Thermodynamic Description of the LCST of Charged

Mar 6, 2014 - The dependence of the lower critical solution temperature (LCST) of charged, thermosensitive copolymers on their charge fraction and the...
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Thermodynamic Description of the LCST of Charged Thermoresponsive Copolymers Jan Heyda,† Sebastian Soll,‡ Jiayin Yuan,‡ and Joachim Dzubiella*,¶,† †

Soft Matter and Functional Materials, Helmholtz-Zentrum Berlin, Hahn-Meitner Platz 1, 14109 Berlin, Germany Department of Colloid Chemistry, Max-Planck-Institute of Colloids and Interfaces, D-14476 Potsdam, Germany ¶ Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany ‡

S Supporting Information *

ABSTRACT: The dependence of the lower critical solution temperature (LCST) of charged, thermosensitive copolymers on their charge fraction and the salt concentration is investigated by employing systematic cloud-point experiments and analytical theory on a macroscopic thermodynamic level. The latter is based on the concept of the Donnan equilibrium incorporated into a thermodynamic expansion of a two-state free energy around a charge-neutral reference homopolymer and should be applicable for weakly charged (or highly salted) polymer systems. Very good agreement is found between the theoretical description and the experiments for aqueous solutions of the responsive copolymer poly(NIPAM-co-EVImBr) for a wide range of salt concentrations and charge fractions up to 8%, using only two global, physical fitting parameters. Our model could be useful as a guide for the optimization of thermoresponsive copolymer architectures in the future design of soft, polymer-based materials.

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INTRODUCTION

guide for the optimization of stimuli-responsive copolymer architectures in the future design of soft functional materials. We emphasize that the aim of this paper is not to provide a full microscopic theory for the LCST, but in contrary the most macroscopic and therefore simple, analytically tractable model for a pragmatic fit of experimental data by thermodynamic parameters. As we will show we have managed to fit a wide variety of experimental LCST points using only two, physically motivated parameters (such as the transition entropy). For a given experimental system, this enables quick predictions of the change of the LCST by charging, and, additionally, provides insight into entropy changes in the system by functionalization. This is complementary to more microscopic models,23−31 which resolve the free energy (landscape) along a microscopic reaction coordinate, such as the local polymer density or the radius of gyration. Microscopic models typically involve more unknown parameters (such as pair interactions or virial coefficients) and therefore less analytically tractable equations. Hence, while our model lacks microscopic insight, is has the advantage to be simple and directly provide experimentally accessible parameters such as the transition entropy change by functionalization. Exploring the microscopic reasons for entropy changes is left as an interesting challenge for future studies.

In the last few years, we have witnessed a massive increase in the development of stimuli-responsive copolymers with various architectures for the potential use in “smart” functional materials, such as drug carriers, antifouling coatings, soft biomimetical tissue, nanoreactors, and carbon nanotube dispersions.1−9 The virtue of stimuli-responsive copolymers is that close to their lower critical solution temperature (LCST), a stimulus can easily switch the copolymer’s hydration properties from being mainly hydrophilic to hydrophobic, leading to substantial changes in its physicochemical properties. Thus, near the LCST the material is easily tunable to possess the functional behavior adequate for a desired application. A very important tool to control the LCST of a polymer is copolymerization and functionalization in order to introduce electrostatically charged groups and the subsequent fine-tuning by salt concentration and the solution’s pH value.10−24 Simple, analytical formulas which describe and predict LCST changes with charge fraction and salt concentration could be highly useful for the guidance of these effects. In this contribution we present a starting point for such a theoretical framework for weakly charged polymers and compare to novel, well-controlled and systematic LCST estimates by cloud-point measurements of the responsive copolymer poly(N-isopropylacrylamide-co-1ethyl-3-vinylimidazolium bromide), in short, poly(NIPAM-coEVImBr).7 We find very good agreement of our theoretical approach with the experiments for a large parameter space covering charge fractions up to 8% and the millimolar to molar range of salt concentrations. Our description may serve as a © 2014 American Chemical Society

Received: December 17, 2013 Revised: January 30, 2014 Published: March 6, 2014 2096

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numbers of chargeable copolymer monomers and total monomers N, will lead to a perturbation of the bimodal free energy distribution. In the following we denote α as charge f raction and χ as the copolymer f raction. As we will show later, for weak charging or large salt concentrations the effects in the compact globule will be much larger than for the expanded state due to the close spatial proximity of the charged groups and the resulting electrostatic repulsion. Thus we can assume that charging elevates the compact state by an electrostatic energy ΔΔGel(α;cs), while the expanded state remains unperturbed, cf. the illustration in Figure 1. The small shift ΔΔGel(α;cs) will be a function of charge fraction α and the salt concentration cs and can be reversed by a small temperature change ΔT, which controls the equilibrium weight between coil and globule states. For small temperature changes and weak copolymerization, ΔT can be obtained by Taylor-expanding ΔG(T,χ,α,cs) viz.

MATERIALS AND METHODS

Experimental Materials and Methods. In our experiments, the copolymers were prepared via free radical polymerization of NIPAM and an ionic liquid monomer, 1-ethyl-3-vinylimidazolium bromide (EVImBr) as detailed in previous work.7 The solution stability of the copolymer was investigated through temperature-dependent UV/vis turbidity measurements at different salt concentrations cs between 0.01 and 1 M and four copolymer fractions χ = 1.2%, 3.1%, 4.8%, and 7.6%. The raw transmission data is presented in the Supporting Information. The LCST is then estimated by the location of the “cloud point”, that is, the temperature above which the polymer coils aggregate and precipitate and the dispersion becomes turbid as estimated by light transmission loss. Since the volume transition from coil to aggregated globules is not always sharp,32 the cloud point depends on the exact criterion how much light is transmitted through the solution. Therefore we employ and compare the two criteria commonly used, i.e., the cloud point is defined by either 50% or 90% transmittance. Furthermore, the pH was fixed to 7.0 in all measurements. As we are treating ionic liquid-like monomers without titratable hydrogens (no pKa values), we can safely assume the fully charged state of this group, that is, a valency of z = +1. The absolute charge fraction is defined as α = |zχ|. Thermodynamic Model. In our theoretical model, we assume that the copolymer volume transition at the LCST can be understood as a transition from a dense collapsed state of the copolymer to an expanded, coil-like state in a bimodal free energy landscape33−35 as a function of the copolymer specific volume v. Schematically, the free energy G(v) of such a two-state model is shown in Figure 1, with the

ΔG(ΔT , χ , α , cs) ≃ ΔG(T0) + +

∂ΔG ∂ΔG ΔT + χ ∂T 0 ∂χ 0

∂ 2ΔG ∂ΔG ΔTχ + cs + ΔΔGel(α ; cs) ∂T ∂χ 0 ∂cs 0

(1)

where the subscript “0” denotes the expansion around the electroneutral and salt-free reference state of the homopolymer at T0, χ = 0, and cs = 0. In our study this reference state is the pure poly(Nisopropylacrylamide) (PNIPAM) homopolymer. We expand only to linear order in T because the changes in temperature are assumed to be small (ΔT ≪ T0) and heat capacity (second order in T) effects are neglected. As will become clearer in the remainder of the manuscript, it is important to expand ΔG also linearly in specific copolymerization χ and salt concentration cs, as well as to the cross term in T and χ. The expansions in χ and cs consider nonelectrostatic effects on ΔG due to the specific change in chemical solvation thermodynamics upon copolymerization or the addition of salt, respectively. In other words, even for a neutral polymer, copolymerization by other neutral monomers or the addition of salt will in general change the LCST and the polymer thermodynamics.14,15,36,37 We have neglected second order terms in the expansion in cs and χ and their cross term because those are not important for the weak perturbation by charge and the simple salt studied here. Also the cross term for the expansion in T and cs can be neglected because experiments do not show a marked dependence of the transition entropy by the addition of simple salts (∂ΔS0/∂cs ≃ 0). The entropy, however, is very sensitive to the even small changes in functionalization.14,15,36,37 Therefore the cross term in T and χ (that is, ∂ΔS0/∂χ), which describes entropy changes by functionalization, is nonzero and needs to be included. These specific effects have to be considered on top of the nonspecific charge effects. The latter, purely electrostatic charging and salt screening effects, are included nonlinearly in the term ΔΔGel(α;cs) in (1), which will be defined further below. For the convenience of the reader the reasoning behind our expansion is illustrated in more detail in the Supporting Information. There also the full expansion including a discussion of heat capacity effects is included. Using the thermodynamic relation ΔS0 = −∂ΔG(T0)/∂T and defining the thermodynamic slopes ∂ΔG/∂χ = mχ, ∂ΔG/∂cs = m, and ∂ΔS0/∂χ = mχ′, and enforcing ΔG(ΔT,χ,α,cs) = 0 at the new LCST with value T0 + ΔT, it is

Figure 1. Schematic free energy landscape G(v) of a copolymer in a two-state model as function of the specific volume v. At the LCST the collapsed (globule) and extended (coil) states at vg and vc, respectively, of the neutral copolymer are equally probable and ΔG = Gg − Gc = 0 (solid blue curve). The free energy of the compact globule state is elevated by a small amount ΔΔGel(α;cs) by charging a small fraction α of copolymer monomers (dashed red curve), thus favoring the coil over the globule state. This perturbation can be compensated by a change ΔT of the temperature leading to a new LCST. In the illustrative conformations, the blue balls denote neutral polymer beads, while red balls are charged beads. corresponding minima in the compact globular (g) and the extended coil (c) states at vg and vc, respectively. Due to the coarse-grained nature of our model, such a view should also be applicable for polymer networks and gels. Right at the LCST of the neutral homopolymer, in the following denoted as T0, the free energy difference (per monomer) ΔG(T0) between collapsed and coil states vanishes, that is, ΔG(T0) = Gg(T0) − Gc(T0) = 0. For the latter equation to hold we assume very narrow free energy minima such that in equilibrium G(v) is dominated by its values right at vc and vg, respectively. The transition is accompanied by a transition entropy per monomer ΔS0 ≡ ΔS(T0). If we identify G with the Gibbs free energy, then the usual thermodynamic relation ΔG = ΔH − TΔS provides ΔS0 = ΔH(T0)/T0 at the LCST, where ΔH(T0) is the corresponding transition enthalpy per monomer. Charging the copolymer by a small fraction α = |zχ|, defined by the charge valency z times the copolymer fraction χ, the ratio between the

(ΔS0 + mχ ′χ )ΔT ≃ ΔΔGel(α ; cs) + mχ χ + mcs

(2)

We rearrange to obtain an explicit expression for the change in LCST

ΔT ≃

ΔΔGel(α ; cs) + mχ χ + mcs ΔS0 + mχ ′χ

(3)

Let us briefly further justify our model with regards to the underlying physical interactions. The physical mechanisms behind the LCST are the hydrophobic attraction which increases with increasing temperature. Our model, however, is a perturbation approach, that is, we are interested only in the first order change of the free energy by 2097

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For high salt concentrations or very small charge fractions (y ≪ 1), eq 8 further simplifies to

weak charge functionalization (at high salt concentrations) on top of the reference polymer free energy. Assuming two-state behavior the model lives on the most macroscopic thermodynamic level and all microscopic interaction effects are effectively described by thermodynamic parameters of the reference polymer, such as the transition entropy ΔS0. By specific functionalization, the average monomer− monomer interactions in the system change, and with that the entropy changes, as expressed by the parameter mχ′. Thus, effectively the perturbation of hydrophobic interactions by functionalization is included. This becomes also clearer in the last section of this manuscript, where we discuss the connection of the thermodynamic slopes m, mχ, and mχ′ to microscopic interactions. The nonspecific, pure charging effects are exclusively modeled by ΔΔGel, as will be detailed in the following. We now estimate the purely electrostatic free energy change ΔΔGel(α;cs) upon charging the copolymer in the solution with monovalent salt concentration cs. We make the mean-field assumption that the collapsed copolymer globules can be modeled as homogeneous, dense compact entities with a mean monomer number density ρg = N/Vg and the salty environment will readily lead to a complete electrostatic neutralization of the globule. Experiments clearly indicate that by crossing the LCST, polymers change their volume by over 1 order of magnitude,14,16,38,39 that is, the monomer density in the globule is much larger than in the coil ρg ≫ ρc. For high salt concentrations or weakly charged polyelectrolytes the entropy due to osmotic pressure in the coil state is on the order of ≃ kBTαρc/2cs,28 which needs to be compared to the one order of magnitude larger ≃ kBTαρg/2cs, as we will see below. Thus, the entropy due to osmotic pressure changes in the highly dilute coil state can be safely neglected. Hence, the free energy change in our model is just based on the changes of ionic osmotic pressure in the globule, called the Donnan pressure.25 Given the globule with volume Vg in contact with the salty reservoir at constant chemical potential, the free energy change per monomer is given by

ΔΔGel(α ; cs) = ΔΠg (α ; cs)Vg /N = ΔΠg (α ; cs)/ρg

ΔT ≃

ΔT ≃

c − = cs exp(eβ Δϕ)

(4)

(5)

are the cation and anion concentrations inside the globule, respectively, determined by the Donnan potential Δϕ. The latter describes the difference between the electrostatic potentials in the globule and the reference bulk solution. It is determined by the electroneutrality condition c+−c−+αρg = 0, which leads to β Δϕ = ln(y +

y2 + 1 )

(7)

The final result for the change ΔT of the LCST in a monovalent salt reads

ΔT ≃

mχ χ 2kBT0cs mcs + ( y 2 + 1 − 1) + ΔS(χ )ρg ΔS(χ ) ΔS(χ )

ΔS(χ )

+

mcs ΔS(χ )

(9)

kBT0ρg α 2 4ΔS0cs

(10)

Table 1. ΔS0 is the Transition Entropy per Monomer of the Pure PNIPAM Homopolymer As Collected from Sources in the Literaturea

(6)

where y = αρg/(2cs). In our perturbation approach, y is the “smallness” parameter, that is, our approach should be valid for small y ≲ 1. We find for the nonspecific, purely electrostatic free energy change

2k T c ΔΔGel(α ; cs) = B 0 s ( y 2 + 1 − 1) ρg

mχ χ

which represents a limiting law for very weak perturbations by charge effects only. It is noteworthy that it has the same scalings in α and cs as mean-field descriptions of the second virial coefficient of polyelectrolyte solutions.41 Parameters and Fitting Procedure. Apart from the electrostatic “external” control parameters, such as the salt concentration cs and the charge fraction α, the other determinants in the theory are intrinsic material constants specific to the neutral reference homopolymer, such as the LCST T0, the transition entropy per monomer ΔS0, the globule density ρg, and the specific changes induced by copolymerization and ions as expressed by mχ, mχ′ and m, respectively. We discuss experimental values of those and our fitting procedure in the following. The nonelectrostatic ion-specific effects for simple, monovalent salts such as KBr are typically linear in salt concentration,42,43 that is ∝ mcs, as already included in our expansion in eq 1. The so-called m-value is the coefficient which describes the change of transition free energy per monomer with salt concentration and is a salt-specific number.44 Thus, higher order ion-specific effects are not important in this work but will be for more complex salts where deviations from linearity occur.42,43 For our salt KBr and pure PNIPAM, which is our reference polymer, this m-value has been measured and is m = −111 J/mol/(mol/L).42,43 For our system of poly(NIPAM-co-EVImBr) we calculated the m-value from the data at high salt concentrations (cs > 0.2 M), where nonspecific charge effects have vanished, and found m = −105 ± 15 J/ mol/(mol/L) in very good agreement with the one in the pure PNIPAM system. The found m-value stays fixed in our fitting procedure. Via subtracting the specific ion effects (∝ mcs) from our experimental data we find that the LCST for the PNIPAM homopolymer reference state is T0 = 306 ± 1 K, independent of the particular choice of the copolymer fraction χ. This LCST value is very close to the experimentally known LCST of pure PNIPAM of about T0 = 305 ± 1 K.4 We conclude that specific copolymerization effects on the free energy are small for our system, that is, mχ ≃ 0, and can be neglected. We keep the value of T0 = 306 K fixed in our global fitting procedure. The transition entropy per monomer ΔS0 for pure homopolymeric PNIPAM chains has been experimentally measured for various chain lengths with data summarized in Table I. The spread in the data is

where kBT = β is the thermal energy, and ΔΠg(α;cs) = Πg(α;cs) − Πb is the net osmotic pressure inside the globule, that is, the difference between osmotic pressures inside the globule and bulk, respectively. For a simple salt like KBr the osmotic coefficient in the studied concentration range cs = 0 and 0.1 M is between unity and 0.93,40 respectively, so that the ions can be well treated as an ideal gas. Therefore we have βΔΠg(α;cs) = c+ + c−−2cs, where

and

4ΔS(χ )cs

+

In the limit of very weak copolymerization and no ion-specific effects we can set mχ = 0, mχ′ = 0, and m = 0, and one obtains a quadratic scaling with charge fraction α and an inversely linear scaling in cs as

−1

c+ = cs exp(− eβ Δϕ)

kBT0ρg α 2

(8)

where we abbreviated by defining the transition entropy of the electroneutral copolymer ΔS(χ) = ΔS0 + mχ′χ. Equation 8 is the key equation derived in this work.

a

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ΔS0 [J/mol/K]

reference/comment

17.0 6.0 12.4 18.4 15.3 17.8 19.2 17.7 21.3

ref 45 20-mer of PNIPAM46 92-mer of PNIPAM47 600-mer of PNIPAM48 ref 14 C18−PNIPAM-C1849 ref 50 microcalorimetry51 end group effects36

Maximum errors are about ±2 J/mol/K. dx.doi.org/10.1021/ma402577h | Macromolecules 2014, 47, 2096−2102

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relatively narrow and we fix ΔS0 in our fitting procedure to the mean experimental value ΔS0 = 15 J/mol/K. Previous experiments, however, show a very sensitive dependence of ΔS0 on functionalization.15,36,37 This change is expressed in the parameter mχ′ = ∂ΔS0/∂χ and depends on the specific chemical nature of the copolymerization. Since measurements are unavailable, we will use mχ′ as one of the global fitting parameters in our study. For pure PNIPAM, the globular density ρg was estimated previously to be in the molar range, roughly 3−6 M, based on the measurements of the hydrodynamic or gyration radius of the polymer in the collapsed state well above the LCST.39,52,53 We find, however, that with those values of ρg the experimental data was difficult to fit in a comprehensive way. We suspect that this deficiency may result from our simplified two-state treatment of the copolymer free energy landscape. Experiments actually indicate that at coexistence the collapsed state may be only partially occupied by monomers due to the presence of ’crumpled coil’ or heterogeneous cluster structures.39,53 Therefore, we use the effective globule number density ρg as the second global fitting parameter, expected to be in the molar range. Under the bottom line our approach has only two fitting parameters: the specific change of the transition entropy ΔS0 with copolymerization χ as expressed by the variable mχ′ and the effective monomer number density ρg in the globular state which is expected to be in the molar range. With those constraints we perform a global fit, i.e., the whole data set in our measured {χ;cs} regime will be described by two single values of mχ′ and ρg, respectively. The reference LCST T0 = 306 K, the salt-specific correction m = −105 J/mol/(mol/L), and the transition entropy ΔS0 = 15 J/mol/K were kept fixed during the fitting. Our experiments provide the change in LCST by an amount ΔT(χ,cs) for a wide range of salt concentrations cs between 0.01 and 1 M and for four copolymer fractions χ = 1.2%, 3.1%, 4.8%, and 7.6%. The latter correspond to the same absolute charge fractions α = |zχ| since in our case z = +1. The changes in LCST are estimated by cloud point measurements (see Materials and Methods). Since the volume transition from coil to aggregated globules is not always sharp,32 (see the raw data in the Supporting Information), the cloud point depends on the exact transmission criterion. Therefore we decided to employ and compare two common criteria in which the cloud point is defined by either 50% or 90% transmittance. Our model is expected to perform well for low charge fractions. Consistently we apply our global fitting only to data below a charge fraction of 5%. The fit for the 7.6% is then a prediction of the theory. The fitting was performed by leastsquares error minimization of the global fit by mχ′ and ρg to the experimental data set {α;cs} = {1.2%, 3.1%, 4.8%; 0.015 M, 0.020 M, 0.040 M, 0.1 M}.

Figure 2. LCST T0 + ΔT for all four copolymer fractions χ (i.e., charge fractions α) as a function of salt concentration cs (upper panel), and for selected salt concentrations cs as a function of charge fraction α (lower panel). Filled circles denote experimental T0 + ΔT(α;cs) data derived with the 50% transmittance criterion. Solid lines represent best theoretical fit based on eq 8 for α < 5%. The fitting parameters are summarized in Table 2.

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RESULTS AND DISCUSSION The experimentally measured ΔT(α;cs) data and the theoretical fits are presented in Figures 2 and 3 for the two cloud point criteria based on either 50% or 90% transmittance, respectively. The corresponding best-fit parameters mχ′ and ρg are summarized in Table 2. As we can see the fits work very well for the lower charge fractions