested in the formation of dense central region formed by less extended chains and sparse outer region formed by more extended chains. In the case of pH-sensitive polyelectrolytes the degree of ionization of the chains in the dense phase is smaller than that of the chain forming the extended phase.23,39,40 The variation of temperature or/and salt

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been partially supported by the Russian Foundation for Basic Research (Grant 11-03-00969a), by Department of Chemistry and Material Science of the Russian Academy of Sciences, and by German Research Foundation (DFG) within SFB 985.

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REFERENCES

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