AdsorptionâAttraction Model for Co-Nonsolvency in Polymer Brushes

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Adsorption−Attraction Model for Co-Nonsolvency in Polymer Brushes Jens-Uwe Sommer* Institute Theory of Polymers, Leibniz-Insitut für Polymerforschung Dresden, Hohe Strasse 6, 01069 Dresden, Germany Institute for Theoretical Physics, TU Dresden, Zellescher Weg 17, 01069 Dresden, Germany ABSTRACT: We study the properties of a polymer brush exposed to a mixture of two solvents where one component, called the co-nonsolvent (CNS), has a stronger preference with respect to the polymer. The concept of preferential adsorption of CNS onto the polymer, as recently proposed by Mukherhi, Kremer, and Marques [Nat. Commun. 2014, 5, 4882], is combined with the mean-field Alexander−de Gennes approach for the polymer brush. The key assumption is that CNS can form bridges between two monomers which is associated with a further gain in free energy, thus leading to an effective monomer−monomer attraction. The adsorption equilibrium of CNS at a given value of the monomer concentration results in a concentration-dependent χ-function for the polymer brush which describes the effective interactions between the monomers in the mixed solvent. This in turn can lead to a discontinuous collapse transition and to a corresponding reentry transition at higher CNS concentrations. The problem can be analytically treated for a minimal model where the increase of self-volume of the monomers due to adsorption of CNS is neglected. In this case the collapse and the reswelling transition have the same signature. For low brush densities in the noncollapsed state we give an analytic approximation for the spinodal of the collapse. This also allows to define two scaling variables instead of the three control parameters which are the grafting density, the CNS−monomer selectivity, and the volume faction of CNS. The proposed effective free energy contribution resulting from the CNS adsorption equilibrium can be transferred to other systems such as to gels or dendrimers.

I. INTRODUCTION When polymers are exposed to mixed solvents, various phase transitions can be induced by variation of solvent composition, temperature, and polymer concentration. A particular interesting scenario is co-nonsolvency which has attracted some renewed attention recently.1−4 Co-nonsolvency occurs if a mixture of two good solvents causes collapse and segregation of the polymer phase in a certain range of compositions of the solvents. For gels or brushes this can lead to volume phase transitions. If one considers perfect miscibility of the two solvent components themselves, only the different affinities of both components with respect to the polymer should be relevant, and we call the component which is preferred by the polymer the co-nonsolvent (CNS), although this notation is somewhat misleading since the CNS is the better solvent here. Very recently the group of de Beer and Vancso5 have investigated a brush made of PMIPAm chains in a methanol−water mixture. For the given parameters in this work a nonmonotonic but smooth behavior of the brush height as a function of the CNS fraction has been observed. Moreover, the authors have discovered a strong nonmonotonic dependency of the friction between the brush and a gold colloid as a function the CNS fraction. One simple way to explain co-nonsolvency is as follows: If the CNS is diluted by solvent, the monomers have to compete © XXXX American Chemical Society

for the preferred solvent species. Sharing CNS by two monomers, i.e., building up monomer−monomer contacts mediated by CNS as indicated on the rhs of Figure 1, is a possible way to gain a higher amount of interaction enthalpy. To reach more binding with the CNS, a collapsed conformation can be preferred, even if the entropy of chain conformations is reduced. Thus, the CNS may be envisioned as a glue for monomers or as boson-mediated particle−particle attraction if one fancies an analogy to particle physics. This concept is supported by atomistic simulations by Mukherji and Kremer,6 who showed that the collapse behavior of single PNIPAm chains in a mixture of water and methanol is connected with methanol excess coordination around the monomers. This is in some contrast to the alternative assumption of strong attraction between the two solvents.3 In order to rationalize the concept of the preferred solvent, two routes can be taken for the case of the polymer brush: First, one can assume the brush is forming a potential trap for the CNS given by its mean density in equilibrium with a bulk phase of given chemical potential (concentration) of CNS. This is sketched on the left-hand side of Figure 1. The second Received: October 17, 2016 Revised: February 21, 2017

A

DOI: 10.1021/acs.macromol.6b02231 Macromolecules XXXX, XXX, XXX−XXX

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the resulting effective Flory−Huggins (FH) model contains strong nonlinearities which can be expressed by an effective concentration-dependent χ parameter. The latter describes the effective interaction between the monomers of the brush in the presence of the two concurrent solvents. It is therefore not straightforward to map this model to a more generic phase transition scenariohowever, such simplification is desirable to rationalize the collapse and reswelling of the polymer−CNS system. The purpose of this work is to consider the extension of the MKM model to the case of a polymer brush where a true volume phase transition can occur and to analyze this transition analytically by considering the simplest case of the CNS− polymer coupling. The rest of this work is structured as follows: In section II we define the adsorption−attraction model in terms of the free energy for the polymer brush and discuss its general consequences. A simplified version of the model, the minimally coupled model which can be treated analytically, is considered in detail in section III. A phase diagram of the brush is discussed as well. In section IV we outline some general consequences of the adsorption−attraction model and show its relation with the “type II” transition proposed for polymers earlier.

Figure 1. Sketch of two models for CNS uptake by the polymer brush. Left: in the a priori mean-field model the CNS is attracted by a potential well which is proportional to the average density of the monomers in the brush and to an energy χx. Bulk and brush (of height H) form two distinct thermodynamic phases in equilibrium, and CNS can be exchanged as indicated by the double-arrow. Right: adsorption−attraction model. CNS adsorbed by the monomers (blue) are in equilibrium with the free CNS (white). Two monomers can share CNS (red) and thus lead to transient loops and bridges, as well to an increase of the adsorption energy by an amount of γϵ.

possibility is to consider an adsorption process where the monomers being the substrate for the CNS. This is sketched on the rhs of Figure 1. The attraction between monomers is then the consequence of contacts between monomers sharing the same CNS molecule. Such contacts also enhance the adsorption by the increase of adsorption energy per CNS. By definition, the latter approach introduces a coupling between adsorption and attraction and in turn leads to an effective nonlinear interaction model beyond the binary coupling, and we will call it the adsorption−attraction model. The similar distinction has been made already by Tanaka et al.,7 who introduced this difference using the terms “spacebound” and “site-bound” adsorption models. However, the meaning of space-bound molecules should be considered with some concern if it is taken on a microscopic level: The preferred species (CNS) is attracted by the monomers via short-ranged interactions. If the CNS is not within the range of interaction, there is no attractive force which should keep this molecule in the close environment of the polymer chains. In fact, there is no thermodynamic or kinetic argument to predict a corona of nonbound CNS around the polymer chains which would correspond to the space-bound state. It needs long-range interactions to achieve the space-bound state (such as electrostatics) in order to force small molecules (then usually counterions) to stay within the otherwise globally charged object (brush, dendrimer, gel). Our distinction according to Figure 1 is meant with respect to the level of approximation (two-component mean-field model or adsorption−attraction model) only. Recently, Mukherhi, Kremer, and Marques1,8 (MKM) proposed an adsorption model for co-nonsovency and showed that this model can explain the nonmonotonic collapse behavior of a single chain as a function of CNS concentration by comparing it to simulation results. Additionally, a simulation study by Heyda, Mudzdalo, and Dzubiella4 indicates the formation of bridges between monomers formed by the CNS to be responsible for the collapse-like behavior in a single chain. Still, in the approach of MKM the level of mathematical complexity is high and numerical solutions have been presented. As we will outline in detail below, this lies in the fact that after considering the adsorption problem of the CNS

II. ADSORPTION−ATTRACTION MODEL FOR CNS IN A POLYMER BRUSH We implement the model sketched on the rhs of Figure 1 for the polymer brush in the following way: solvent and CNS being both athermal solvents for the polymer, and all components are assumed to be perfectly miscible. Following the original model by Flory and Huggins,9,10 this will be expressed by the mixing contribution to the free energy in the volume V given Fmix = kBTV(1 − cm) ln(1 − cm), where cm denotes the volume fraction taken by the monomers and 1 − cm represents the common volume fraction of solvent and CNS. Since both solvent components are noninteracting, there is no other contribution to the free energy from the solvents. So all the usual Flory− Huggins solvent−solvent and polymer−solvent χ parameters are zero. The only difference between solvent and CNS is the preference of the monomers for the CNS. This can be expressed by an energy gain (selectivity), ϵ, for a CNS in contact with a monomer; see rhs of Figure 1. Thus, the preference of the polymer for the CNS results in adsorption or “wetting” of the chains with CNS, which is controlled by a balance between the entropy gain of the free CNS and free energy gain by the adsorption. So far we follow the original idea of MKM for the description of a single chain as presented in ref 1. The preference of the CNS leads further to a “stickiness”, which induces contacts between monomers sharing CNS. For a homogeneous brush we will treat this effect in a mean-field approach by defining a free energy of attraction which is the product of the probability to find a monomer which has adsorbed CNS close to a bare monomer. Finally, for the polymer brush a decrease in density can only be realized by stretching of the chains which reduces the conformational entropy, and we assume Gaussian elasticity here. All this results in the following free energy expression of the brush−solvent− CNS system: B


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Macromolecules f (ϕ , c) = fads + fattr + fbrush

and CNS with respect to the miscibilityboth species are solvent-occupied sites in the FH model with a total fraction given by 1 − cm = 1 − c − v1ϕc. The increase of the excluded volume due to adsorbed CNS breaks the symmetry ϕ → 1 − ϕ of the free energy. The essential physics of this model, in particular the collapse transition, however, is contained in the symmetric case, which is formally obtained for v1 = 0. Being a mean-field approach, the prefactor of the elasticity term differs from 1/2 we have chosen here for convenience, but this can be included into the definition of the grafting density. We note the relation between the volume fraction and the height of the brush, H = σN/c. The chain length does not explicitly enter the free energy model of the polymer brush. That means that all brushes having the same grafting density display universal behavior. This holds as long as the brush is well above the overlap threshold. Minimizing f(ϕ,c) with respect to both arguments gives the simultaneous equilibrium for both CNS adsorption and the monomer concentration in the brush. Then, we obtain the equilibrium density/height of the brush as a function of the external control parameters such as μ and ϵ. In the upper left panel of Figure 2 we display the result for the equilibrium density of the brush as a function of the chemical potential, μ, for several parameters of the interaction constant ϵ, and for γ = 1 and v1 = 0.25. In the lower left panel of Figure 2 we display the same results for the height ratio H/H0 vs the chemical potential of the CNS, where H0 corresponds to the solution for the brush without CNS, i.e., ϵ = 0. Here, the

with

fads = ϕ ln ϕ + (1 − ϕ) ln(1 − ϕ) − μϕ − ϵϕ fattr = −2ϵγϕ(1 − ϕ)c fbrush =

⎛1 ⎞ 1 σ2 + ⎜ − 1 − v1ϕ⎟ ln(1 − c − v1ϕc) 2 ⎝c ⎠ 2 c

(1)

Here, f = F/MN is the free energy per monomer unit where N denotes the number of (Kuhn) monomers in the polymer chain, M the number of chains, and c the density of monomers. The energy units are taken as kBT throughout this work, and the length scale is given by the Kuhn segment length. The first part, fads, corresponds to the adsorption, or binding, of adsorbed CNS on the monomers of the chains. The number of CNS per monomer is denoted by ϕ. The preferential adsorption is expressed by the energy gain ϵ per adsorbed CNS. The first two terms in fads in eq 1 correspond to the lattice gas state of CNS on the polymer chains as a substrate. The chemical potential of the CNS can be denoted as ⎛ ρ ⎞ μ = ln⎜ ⎟ ⎝1 − ρ ⎠

(2)

where ρ is the fraction of CNS in the solvent bulk. The term fattr denotes the mean-field attraction between monomers caused by forming a bridge due to CNS; see rhs of Figure 1. This term is proportional to the average probability that a given monomer (ϕ or (1 − ϕ) state) is in contact with a monomer of the other state (1 − ϕ or ϕ). Here, the state of the monomer is defined by either having adsorbed a CNS (with probability ϕ) or being empty (only surrounded by solvent, with probability 1 − ϕ). The strength of this additional attraction (“glue”) is given by γϵ, which takes into account bridging and size effects of the CNS by considering γ ≠ 1. Note that for the case of a single chain as investigated by MKM such bridges formed by CNS were taken into account differently. Here, an ad hoc penalty for an average loop has been introduced.8 In a dense brush, however, it is far more likely to form bridges between different chains which are interpenetrating each other instead of forming loops in a single chains. This motivates us to apply a mean-field concept as given by fattr. We can formally rewrite the attractive free energy in eq 1 as fattr = −χc by defining an FH interaction f unction as

χ = 2ϵγϕ(1 − ϕ)

(3)

understanding that ϕ depends on ρ and c. Let us note that the analogy with the FH parameter is formal only, and the χ function defined here is an extension of the concept of the interaction parameter introduced by Flory. We will come back to this point further below where we can present an analytic solution for χ(c). The last group of terms, f brush, correspond to the conformational free energy of the brush as a function of the volume fraction of monomers and of the grafting density, σ, within the Alexander−de Gennes approach11,12 together with the mixing free energy of the miscible solvents. The first term in f brush is the free energy for stretching in Gaussian approximation, and the second term is the FH free energy per monomer of the brush. Here, we took into account the effective increase of volume by adsorption of CNS, where v1 denotes the added volume fraction by full saturation of the polymer by CNS. As noted already above, we make no difference between solvent

Figure 2. Upper panels: density of the brush as a function of the chemical potential, eq 2, of the CNS for the case v1 = 0.25 and v1 = 0, respectively, as obtained from the numerical minimization of eq 1. The grafting density has been chosen as σ = 0.005. The complementary volume fraction, 1 − c, is occupied by an ideal mixture of the two solvents with the volume fraction ρ of the CNS in the solution. Lower panels: relative brush height for the same parameters. The double arrow highlight jumps in the density plots which correspond to collapse (asymmetric case) and to the reentry (symmetric case). C


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Table 1. Table of Essential Symbols Used in This Worka

discontinuous transition (jump in height) is easier to discriminate. We can clearly see a nonmonotonic behavior which displays a discontinuous transition for ϵ = 2. On the rhs of Figure 2 we show the corresponding results for the case, v1 = 0. Here, the value of ϵ which is necessary to obtain a discontinuous phase transition is about 1.6. The symmetric case, v1 = 0, displays all relevant features of the transition. Besides breaking the symmetry with respect to the maximally collapsed state, the additional excluded volume shifts the critical point of the transition toward higher values of ϵ. We note that the results found by the group of de Beer and Vancso5 for PNIPAm using methanol as CNS in water, presented in Figure 1 in the referenced work, is very similar to the behavior displayed for the lowest CNS selectivity, ϵ = 1.2 in the symmetric case; see lower right panel of Figure 2. The height shrinkage here is about a factor of 2, and the minimum corresponds to a volume fraction of about 0.25. However, to make a more systematic connection with the experimental system, a systematic variation of the experimental grafting density and degree of polymerization would be necessary. For the sake of comparison, we consider the two-phase mean-field model according to the sketch on the lhs of Figure 1 in the Appendix. This model has been originally introduced by Opfermann, Coalson, Jasnow, and Zilman (OCJZ)13 for the case of attractive nanoparticles instead of CNS. A major difference between the two approaches is that the OCJZ model predicts a very high value of the attraction energy between the CNS and the brush of the order of 10 kBT to reach a discontinuous transition, while this value being of the order of 2 kBT for the adsorption−attraction model.

symbol f c ϵ γϵ ϕ ρ μ z σ χ(c) v0 V κ y s a

All energies are given in units of kBT; all length scales in units of the Kuhn length.

maximum. We introduce the deviation form this halfoccupation of the chains by CNS by the variable δ defined by 1 ϕ = (1 − δ) (6) 2 2 In the limit of δ ≪ 1 we obtain (ignoring constant terms) 1 1 1 f (δ , c) = δ 2 + δ ln z − ϵγ(1 − δ 2)c + fb (c) (7) 2 2 2

III. SOLUTION OF THE SYMMETRIC CNS MODEL AND PREDICTION OF A DISCONTINUOUS COLLAPSE TRANSITION The general problem of the adsorption−attraction model is the same as for previous approaches to the co-nonsolvency problem: The equations minimizing the full free energy are analytical nontractable. This hampers a rational understanding of the nature of the collapse transition. We have already noted that the essential physics of the nonmonotonic behavior of the brush density/height, and also the occurrence of the discontinuous collapse transition as a function of the system parameters, is contained in the symmetric model (see also Figure 2). The “thickening” of monomers due to adsorbed CNS leads to repulsive contributions only and shifts the height of the brush to larger values if CNS dominates. On the other hand, by restricting to the symmetric case, v1 = 0, the analytical treatment is greatly simplified. Thus, we will consider this case in this section in detail. For a quick overview over the symbols used, see Table 1. For the case v1 = 0 we rewrite eq 1 defining the symmetric model:

Here, fb (c) = fbrush (v1 = 0) =

⎛1 ⎞ 1 σ2 + ⎜ − 1⎟ ln(1 − c) 2 ⎝ ⎠ 2 c c

δ=−

+

( 1c − 1) ln(1 − c) (see

1 2

ln z

1 + ϵγc

(8)

Resubstituting this result into eq 7, we obtain the free energy of the brush at a given density in equilibrium with the CNS: 1 f (c) = − ln 2 z − χ (c)c + fb (c) (9) 8 with the χ function χ (c ) =

ϵγ ⎡ 1 (μ + ϵ)2 ⎤ ⎥ ⎢1 − 2⎣ 4 1 + ϵγc ⎦

(10)

A detailed derivation can be found in the Appendix. From this expression we see that the maximum possible value of the χ function is χ = ϵγ/2. Thus, only for ϵγ > 1 a collapsed state of the brush is possible. The concentration dependence of the χ function is key to the discontinuous collapse for the case γϵ > 1, since larger values of c can drive the effective solvent quality from good to poor. Thus, we can already identify the necessary condition for a discontinuous collapse transition to be γϵ > 1. The concentration-dependent χ function expresses the fact that CNS adsorption in equilibrium with the brush has an impact on higher viral coefficients and not only on the second (as the classical χ parameter). This can be seen by expanding χ(c) with

(4)

In eq 4 the symmetry ϕ → 1 − ϕ is only broken by the logarithm of the adsorption constant z = e ϵ+ μ

1 σ2 2 c2

eq 1). The coupling of CNS with the brush, third term in eq 7, favors half-occupation of the polymers by the CNS (smaller values of δ2) because this leads to a maximum energy gain by sharing CNS. The equilibrium condition with respect to δ leads to

f (ϕ , c) = ϕ ln ϕ + (1 − ϕ) ln(1 − ϕ) − ϕ ln z − 2ϵγϕ(1 − ϕ)c +

free energy per monomer concentration (volume fraction) of monomers energy gain for contacts between CNS and monomers energy gain for a bridge formed by monomers sharing CNS fraction of monomers occupied by adsorbed CNS volume fraction of CNS in the bulk chemical potential of CNS in the bulk, μ = ln(ρ/(1 − ρ)) adsorption constant, ln z = μ + ϵ grafting density χ function, concentration-dependent χ “parameter” excluded volume constant: v0 = (1/2)(1 − γϵ) external field caused by interaction of CNS: V = (1/8) ln2 z parameter describing the effective strength of CNS attraction: κ = (γϵ − 1)/2γϵ rescaled brush density: y = γϵc rescaled grafting density: s = γϵσ

(5)

Obviously, the maximum contraction of the brush is reached for the case ϕ = 1/2, where the χ-function (see eq 3) takes its D


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Figure 3. Left: the function V(y) according to eq 13 for various CNS−monomer interaction parameters ϵ. Here, γ = 1 has been chosen. The horizontal lines indicate the Maxwell construction for the discontinuous transition. The value of ϵ = 1.5 is close to the critical point. The scaling parameter s = ϵσ was fixed to s = 0.005. Right: the comparison between the full FH equation of state including all higher virial coefficients and the approximation by the second virial only. The arrows indicate the spinodal points for collapse, y1, and swelling, y2.

Figure 4. Chemical potential of the CNS as a function of the volume fraction of the brush for γ = 1. Left: the behavior of the chemical potential of the CNS volume fraction vs the concentration of the brush, c, for a fixed grafting density for various values of the CNS−monomer interaction. The curves correspond to the isotherms of the system. Horizontal lines mark the maximum of the collapse which is shifted under variation of ϵ according to eq 16. Right: the same quantity, μ = ln(ρ/1 − ρ), for the sake of clarity only for the upper leave of the solution (reentry). Here, the grafting density of the brush has been varied for a fixed value of the interaction constant. The arrow indicates the shift of the spinodal point of collapse (minimum of the isotherm) with respect to σ.

respect to c for small concentrations of the brush. In fact, the third virial coefficient can become negative, while the second, given by χ(c = 0), remains positive, and this can be considered as the generic cause of the discontinuity of the transition. Generally, this bears some similarity with the n-cluster model as proposed by de Gennes,14 which has been subsequently related with a concentration-dependent χ parameter; for a recent discussion, see ref 15. We will outline this point in more detail in the Discussion section. Minimizing the free energy in eq 7 with respect to the brush density reads 0=

2 ln (1 − c) σ2 1 1 (1 − γ ϵ) + γ ϵδ 2 − 3 − 2 2 c c2

where the symbol

2

1

ln (1 − c) = ln(1 − c) + c + 2 c 2 de-

notes all higher than second-order contributions to the logarithm, and the corresponding term in eq 11 thus contains all higher virials of the brush’s equation of state. We denote the excluded volume constant by

v0 =

1 (1 − γ ϵ) 2

(12)

which is smaller than zero for the potentially collapsed state. The formal solution for the equilibrium concentration of the brush can be obtained implicitly using eqs 11 and 8 by introducing the (symmetric) adsorption field, V, in the following form:

(11) E


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Figure 5. Left: phase diagram for the spinodal according to eq 19 toward the collapsed state obtained in the approximation of low brush density. Right: the critical line as obtained using the third virial expansion according to eq 20. The hatched region indicates the region of a discontinuous transition where the phase diagram on the left can be applied. Above the critical line the CNS induces a continuous crossover only.

⎧ 2 ln ⎪ 1 2 s2 2 ln z = (1 + y) ⎨κ + 3 + V (y) = 8 y ⎪ ⎩ def

(1 − )γ ϵ ⎪⎬⎫

Being independent of all other parameters, eq 16 gives a very simple test of the model and is indicated by the horizontal lines on the lhs of Figure 4. The scaling introduced with the variables y and s in eq 13 is broken by the higher virial coefficients. On the rhs of Figure 3 we display again the function V(y) for a specific interaction parameter together with the approximation disregarding all higher but second virials. We note that the approximation of the spinodal point (where the density becomes unstable toward the collapse) disregarding the stabilizing terms is rather good, indicated by y1 in the figure. Thus, at least for low values of the brush density, y ≪ 1, the second virial is sufficient to estimate the spinodal. This, together with the preserved scaling in this approximation, simplifies the complex phase diagram considerably and makes it accessible to simple analytical solutions. The spinodal point is given by finding the zero-point of the first derivative of V(y) with respect to y. In the approximation of the second virial, i.e., disregarding the contribution of 2ln(1 − c) and assuming y ≪ 1, we obtain the relation

y γϵ

y2

⎪ ⎭

(13)

where the scaling variable of the order parameter and the rescaled grafting density have been introduced: y = γ ϵc

and

s = γ ϵσ

(14)

The parameter κ is defined as κ = |v0| /ϵ = (γ ϵ − 1)/2γ ϵ

(15)

where the last relation is valid for γϵ > 1, which we have identified as the necessary condition for a discontinuous collapse. In Figure 3 we display the function V(y) for various parameters of the CNS−monomer interaction ϵ and for a fixed parameter of rescaled grafting density s = 0.005. Increasing the interaction strength, we reach a discontinuous transition which is indicated by a nonmonotonic behavior of V(y) where minima and maxima indicate instable (spinodal) points. For the given choice of parameters the critical point is close to ϵ ≃ 1.5. We note that the phase diagram is three-dimensional (σ, ϵ, μ), and a line of critical points under variation of the grafting density is obtained. The symmetry of the model with respect to high and low CNS density transitions is captured by the fact that V(y) represents the square of the logarithmic adsorption constant with two solutions μ±. This becomes explicit if we solve eq 13 for the chemical potential of CNS numerically. The result is displayed on the lhs of Figure 4. Here, also the unscaled value of the grafting density is used as a parameter. The true equilibrium results can be obtained by the Maxwell construction through the instabilities (not shown). The shift of the maximum density (minimum height) of the brush with respect to ϵ is exactly given by the solution for δ = 0 (maximum effective χ-parameter) and reads μ+ϵ=0→

ρmax 1 − ρmax

4

y s2 2 ≃2 1 ≃ y14 κ 3 + y1 3

(17)

Transforming this relation back to the original variables, we obtain c ∼ ϵσ . This, interestingly, corresponds to the Θ-point brush scaling. In the same level of approximation we obtain directly from eq 13 1 2 s2 ln z ≃ 3 8 y

(18)

which leads to the approximate equation for the spinodal in the reduced parameter space: sκ 3/2 ≃ A(μ + ϵ)4

(19) 3/2

with the numerical constant A = (3/2) /64. The resulting phase diagram is displayed on the lhs of Figure 5. With increasing grafting density the collapse transition is shifted to smaller values of the CNS density. Equivalently, the reentry transition is shifted to larger values of the CNS densities.

= e−ϵ (16) F


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⎛1 ⎞ ϵγ 1 ln 2 z − c + ⎜ − 1⎟ ln(1 − c) ⎝c ⎠ 8 (1 + ϵγc) 2 1 1 2 1 3 = f0 + v2c + v3c + v4c + O(c 4) (21) 2 6 12

This is directly visible on the rhs of Figure 4, where the chemical potential of CNS is displayed vs the brush density for various values of the grafting density for the upper leave of the solution, i.e., for the reentry zone. Here it is also visible that with increasing grafting density the transition becomes weaker and finally fades out via a critical point. Then, the approximation of small volume fractions of the brush at the spinodal point becomes invalid, and the reduction of the parameter space by the above introduced scaling seizes. In order to obtain an estimate for the critical line, we consider the third-order virial expansion in eq 13. First, to obtain the spinodal for the instability of the collapsed state toward swelling (see rhs of Figure 3), we neglect the brush elasticity being in the high-density phase now. With this approximation we obtain for the position of the maximum of the truncated function V(y): y2 ≃ 2κ(γϵ)2 − 1/3 ≃ 2κ(γϵ)2 for γϵ > 1. Now, a rough estimate for the critical line can be obtained by equating the two values for the points of instability, y1 = y2. Using the result of eq 17, the critical line is then given by

fs (c) = −

with v2 = 1 − γ ϵ +

1 γ ϵ ln 2 z 4

3 2 2 2 γ ϵ ln z 4 3 v4 = 1 + γ 3ϵ3 ln 2 z 2

v3 = 1 −

(22)

giving the expansion coefficients with respect to the density which corresponds to a virial expansion. For the derivation see the Appendix, eq 30. The limit of long chains means to neglect the translational entropy of the chains, i.e., neglecting the term (c/N)ln c in the free energy of the FH model. This is strictly valid for fixed chains such as in brushes or in networks. Note that under this assumption there is no phase transition (coexistence) with a constant χ parameter in the standard FH model, since there is no entropic driving force toward a diluted (mixed) state (for infinitely long or tethered chains) but there exists a well-defined state with a finite concentration for χ > 1/2. The series expansion for small values of the polymer concentration, c, up to the fourth order (f 0 denotes the concentration-independent contribution) is also given in eq 21, and this defines the coefficients vk where the index k corresponds to the order of the viral expansion. Note that all coefficients in eq 22 depend on the temperature (via ϵ, due to our convention of kBT units) and on the chemical potential of the CNS given by the parameter z (see Table 1). If we assume as an example ln2 z ≃ 2 and γ, ϵ = 1, which correspond to a rather moderate choice of parameters, we reach a state where the third viral coefficient, v3 = −2, is negative while the second virial coefficient, v2 = 1, remains positive. In this approximation the model is similar to the cluster model originally introduced by de Gennes.14 Apparently, a coexistence exists between a highly diluted phase (c = 0) and a condensed phase (c > 0) if a characteristic value of either z or ϵ is reached, as sketched in Figure 6. This transition is sometimes denoted as a “second type” or “type II” transition to discriminate it from the conventional demixing transition.15−17 Note that the free energy of the brush is given by adding the elasticity contribution: f = fs + σ2/2c2; see also eq 9. Thus, the discontinuous collapse/reentry transition can be traced back to the sign change of the effective three-monomer interaction, v3, while v2 remains positive. This is induced by the adhesion between monomers caused by the CNS. It is important to note that it is not the effective two-monomer interaction and which causes the discontinuity as one might expect from the naive picture of the formation of “bridges”. The change of sign in v2 alone causes a continuous, θ-point-like, transition. This emphasizes the role of the nonlinear χ function and justifies the notation of a “second type” of transition. The elastic contribution of the polymer brush, σ2/2c2, suppresses states with low concentrations as sketched in Figure 6. This shifts the transition toward lower values of the control parameters, ϵ and z, but can also completely suppress the

σc ≃ (32/3)1/2 (κγ ϵ)5/2 (γ ϵ)1/2 = (32/3)1/2 |v0|5/2 (γ ϵ)1/2 (20)

In the last equation we used the definition of the excluded volume constant, eq 12. Above σc for the variation of the CNS density only a smooth crossover can take place, such as displayed on the lhs of Figures 3 and 4 for ϵ = 1.2. The solution for the critical line is displayed on the rhs of Figure 5. This condition limits the validity of the phase diagram in Figure 5; i.e., only the points (σ, ϵ) below the critical condition can be considered for the spinodal line. We note that the full phase diagram is three-dimensional (μ, ϵ, σ), and only the scaling valid in the second virial approach allows a simplification to the 2D phase diagram. The critical condition can be used to construct the 3D phase diagram for the spinodal adding of the third axis given by ϵ. Then, the spinodal becomes a surface which extends up to a limiting grafting density given by the critical condition for a given interaction parameter.

IV. DISCUSSION AND CONCLUSIONS The essential idea of the adsorption−attraction model is that monomers tend to share CNS which gives rise to an additional free energy gain per CNS unit. This leads to an effective attraction between the monomers of the brush, which in turn causes a nonmonotonic shrinking of the brush height ultimately leading to a discontinuous transition. The aim of this work was to establish the corresponding free energy model in a meanfield approximation and to analytically discuss some of its essential features by further simplification. This is obtained by ignoring the effect of the CNS to the excluded volume of the total system. In this case the model obeys a symmetry with respect to low and high CNS concentrations. The equilibrium between CNS and polymer results in a concentration dependent χ function (effective χ parameter) which attains a maximum, χmax = γϵ/2, if the polymers are half-occupied by CNS. A collapsed state is reached for γϵ > 1. Away from the half-filled state the χ function increases with increasing polymer concentration. The general physics behind the CNS-induced discontinuous collapse transition can be understood by considering the thermodynamic equilibrium between polymers and the CNS in a polymer solution of long chains. The free energy per monomer unit of the solution can be denoted as G


Article

Macromolecules

mapping between the co-nonsolvency effect and the type II transition. To conclude, we have shown that taking into account the short-range interaction of a preferred solvent species (CNS) in a Flory−Huggins-type model leads to nonlinear coupling (concentration dependent χ function) in the free energy of the polymer system in equilibrium with the CNS. This gives rise to discontinuous collapse/reentry transition scenarios for a homogeneous polymer brush. In the approximation of a minimally coupled model the CNS−brush equilibrium can be studied analytically. The minimally coupled model leads to a symmetry between the collapse and the reswelling (reentry transition). The increase of grafting density, or adsorption strength, shifts the collapse toward lower values of the CNS volume fraction. We predict that the discontinuous transition ends in a critical line with increasing grafting density. The discontinuous nature of the collapse/reentry transition can be related to a “second type” of phase transition as proposed originally by de Gennes. Our model for the CNS equilibrium provides a rational basis for the concentration-dependent χ function. A detailed test of the model can be provided by coarse-grained simulations of brushes interacting with the mixture of concurrent solvents. We have demonstrated that the essential features of cononsolvency for polymer brushes can be rationalized within the framework of the adsorption−attraction model and the Alexander−de Gennes-type mean-field model and does not need extraordinary strong attractive interaction parameters neither between the solvent compoments nor between CNS and monomers. If instead of CNS attractive nanoparticles having a diameter of the order of the Kuhn segment are considered, all the conclusions remain the same. Nanoparticles can lead to larger adsorption strength, ϵ, and thus can be more effective to cause a discontinuous transition of the brush. Such effects have been already discussed as possible mechanism for the operation of nuclear pores.20,21 We note that the nonlinear coupling mechanism induced by an energetically preferred solvent component can be transferred to various polymer systems such as networks or dendrimers by only changing the elasticity term in the free energy. Moreover, it can lead to interesting adsorption and self-assembly scenarios if combined with an adsorption free energy of the polymer phase or with segregation models. Finally, the generalized Flory−Huggins model itself leads to coexistence scenarios between CNS-rich and CNS-poor phases.

Figure 6. Blue curve is a sketch of the free energy per monomer unit of a polymer solution, fs, in equilibrium with CNS at the coexistence point according to eq 21. The type II transition caused by the formation of CNS-mediated adhesive contacts between monomers is driven by an attractive three-monomer interaction, v3, while the twomonomer interaction, v2, is still repulsive. The elastic free energy of the polymer brush (red line) disfavors low concentrations and shifts, or, depending on the grafting density completely suppresses, the discontinuity.

discontinuity for larger grafting densities resulting in a smooth crossover. The consideration of the finite excluded volume contribution of the CNS leads to an asymmetric behavior with a larger equilibrium height at pure CNS conditions. It can be considered approximately by adding an additional excluded volume parameter which is given by 2v1ϕ = v1(1 − δ) which causes a further concentration dependence via the adsorption equilibrium for δ in eq 8. The approximation of the free energy by the leading order expansion in the concentration is a possible route to analyze the asymmetric case as well. In order to compare the adsorption−attraction model with computer simulations or experimental results three parameters are characteristic for the co-nonsolvency effect: the value of ϵ which should be understood as the selectivity of the CNS per Kuhn segment of the chain, the parameter γ which reflects the coupling between monomers sharing CNS, and the parameter v1 which corresponds to the addition excluded volume of the chains in the fully saturated CNS state. The latter can be defined by the difference between the excluded volume parameter for the pristine CNS state and the pure solvent state, i.e., v1 = v(ρ = 1) − v(ρ = 0). Additionally, the prefactor of the elasticity in the mean-field brush model should be considered which, however, might be obtained by comparing results for the brush height at different grafting densities or degree of polymerization in one of the pure solvent states. In order to compare with the present model, a systematic variation of grafting density and degree of polymerization of the chains is desirable. Our approximation relies on the simplest free energy model of the polymer brush assuming a homogeneous density profile. The extension of our model to a nonhomogeneous profile such as the so-called parabolic profile18,19 may have an interesting consequence: As we can map the CNS−polymer equilibrium to a type II phase transition, we can refer to the study of Baulin, Zhulina, and Halperin, who considered a generic model for a type II transition for the case of a polymer brush.16,17 It has been shown in these works that the discontinuous collapse transition of the homogeneous model can lead to vertical phase separation which realizes a type II coexistence state within the same brush (see also ref 15). Experimental or simulational evidence for such a two-state profile could be a proof for the

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APPENDIX

Two Phase Mean-Field Model of Co-Nonsolvency in Polymer Brushes

Here, we re-reconsider the model introduced by Opfermann, Coalson, Jasnow, and Zilman (OCJZ),13 who used the twophase approach which corresponds to the situation on the lhs in Figure 1. One might consider this model as a realization of the “space-bound” concept according to the classification by Tanaka et al.7 The free energy per monomer is defined by 1 1 ψ ln ψ + (1 − ψ − c) ln(1 − ψ − c) c c ψ 1 1 − (1 − c) ln(1 − c) + ψχx − μ + Π c c c

fOCJZ =

+ fbrush (v1 = 0) H

(23) DOI: 10.1021/acs.macromol.6b02231 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules We chose the symbol ψ for the volume fraction of CNS in the brush. We note that OCJZ considered nanoparticles instead of CNS which have the same size as the monomers. The symbol Π denotes the osmotic pressure. The coupling between brush and CNS is now included by the mean-field expression ψχx, where χx has to be negative. Thus, the brush is a potential trap for the CNS. As we have pointed out already in the Introduction, this is a purely technical approach since the short-range interactions between monomers and CNS do not give rise to any volume excess of the CNS inside the brush without adsorption. We note that the coupling in the mixing term, second term in eq 23, between ψ and c is only repulsive and not important for the collapse transition similar to our discussion above. What we called the symmetric case in the adsorption−attraction model (v1 = 0) corresponds to the decoupling approximation here:

We close this comparison by noting that the numerical solution for the minimum of f OCJZ with respect to ψ and c in the general case predicts a qualitatively similar non-monotonic behavior for the height ratio as a function of the CNS concentration as the adsorption−attraction model. However, the χx parameter necessary to reach the discontinuity has to be of the order χx ≃ −10,13 which can be traced back to the underestimation of the CNS−monomer coupling in the a priori mean-field approach. Derivation of the χ Function in the Symmetric Model. Starting with the free energy function for the symmetric model given in eq 4 and using the substitution eq 6, we obtain without any approximation the following expression: f (δ , c ) =

1 1 (1 − ψ − c) ln(1 − ψ − c) − (1 − c) ln(1 − c) c c 1 → (1 − ψ ) ln(1 − ψ ) (24) c

−

Π = −ln(1 − ρ) = ln(1 + z 0)

f (δ , c ) =

We note the equation of state of the lattice gas model which is used consequently in this work: z0 with the fugacity z 0 = e μ ρ= 1 + z0 (25)

−

ϵγ 1 ln 2 z − c + f b (c ) 8 (1 + ϵγc) 2

(30)

The first two terms correspond to the equilibrium contribution of the adsorption−attraction coupling at a given concentration of the brush. The second could be understood as the χ parameter contribution of the standard Flory−Huggins model in the case of half a dsorbed conditions (ln z = 0). Note that we consider the free energy per monomer (and not per volume unit) so that the coupling of each monomer is proportional to c. If the system is driven away from that maximum, i.e., ln z ≠ 0, the effective attraction between the monomers caused by the bridging of CNS becomes nonlinear with respect to c. This means that also higher-order interactions such as c2, c3, ... (3particle, 4-particle, ...) become relevant. This can be formally expressed by a concentration dependent χ parameter which we call the χ function to clearly distinguish from the case of a simple FH model. In order to see this, we use the identity

1 1 1 ⎛ 1 + z1 ⎞ ln(1 − ψ ) + Π = − ln⎜ ⎟ c c c ⎝ 1 + z0 ⎠ (27)

which defines the effective interaction parameter, χ(c), between the brush and the solvent mixture and which depends on the concentration itself. We can identify the nonlinearity of the χ function again as the origin for the collapse transition obtained in ref 13. Here, we have used the equation of state, eq 25, and the expression q = e−χxc − 1, which goes linearly to zero for c → 0. Considering the leading order with respect to c, we obtain 1 χ2 = χx 2 ρ(1 − ρ) 2

1 2 1 1 δ + δ ln z − ϵγ(1 − δ 2)c + fb (c) 2 2 2

1 ln 4 2

f (c ) = −

Resubstitution into eq 23 provides the effective free energy for the brush as a function of c. This gives the free energy for the equilibrium between the two phases as a function of the volume fraction, c, of the brush:

1 ln(1 + ρq) = χ (c)c c

(29)

which is eq 7 in the main text when the constant term is ignored. Substitution of the minimal value of δ = −[((1/2) ln z)(1 + ϵγc)], see eq 8, leads to the free energy of the brush in equilibrium with the adsorbed CNS

The last term, f brush(v1 = 0), in eq 23 is the brush free energy for the (symmetric) case v1 = 0 in eq 1. To obtain the essential physics, we ignore the repulsion between CNS and monomers and apply eq 24. The thermodynamic equilibrium with respect to uptake of CNS is then given by z1 ψ= with z = z 0e−χx c 1 + z1 (26)

=−

ϵγ (1 − δ 2)c + fb (c) 2

Here, the minimization with respect to δ cannot be done exactly. Since we interested in the collapse which occurs near δ = 0, we expand the free energy up to the order δ2. This results in

The osmotic pressure of the CNS in solution is given by

f − fbrush =

1 δ ⎛ 1 + δ ⎟⎞ δ (1 − δ 2) + ln⎜ + ln z 2 2 ⎝1 − δ ⎠ 2

ln 2 z ln 2 z = ln 2 z − ϵγ c 1 + ϵγc 1 + ϵγc

and we can again ignore terms which do not depend of c (since we are interested in the equilibrium with respect to c only). Then we can rewrite eq 30 as

(28)

Higher order terms would provide the c dependence of χ. Comparing this with eq 3, we see that the enrichment of CNS due to adsorption increases the coupling, since ϕ ≫ ρ for ϵ > 1. Therefore, eq 28 underestimates the impact of CNS as compared to the adsorption−attraction model.

f (c ) = −

ϵγ ⎛ 1 ln 2 z ⎞ ⎟ c + f b ( c ) = − χ (c )c + f b (c ) ⎜1 − 2⎝ 4 1 + ϵγc ⎠

which correspond to eq 9 with the definition of the χ function, eq 10 in the main text. I


Article

Macromolecules Let us note that formally the introduction of χ(c) is not necessary, and of course all results can be obtained directly from eq 30. Here we make a connection with the general discussion in the literature where a c-dependent χ parameter is associated with unusual phase behaviorparticularly in polymer brushes.

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(16) Baulin, V.; Zhulina, E.; Halperin, A. Self-consistent field theory of brushes of neutral water-soluble polymers. J. Chem. Phys. 2003, 119, 10977. (17) Baulin, V. A.; Halperin, A. Signatures of a ConcentrationDependent Flory chi Parameter: Swelling and Collapse of Coils and Brushes. Macromol. Theory Simul. 2003, 12, 549. (18) Semenov, A. Contribution to the theory of microphase layering in block-copolymer melts. JETP Lett. 1985, 61, 733. (19) Milner, S.; Witten, T.; Cates, M. Theory of the Grafted Polymer Brush. Macromolecules 1988, 21, 2610. (20) Osmanovic, D.; Ford, I. J.; Hoogenboom, B. W. Model Inspired by Nuclear Pore Complex Suggests Possible Roles for Nuclear Transport Receptors in Determining Its Structure. Biophys. J. 2013, 105, 2781. (21) Coalson, R. D.; Nasrabad, A. E.; Jasnow, D.; Zilman, A. A Polymer-Brush-Based Nanovalve Controlled by Nanoparticle Additives: Design Principles. J. Phys. Chem. B 2015, 119, 11858−11866.

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. ORCID

Jens-Uwe Sommer: 0000-0001-8239-3570 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author thanks André Galuschko for valuable discussions and the Deutsche Forschungsgemeinschaft (DFG) for funding under Grant SO 277/12-1.

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REFERENCES

(1) Mukherji, D.; Marques, C. M.; Kremer, K. Polymer collapse in miscible good solvents is a generic phenomenon driven by preferential adsorption. Nat. Commun. 2014, 5, 4882. (2) Scherzinger, C.; Schwarz, A.; Bardow, A.; Leonhard, K.; Richtering, W. Cononsolvency of poly-N-isopropyl acrylamide (PNIPAM): Microgels versus linear chains and macrogels. Curr. Opin. Colloid Interface Sci. 2014, 19, 84−94. (3) Dudowicz, J.; Freed, K. F.; Douglas, J. F. Communication: Cosolvency and cononsolvency explained in terms of a Flory-Huggins type theory. J. Chem. Phys. 2015, 143, 131101. (4) Heyda, J.; Muzdalo, A.; Dzubiella, J. Rationalizing Polymer Swelling and Collapse under Attractive Cosolvent Conditions. Macromolecules 2013, 46, 1231−1238. (5) Yu, Y.; Cirelli, M.; Kieviet, B. D.; Kooij, E. S.; Vancso, G. J.; de Beer, S. Tunable friction by employment of co-non-solvency of PNIPAM brushes. Polymer 2016, 102, 372. (6) Mukherji, D.; Kremer, K. Coil-Globule-Coil Transition of PNIPAm in Aqueous Methanol: Coupling All-Atom Simulations to Semi-Grand Canonical Coarse-Grained Reservoir. Macromolecules 2013, 46, 9158. (7) Tanaka, F.; Koga, T.; Kojima, H.; Xue, N.; Winnik, F. M. Preferential Adsorption and Co-nonsolvency of Thermoresponsive Polymers in Mixed Solvents of Water/Methanol. Macromolecules 2011, 44, 2978−2989. (8) Mukherji, D.; Marques, C. M.; Stuehn, T.; Kremer, K. Co-nonsolvency: Mean-field polymer theory does not describe polymer collapse transition in a mixture of two competing good solvents. J. Chem. Phys. 2015, 142, 114903. (9) Huggins, M. Solutions of Long Chain Compounds. J. Chem. Phys. 1941, 9, 440. (10) Flory, P. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1941, 9, 660. (11) Alexander, S. Adsorption of chain molecules with a polar head a scaling description. J. Phys. (Paris) 1977, 38, 983. (12) de Gennes, P.-G. Conformations of Polymers Attached to an Interface. Macromolecules 1980, 13, 1069. (13) Opferman, M. G.; Coalson, R. D.; Jasnow, D.; Zilman, A. Morphological control of grafted polymer films via attraction to small nanoparticle inclusions. Phys. Rev. E 2012, 86, 031806. (14) de Gennes, P.-G. A second type of phase separation in polymer solutions. C. R. Acad. Sci. II 1991, 313, 1117−1122. (15) Halperin, A.; Kröger, M.; Winnik, F. M. Poly(N-isopropylacrylamide) Phase Diagrams: Fifty Years of Research. Angew. Chem., Int. Ed. 2015, 54, 15342. J


AdsorptionâAttraction Model for Co-Nonsolvency in Polymer Brushes

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