Gluonic and Regulatory Solvents: A Paradigm for Tunable Phase

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Gluonic and Regulatory Solvents: A Paradigm for Tunable Phase Segregation in Polymers Jens-Uwe Sommer*,†,‡ †

Leibniz-Institut für Polymerforschung Dresden, Institute Theory of Polymers, Hohe Strasse 6, 01069 Dresden, Germany Institute for Theoretical Physics, TU Dresden, Zellescher Weg 17, 01069 Dresden, Germany

‡

ABSTRACT: A theoretical concept for phase segregation of polymers in the presence of multicomponent selective solvents is presented. Phase separation is caused by nonspecific attractive interactions between the polymers and a smaller component in the solution instead of repulsion between monomers and solvent molecules. We call the component that adsorbs on the polymers and thus causes condensation “gluonic”. It is shown that a discontinuous phase transition from a diluted or semidiluted state to the condensed state takes place if the fraction of the gluonic component is increased. The location of the phase coexistence can be shifted by influencing the binding efficiency of the gluonic component. This can be achieved by adding a regulator component to the solution. The latter is assumed to bind specifically to the polymer and interferes with the gluonic component. In this way a switching of the state of the polymer from dissolved to condensed can be achieved without changing the chemical properties of the system. Applications of this model range from co-nonsolvency in synthetic polymers, over polymer−nanoparticle systems, to biological systems such as the formation of protein−RNA droplets.

1. INTRODUCTION Segregation effects in polymers are usually caused by repulsive interactions between monomers and solvent. This is rationalized by the Flory−Huggins (FH) mean-field model where the interaction between the different components such as solvent and monomers is represented by the so-called χ-parameter.1−3 This model can be written in the following form for a simple polymer solvent mixture: FFH =

1 c ln(c) + (1 − c) ln(1 − c) + χc(1 − c) N

attractive component in the solution and that even operates if the classical FH interaction parameter χ can be completely neglected, i.e., when all components of the solution are perfectly miscible. The effect we are going to describe here is the result of solvent-mediated effective coupling between the polymers. This resembles the effect of a glue; that is why we call this component gluonic solvent (GLS). The reason to introduce this term is that the theory discussed in this work can be applied to quite different systems including polymers, nanoparticles, or biological systems, and thus the specific terms commonly used in different contexts can be considered from a unified perspective. To illustrate the concept of gluonic solutions, we consider a few potential examples for gluonic solvents which are discussed in the literature. The first example, which has attracted some renewed interest recently, is co-nonsolvency.4−8 Co-nonsolvency occurs if a mixture of two good and mutually miscible solvents causes segregation of the polymer phase in a certain range of compositions. For gels this can lead to jumplike volume phase transitionsin stark contrast to the abovementioned absence of a discontinuous transition according to the simple FH model. A well-studied case is poly(Nisopropylacrylamide) (PNIPAm) in a methanol−water mixture where methanol can be considered as the gluonic component.9 Isolated chains were studied by atomistic and adaptiveresolution computer simulations, and evidence for preferential

(1)

Here, the volume fraction of the polymers consisting of N monomers in the solution is denoted by c, and the free energy per volume, FFH, is taken in units of kBT. In this representation the χ-parameter is defined by χ0/kBT, and χ0 corresponds to the effective (typically repulsive) interaction between the monomers and the solvent. A discontinuous transition and coexistence between a condensed phase and a diluted phase of the polymer is the consequence of the balance between the entropy of translation of the polymers in the diluted state and the interaction enthalpy in the condensed state. For polymer chains immobilized on a substrate (brushes), by cross-linking (gels), or in the semidiluted state such as caused by a finite osmotic pressure, the first term in eq 1 vanishes, or can be ignored, and thus no phase coexistence can occur. In these cases the degradation of the solvent quality, usually controlled by lowering the temperature, leads to a monotonous and smooth increase of density only. In this work we consider another generic mechanism for phase segregation of polymers that relies on the presence of an © XXXX American Chemical Society

Received: February 16, 2018 Revised: March 26, 2018

A

DOI: 10.1021/acs.macromol.8b00370 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules adsorption of methanol on PNIPAm was obtained.10 This result is in contrast to the alternative explanation of strong attraction between the solvent components.6 The concept of preferential adsorption was further investigated in coarsegrained simulations where evidence for the adsorption-induced collapse behavior was given.5,7 Mukherji, Marques, and Kremer could explain their simulation results by taking into account the simultaneous adsorption of a methanol molecule by two monomers which gives rise to the formation of loops within the chain. The contribution of loops has been estimated using the average free energy for return events in self-avoiding walks, and thus a prediction for the shrinking of the radius of gyration of the polymer chain was made.5 It should be noted that first indication for a contraction of a single chain due to preferential adsorption of one component in a mixed solvent has been obtained in lattice-based Monte Carlo simulations by Magda et al.11 Inspired by the operation of the nuclear pore complex, Zilman et al.12,13 have proposed a model where nonspecifically adsorbed nanoparticles cause a collapse transition in a polymer brush. Here, the nanoparticles play the role of the gluonic component which forms multiple contacts with the polymers, which are disordered proteins in this case. The authors conducted simulations and proposed a theoretical model that displays a jumplike collapse transition of the brush−nanoparticle system. This model has been reconsidered analytically in our previous work.8 Yet another example can be related to phase segregation in biological cells. Here, certain proteins are known to spontaneously form droplets together with mRNA, i.e., segregate from the cytoplasm, although all components are water-soluble. An intensively studied case is so-called Pgranules which are droplets emerging in embryonic cells of the nematode Caenorhabditis elegans.14 Recently, it was shown that only two components are sufficient for phase segregation, which are mRNA and the protein PGL-3. Here, the formation of droplets in aqueous solution could be found in vitro. In this case, PGL-3 can be considered as the gluonic component, which is known to exhibit several (here six) nonspecific RNAbinding sites.14−16 These examples indicate the variety of systems that display discontinuous phase segregation although all components are miscible with each other if considered pairwise. It strongly suggests that there is a generic physical mechanism behind such transitions which can apparently not be predicted by the FH model using direct binary interactions between the components. It is the aim of this work to outline an extension of the FH model which is able to describe such phase transitions and which reveals the overarching physical principles operating in the very different systems as described above. In our previous work,8 we have proposed the “adsorption− attraction” model for co-nonsolvency in polymer brushes which takes into account the adsorption of co-nonsolvent onto polymer chains, thus extending the concept of Mukherij et al.5 to many chain systems. The key concept was to introduce a mean-field coupling between monomers due to multiple (or nonspecific) adsorption of co-nonsolvent. We could show that this results in a concentration-dependent χ-function, and we predicted a discontinuous collapse and reentry transition for the brush. The latter is driven by negative higher virial coefficients in the presence of a positive effective excluded volume. Such a scenario can be termed as type II phase separation.17,18 In this work we will first extend our concept of adsorptioninduced attraction to polymer solutions where the chains are

not immobilized. The translational degrees of freedom of the chains give rise to a demixing transition of ordinary type in gluonic solution with an effective negative second virial coefficient at very low concentrations. At higher concentrations a type II scenario appears, so that a phase coexistence can be established even in the semidiluted state. Generally, and in contrast to the corresponding transition in polymer brushes, the demixing transition becomes chain length dependent. In particular, we derive an analytic expression for the closed-loop spinodal phase diagram of the polymer solution controlled by the concentration of the gluonic solvent and the chain length. We will further show that the demixing transition in gluonic solutions can be controlled by a third solvent component which we call regulatora term borrowed from the study of Pgranules in biological cells. Similar to the gluonic solvent, the regulator binds to the polymer component but does not lead to bridging between monomers. This is most likely the case for Pgranules (mRNA/PGL-3) in the presence of the MEX-5 protein.15 The latter has a strong specific RNA-binding site and thus interferes with the weaker and nonspecific adsorbing PGL3 protein. The interference of gluonic and regulator components which both compete for adsorption onto the polymer leads to a shift of the phase coexistence. Usually the phase diagram is fixed by the chemistry of the components. The composed gluonic−regulator system, by contrast, is able to control the phase transition by physical preparation, i.e., by adding a minor amount of a third component. This might open interesting applications for the design of materials which act as chemical-logical switches or to localize the condensed polymer phase by application of a gradient of the regulator component. In this work we propose a theoretical concept that is able to describe this interference effect on the phase coexistence from a common point of view and in a mathematically tractable form. Our theory is kept as simple as possible to highlight the generic thermodynamic features of these complex polymer systems. The rest of this work is structured as follows: In section 2 we extend our previously proposed model for a polymer brush8 toward a generic model for polymer solution containing a gluonic component. In section 3 we provide an analytical solution of this model, discuss the physical origin of the phase transition, and provide a solution for the phase diagram. In section 4 we introduce the concept of regulatory components which bind to the polymers and extend our theory accordingly. For the case of strong regulators we give an analytical solution and show that strong regulators simply shift the phase coexistence toward higher concentrations of the gluonic component. We discuss and conclude our results in section 5.

2. ADSORPTION−ATTRACTION MODEL FOR POLYMER SOLUTIONS IN THE PRESENCE OF GLUONIC COMPONENTS In this section we consider a binary solution consisting of polymer chains and GLS as sketched on the left-hand side in Figure 1. The polymer is considered under a constant osmotic pressure, Π, while the other components can be freely exchanged with a reservoir. The GLS is controlled by a bulk concentration, ρ, which realizes a chemical potential. The latter can be approximated by a lattice gas equation of state such as μ = ln(ρ/(1 − ρ)), but any other equation of state, μ(ρ), can be considered as well. The mixture of solvent and GLS is considered as athermal. The same is assumed for the pure polymer solution in the absence of GLS. Thus, the osmotic pressure of the polymers in solution and the chemical potential B

DOI: 10.1021/acs.macromol.8b00370 Macromolecules XXXX, XXX, XXX−XXX

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

with

fads = ϕ ln ϕ + (1 − ϕ) ln(1 − ϕ) − μϕ − ϵϕ fattr = − 2ϵγϕ(1 − ϕ)c fsol = Figure 1. Left: sketch of the thermodynamic model. Right: schematic of adsorbed GLS and bridging between monomers due to GLS. The polymer solution with a given number of polymer chains is considered under constant osmotic pressure, Π, while solvent (background) and GLS (white circles) can be freely exchanged. The concentration of the GLS, ρ, is fixed in the bulk which defines the chemical potential μ. The GLS can adsorb on the polymer as indicated by blue circles. The adsorption (weak binding) is driven by an energy gain of −ϵ. Because of the capacity of multiple adsorption GLS can form bridges between the monomers. This is related to an additional energy gain which might be altered by local conformation properties or by the nature of the multiple binding of the GLS, which is indicated by a value of γ ≠ 1.

⎛1 ⎞ Π 1 ln(c) + ⎜ − 1⎟ ln(1 − c) + ⎝c ⎠ N c

(2)

Here, fads denotes the free energy which corresponds to the adsorption of GLS onto the polymer. The symbol ϕ denotes the coverage of the polymer by GLS. A value of ϕ = 1 means that the polymer is completely covered by GLS. The symbol ϵ denotes the energy gain per GLS if bound to the polymer (remember that ϵ is given in units of kBT; thus, ϵ can be reduced by increasing the temperature). The expression for fads is obtained from an exact statistical model for the adsorption isotherm, the only specific assumption being the lattice-gas-like equation of state of the adsorbate on the substrate (polymer chains), i.e., the first two terms of fads. The expression for fattr is a model for the coupling caused by the “sandwich-like” adsorption of GLS by two segments of the polymers, as sketched on the right-hand side of Figure 1. It does not matter whether these segments belong to the same chain or to different chains. The concentration of segments (in units of the volume of GLS) is denoted by c. The term fattr is a mean-field approximation for the coupling of two segments induced by the GLS and should be read as follows: The coupling is proportional to the product of the probability that the given segment is at the same place as some other segment, c, times the probability that only one segment has bound to a GLS: 2ϕ(1 − ϕ). In the spirit of the mean-field approach the density of the polymers is considered to be homogeneous. The energy gain due to coupling is given by ϵγ, where γ denotes any difference between the strength of the additional binding as compared to simple binding. We will usually set γ = 1 in most examples for simplicity. The expression fsol stands for the free energy per segment of the polymer solution at constant osmotic pressure. The choice of constant osmotic pressure (and thus variation of c) allows for a simpler description of the phase behavior. We note that for the

of the GLS fix the thermodynamic equilibrium states at a given temperature. In the following we will consider energetic units of kBT. All thermodynamic forces such as μ and Π are then understood as given in units of the absolute temperature. Furthermore, the length scale will be considered in units of the size of GLS particles, l. We call a sequence of length l along the polymer chains a (coarse-grained) segment. This simplifies the presentation of the theory by omitting the repeated notation of these constants in the mathematical expressions. In a recent work a model has been proposed to explain the collapse and reentry behavior of polymer brushes in the presence of co-nonsolvents as a possible realization of GLS for synthetic polymers.8 Here, this will be extended to the case of a polymer solution where polymer chains are not immobilized on a substrate. The free energy per polymer segment is proposed as follows:a

Figure 2. Numerical solution of eq 2 at a given osmotic pressure Π = 0.001 and γ = 1 The polymer concentration is displayed as a function of the chemical potential of the GLS which is given by the bulk density of the GLS. Generally a nonmonotonous behavior is obtained. The solution is symmetric with respect to the maximum density located at μ = −ϵ. The left-hand side displays the results for N = 100 and various values of the CNS selectivity ϵ. Above a certain selectivity, here ϵ ≃ 1.6, a discontinuous transition is observed. The jump in density defines the coexistence point of the two phases. This is indicated by the double arrow in the figure. The right-hand side displays the behavior under the change of the degree of polymerization, N, at a constant value of the selectivity. For smaller values of N like N = 10 here, the GLS does not cause a detectable effect. C

DOI: 10.1021/acs.macromol.8b00370 Macromolecules XXXX, XXX, XXX−XXX

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broadens its window but eventually smooths again the collapse behavior as a function of μ. The behavior of the maximum collapsed state can be understood analytically. The maximum of density in Figure 2 corresponds to the field-free case ln z = 0 and to ϕ = 1/2. In this case eq 2 can be rewritten into the standard FH model:

case of immobilized chains such as polymer brushes the contribution of the translational entropy, (ln c)/N, is absent which excludes the possibility of a standard phase coexistence driven by a negative second virial. Equation 2 displays a symmetry in the field-free case, μ + ϵ = 0, with respect to ϕ = 1/2, i.e., ϕ ↔ 1 − ϕ. Here, we denote the linear contribution in the free energy as the “field” term in accordance with the general notation in the physics of phase transitions. This leads to a symmetry for the equilibrium solution with respect to mirror operation of the field: ±(μ + ϵ). The practical consequence of this symmetry is the equivalence of the condensation and reentry transition as we will discuss below. The free energy model in eq 2 can be extended to take into account the additional excluded volume effect of the CNS. This would affect the mixing entropy, the second term in fsol, and breaks the symmetry between condensation and reentry behavior. While the generalized model can only be treated numerically, the symmetric model has the advantage to allow for an analytic solution. Since this model contains the essential features of the phase diagram, we will restrict ourselves to it in the following. The equilibrium equation of state for polymer, c(Π,μ,ϵ), is obtained by the global minimum of the free energy with respect to both the binding (adsorption) of GLS, given by ϕ, and the density of the polymer. The numerical solution is displayed in Figure 2 for a given set of parameters. A general feature of the density isotherm c(μ) is its nonmonotonous behavior. Increasing the density of the GLS leads to larger adsorbed fraction, ϕ, of the CNS on the polymers and thus to a larger coupling between monomers, according to the expression for fattr in eq 2. Because the coupling is maximal if the polymers are half-occupied with adsorbed GLS, i.e., for ϕ = 1/2, further adsorption reduces the coupling again. As noted already above, the solution is symmetric with respect to the state ϕ = 1/2, which is reached at the point of zero field ln z = μ + ϵ = 0

⎛1 ⎞ 1 1 Π f0 = − ϵγc + ln c + ⎜ − 1⎟ ln(1 − c) + ⎝ ⎠ 2 N c c

(4)

Here, we can introduce the effective FH parameter χ0 =

1 ϵγ 2

(5)

Fixing the GLS volume fraction at μ = −ϵ leads to a FH model of the polymer in an effective thermal solvent. In particular, poor solvent behavior can be induced by the GLS only for 2χ0 = γ ϵ > 1

(6)

For the spinodal point for the collapse (away from the critical 1 1 point) we obtain the approximation of c0 ≃ N |v | and 0

Π0 ≃

1 1 N2 |v0|

with v0 = 1 − ϵγ. Thus, by variation of the osmotic

pressure the solution becomes unstable at the condition ΠN2 = const, which explains the dramatic behavior of the right-hand side of Figure 2 with respect to N. However, only the particular case of half-coverage of chains with GLS can be mapped to a FH model with an effective and constant χ-parameter. The discontinuous collapse and reentry transition with respect to the variation of the GLS volume fraction, in particular for larger values of N2Π, cannot be explained. Here, the coupling between the condensation of the polymer and an increased adsorption of GLS mediated by fattr in eq 2 becomes important. This can be understood by formally introducing a FH function χ(ϕ(c)) = 2ϵγϕ(1 − ϕ). Increasing χ induces a higher concentration c, which in turn increases the adsorption by bridge formation and increases ϕ. This feedback cycle can eventually lead to discontinuous condensation transition of the polymer if μ reaches a given value. In the next section we will outline an analytical model for this transition.

(3)

Here, we have introduced the adsorption fugacity, z = exp(μ + ϵ), a definition which will become useful. This equation defines the maximum of the density which corresponds to the maximum coupling at ϕ = 1/2. Thus, for larger values of ϵ, the maximum of the density is shifted to smaller values of μ = −ϵ and thus to smaller values of ρ. Most remarkable is the jump of the polymer density under variation of the volume fraction of GLS for higher values of ϵ. For the parameters chosen on the left-hand side of Figure 2, this is given by ϵ > 1.6. This indicates a discontinuous phase transition (condensation) of the polymer/GLS phase at the given osmotic pressure. This defines a coexistence point where the two phases of different density are in equilibrium at the same osmotic pressure. The corresponding densities can be read off from Figure 2; an example for the case ϵ = 2.2 is indicated by the double arrow. The density of the polymer phase at low coupling induced by the GLS and for c ≪ 1 is given by the approximate ideal gas law: c ≃ N·Π. This defines the limiting value of the polymer density in the case of a dilute solution. The right-hand side of Figure 2 shows the effect of chain length. Here, a dramatic difference can be observed for smaller and larger values of N. For the given selectivity of the GLS of ϵ = 1.8 there is no visible transition for N = 10. Increasing the value of N induces a jumplike transition to a condensed state in a certain window around μ = −ϵ. Further increase of N increases the density in the condensed state and

3. CONCENTRATION-DEPENDENT χ-FUNCTION AND TYPE II TRANSITION To understand the phase behavior of the polymer under variation of the volume fraction of the gluonic solvent, the free energy in eq 2 should be minimized with respect to ϕ. Then, the replacement of the solution ϕ(μ,c) into the free energy leads to an effective free energy for the polymer where the effect of GLS is mapped onto an effective monomer−monomer interaction which will depend on the GLS concentration. We can follow here our recent work.8 Because of the symmetry with respect to the maximum coupling at ϕ = 1/2, the deviation from this value is introduced as δ according to 1 ϕ = (1 − δ) (7) 2 The adsorption equilibrium of the GLS is therefore given by the equation ⎛1 + δ ⎞ ⎟ + 2δy = 0 ln(z) + ln⎜ ⎝1 − δ ⎠

(8)

where the rescaled concentration D

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y = γ ϵc

(9)

has been introduced. This equation has no explicit solution 1−δ δ(y,ln z) but the inverse solution z = 1 + δ exp( −2δy) displays a monotonous behavior, and thus a unique solution does exist. As the most interesting region will be close to the maximum coupling, an expansion for δ ≪ 1 can be made. This results in the approximate solution δ=−

1 ln z 21+y

(10) Figure 3. Sketch of the various contributions to the free energy f(c) (see eq 13) for the case of a negative third but a positive second virial coefficient. The function f 2−4 denotes the virial contributions up to the fourth order. A finite osmotic pressure lifts the free energy for small values of the concentration and can induce a discontinuous condensation of the solution. The entropy of the center of mass of the polymers has the opposite effect, favoring the low-density state.

Replacing this into eq 2, one obtains f (c ) = −

1 2 ln z − χ (c)c + fsol (c) 8

(11)

with the concentration-dependent χ-function ⎡ 1 (μ + ϵ)2 ⎤ χ (c) = χ0 ⎢1 − ⎥ 4 1 + ϵγc ⎦ ⎣

(12)

z > 4/3γ2ϵ2, which is a moderate condition not violating the approximation used in eq 10. Furthermore, we can expect a crossover from the type II transition to the normal demixing transition if either N, the concentration/pressure, or the value of the field, ln z, becomes smaller (particularly for the case z = 1, as discussed at the end of section 2). In such a case an effective second virial coefficient competes with the translation entropy as in the classical FH scenario. In order to discuss the phase transition, the pressure isotherm Π(c, ln z, γϵ) can be calculated from eq 11 by the condition ∂f(c)/∂c = 0, which leads to

where we have used eq 5. If we are away from the point μ = −ϵ (half-covered state of the polymers), the χ-function increases with increasing concentration of the polymer. We note that the concentration dependence of the χ-function forbids a simple interpretation as a χ-parameter in the FH model. For instance, the differentiation of f(c) with respect to c in eq 11 leads to two terms: χ and also c·χ′(c). In fact, our model gives rise to a modification of all virial coefficients, in contrast to the classical FH model where only the second virial coefficient is concerned. The expansion of f(c) up to the fourth virial and omitting the c-independent term reads f (c ) ≃

1 1 1 1 3 Π ln c + v2c + v3c 2 + v4c + N 2 6 12 c

Π=

(13)

with8 1 v2 = 1 − γ ϵ + γ ϵ ln 2 z 4 3 2 2 2 v3 = 1 − γ ϵ ln z 4 3 v4 = 1 + γ 3ϵ3 ln 2 z 2

⎞ c 1 ⎛ 2γ ϵψ + ⎜ − β ⎟c 2 + V3(c) 2 N 2 ⎝ (1 + γ ϵc) ⎠

(15)

Here, the function V3 denotes all higher virial coefficients of the athermal polymer solution starting from the third viral and is given by V3 = −ln(1 − c) − c −

1 2 c 2

(16)

for the FH lattice model. The parameter β is defined by (14)

β = γϵ − 1

(17)

Further, we have introduced the strictly positive field

Obviously, there exists a parameter window for which the third viral coefficient is negative but the second virial coefficient stays positive. This situation is sketched in Figure 3. In this case a discontinuous transition can occur even in the limit of N → ∞ or, more precisely, if the polymer solution has a higher concentration as necessary for FH coexistence. This explains the existence of a discontinuous transition in cases where the translation entropy is absent such as for polymer brushes, as we have studied in our previous work.8 This justifies the notation of a second type of (or type II) phase transition as it has been introduced ad hoc by de Gennes and was discussed in earlier works aimed to particularly explain the lower critical solution behavior of PEO.17−20 We denote the necessary window of parameters for the existence of the type II transition by the condition γϵ > (1 + 7/3 )/2, which is only slightly larger than the necessary condition for the existence of the discontinuous transition given by eq 6. The lowest possible field is given by ln2

ψ=

1 2 1 ln z = (μ + ϵ)2 8 8

(18)

We note that the prefactor of the second term in eq 15 corresponds to v2/2 in eq 14 for c → 0. For the case of discontinuous phase transition the pressure must display an unstable region of negative compressibility given by ∂Π/∂c < 0. In Figure 4 we display an example which shows the pressure isotherm for a discontinuous condensation transition given by eq 15. The coexistence region is defined by the Maxwell construction as indicated by the horizontal isopressure line in the figure. This cannot be obtained analytically. However, we are able to calculate the spinodals at which the solution becomes unstable and the existence of which is the necessary condition for a discontinuous transition scenario. The spinodals are given by ∂Π/∂c = 0 and can be written in the following form: E

DOI: 10.1021/acs.macromol.8b00370 Macromolecules XXXX, XXX, XXX−XXX

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This corresponds to the FH result and thus to the smooth transition which occurs in this case. For stronger fields, for instance for lower GLS concentrations, the type II transition causes a coexistence of condensed and dissolved phases. The onset of the type II transition is clearly shown by the kink of the spinodal for N = 1000 as indicated by the filled circle. This corresponds to the change of sign of v2 in eq 14. To discuss the influence of the chain length on the condensation transition, we restrict ourselves to the case β = ϵ − 1 = O(1), i.e., β not small. That means that we are far away from the FH critical point. Further we are interested in the condensation only, i.e., in the case c ≪ 1. From eq 19 it becomes obvious that the condition 1/N ≪ c should be valid in order to neglect the influence of the chain length. Thus, for small enough concentrations the chain length effect is always dominant. However, we note that c ≪ 1/N is far below the semidilute limit. This corresponds to a coexistence between a highly diluted and the concentrated solution deep in the miscibility gap of the classical FH case. For the case c ≫ 1/N the N dependence in eq 19 can be ignored, and the spinodal is determined by type II behavior.

Figure 4. Osmotic pressure of the polymer solution as a function of the specific volume for the given values of parameters according to eq 15. The coexistence pressure is indicated by the horizontal line and the spinodal points by filled circles. The pressure was multiplied with the chain length to provide natural units.

ψs(c) =

3 (1 + γ ϵc) ⎧ 1 c2 ⎫ ⎬ ⎨βc − − 2γ ϵc N 1 − c⎭ ⎩

(19)

This defines the (spinodal) phase diagram in the μ, c space with the two parameters γϵ and N. We note that a zero value of the curly brackets defines the FH spinodal with χ0 = γϵ/2, i.e., β = −v0, which is the only solution for ψ = 0. In Figure 5 we display the spinodal phase diagram for γ = 1 and ϵ = 2. The two solutions at the same value of μ(c) correspond to the two extrema of the pressure isotherm. The lower part of μ(c) defines the condensation transition (indicated by the blue arrow in Figure 5). The region of demixing is topologically closed. For our model we obtain a symmetric collapse and reentry transition. Noticeable is the strong dependence of the chain length. A state which is condensed at a given concentration of GLS can be dissolved if the chain length is reduced. For the limiting case of infinite chain lengths there is no stable solution for small values of ψs, i.e., close to the optimally GLS-loaded state of the polymer.

4. REGULATORY SOLVENTS The essential mechanism of GLS is the induced coupling between monomers by forming bridges. This mechanism can be controlled/interfered by additional components in the solution which we call regulatory solvents (RGS). Here, we restrict ourselves to RGS which also adsorb on the polymer, but in contrast to GLS, it is not able to induce bridges between monomers. This can be also understood as a small value of γx ≪ 1 for the case of the RGS. We denote the new component by the index “x”. The free energy per monomer in eq 2 has to be extended as follows: f (c , ϕ , ϕx) = ϕ ln ϕ − μϕ − ϵϕ + ϕx ln ϕx − μx ϕ − ϵxϕ + (1 − ϕ − ϕx) ln(1 − ϕ − ϕx) − 2γ ϵϕ(1 − ϕ − ϕx) + fsol (20)

Figure 5. Spinodal phase diagram of a polymer solution in GLS for various values of the chain lengths and for γ = 1 and ϵ = 2. The blue arrow indicates an increase of concentration of GLS which causes a discontinuous condensation transition for sufficiently large values of N. The filled circle and the dashed line display the change of sign of the second virial, v2, given in eq 14, and is located at μ = −2 − √2 for the present choice of parameters. F

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Figure 6. Illustration of the effect of RGS on the phase behavior of a polymer solution in the presence of GLS. The RGS (red circles) adsorbs on the polymer and blocks monomers for GLS as sketched on the right-hand side. The interference of RGS causes a reduction of the effective chemical potential of the GLS according to eq 22 and thus reduces the bridging between the monomers. This shifts the coexistence region to higher values of the concentration of GLS as shown in the spinodal phase diagram for the case of N = 100 and ϵ = 2. As a consequence, condensed droplets of polymer and GLS (illustrated as black circles) which are formed close to the phase coexistence line are dissolved due to the action of the RGS (illustrated by hatched black circles).

transitions within the solvent mixture can dramatically enhance the regulatory behavior to note just a few interesting extensions of our model.

The chemical equilibrium with respect to RGS is obtained by ∂fads/∂ϕx = 0. Since there is no nonlinearity in the coupling term for RGS, its solution can be obtained exactly: ϕx =

Zx (1 − ϕ) 1 + Zx

with Zx = exp{μx + ϵx + 2γ ϵϕc}

5. DISCUSSION AND CONCLUSIONS An important consequence of the concepts of gluonic and regulatory solvents is the possibility to control a discontinuous condensation transition in polymer solutions without changing temperature or molecular interactions. Moreover, even in the absence of translational entropy a discontinuous transition related with a jumplike switch of polymer density takes place as a result of the nonlinear character of the effective coupling caused by GLS. This is relevant for polymers that are grafted to substrates (brushes) and cross-linked (gels) or for transitions inside larger molecular structures such as dendrimers or singlechain nanoparticles. An example that displays both gluonic and regulatory factors is the formation and the control of RNA−protein droplets in biological cells. For the case of P-granules, for instance, the formation of droplets formed by mRNA and the protein PGL-3 can be suppressed in the presence of another protein called MEX-5.14,16 Here, PGL-3 can be considered as the GLS which is reported to have six nonspecific RNA-binding sites which have been identified as RGS repeats.15 The third component, MEX-5, is known to bind specifically and much stronger to RNA (the dissociation constant with RNA being about 20 times smaller than that of PGL-315). Thus, MEX-5 can be considered as RGS. The control of the formation of RNA− protein droplets, most likely by a gradient of MEX-5, in the embryonic cells of C. elegans is vital for the delivery of these droplets (P-granules) to only one of the daughter cells after cell division. We would like to note that the difference between gluonic and regulatory action in biological systems can be reflected by the difference between nonspecific and specific binding of proteins to RNA. In the physics of synthetic polymers the concept of GLS can be mapped to the co-nonsolvency effect which has gained renewed interest recently. The most intensively studied system here is PNIPAm in a mixture of water and alcohol.4 Most interesting would be to explore possible candidates for regulatory components. In this work we have focused on the scenario where the RGS binds to the polymer and is thus blocking the monomers for GLS. While the concept of specific

(21)

Substituting this back into eq 20 gives the equilibrium free energy with respect to adsorption of RGS: f x (c , ϕ) = f (c , ϕ) − (1 − ϕ) ln(1 + Zx)

where f(c,ϕ) is given by eq 11. Because of the dependence of Zx from ϕ and c, this does not allow a straightforward solution for the total system. However, there is a limiting case which allows for a simple solution: If the RGS is binding much stronger to the polymer as compared to the GLS, i.e., for ϵx ≫ ϵ, we can ignore the dependence of Zx from ϕ and from c, and we obtain f x (c , ϕ) = f (c , ϕ) − (1 − ϕ) ln(1 + zx)

with zx = e μx +ϵx

Thus, the effect of the RGS in this limiting case can be mapped to a shift of the chemical potential of the GLS: μ′ = μ − ln(1 + e μx +ϵx)

(22)

The action of the RGS is therefore equivalent to the reduction of the concentration of the GLS and can thus trigger the transition. The physical reason is the blockage of monomers by RGS which are then disabled to adsorb GLS and thus to induce attractive interactions via the formation of GLS bridges. In Figure 6 we illustrate the effect of RGS on the phase diagram. For a state close to the coexistence line only a small increase in concentration of RGS can cause a transition from the condensed to a dissolved polymer phase. It is worth noting that the same concept, namely selective adsorption of a cosolvent onto the polymer chain, can explain both effects: condensation and dissolution. We add that the dissolution of the condensed phase by the RGS will also depend of the chain length of polymers. Besides the competition for preferential adsorption between GLS and RGS there are other possibilities to shift the adsorption-induced transition meditated by the GLS. Complexation induced by attraction between GLS and RGS would reduce the chemical potential of the GLS via the resulting equation of state of the GLS−RGS−solvent mixture. Phase G

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Macromolecules

To conclude, we have shown that components of a solution that has the tendency to associate with the polymer, which we termed “gluonic” here, give rise to discontinuous condensation transitions into a polymer−GLS-rich phase via a generic mechanism of forming bridges between monomers. This is reflected in theory by a concentration-dependent χ-function. The resulting phase transition persists in the absence of translational degrees of freedom of the polymer. It is caused by negative higher virial coefficients and can be termed as type II transition according to a conjecture by de Gennes.17 A crossover into the normal demixing transition driven by a negative second virial coefficient is predicted for very low concentrations of the polymer. We have shown that the transition can be tuned by solvent components which interfere with the adsorption of gluonic solvents and which we termed “regulatory” therefore. Complex polymer solutions involving associating components being adsorbed or bound by the polymer offer phase transition scenarios which are controlled by the interplay of several components and not by the chemistry of the system only. Complex feedback cycles may be constructed in such solutions where associating components are themselves regulatory to other parts of the system. Nonequilibrium states driven by flow or shearing can influence the local compositions and phase behaviors leading to flowinduced phase transitions. The theory presented here offers a unified treatment of demixing phase transitions in associating polymer solutions based on a mean-field argument.

binding is very common in biological systems, on a molecular level this might be related to hydrogen bonds or to the formation of complexes with larger molecules thus acting as RGS. Interestingly, there is some similarity with the model proposed by Beriakov, Bruisma, and Pincus to explain the phase behavior for LCST polymers.19 Here, the strong binding of water itself to the polymer by H-bonds can explain the solvation of an otherwise hydrophobic polymer which is (poly(ethylene oxide)), PEO. In our terms this corresponds to the situation of a regulatory component operating on a polymer that has otherwise a classical FH demixing behavior, i.e., in the absence of gluonic components. Simply speaking, PEO is assumed to have a large FH parameter with water (being hydrophobic) in the absence of H-bonds. Formation of the latter shields the monomers from their native interaction and induces dissolution if the fraction of H-bonds is high enough. In terms of our free energy model in eq 2 this means to replace the coupling function with the expression fattr = −χ(1 − ϕx)2c, where χ denotes the classical FH parameter for the polymer (PEO) in poor solvent, i.e., χ ≫ 1/2. The factor (1 − ϕx)2 accounts for the non-H-bonded monomers to interact with each other only. We note that this regulator-only system is nonlinear in its coupling with the polymer because of the new expression for fattr which forbids a simple solution. This nonlinearity is also the cause of the discontinuous transition window shown in this case as well.19 This leads us to possible extensions of the model presented in this work. For the sake of clarity we have chosen the simplest realization where all components are perfectly miscible among each other, and the adsorption of selective solvents is the only energetic contribution. In practice, there can be a tendency for the polymer to segregate from the pure solvent even in the absence of any additional components. This would manifest itself by an additional term in eq 2 of the form −χ·c. Other generalizations concern the effect of the additional solvent components on the mixing entropy. In this work we have ignored this contribution which implicitly restricts our results to small volume fractions of the GLS/RGS components. The influence of additional solvent components to the mixing entropy causes an asymmetry between the condensation and the reentry transition as discussed in our previous work about the co-nonsolvency effect in polymer brushes.8 In principle, taking these contributions into account makes a closed analytical solution virtually impossible. However, in practical cases the reentry transition, i.e., the supersaturated coverage of polymer with selective solvent components, will be of less interest. Moreover, switching the phase behavior of the polymer solution with a minimum amount of additional components will most likely be desirable. Thus, the simplest model presented in this work shall be sufficient to predict these scenarios. Another generalization of our model is to take into account correlation effects between the adsorbed cosolvents related with a coupling energy, −J, when two cosolvent molecules are adsorbed on neighboring monomers along the chain. Since linear polymers are 1D substrates for the GLS or RGS this could be taken into account in formal analogy with the 1D-Ising model, and the term fads in eq 2 should be substituted accordingly. Because of the 1D problem one should, however, not expect a new (adsorption) phase transition, but rather an enhancement of the discontinuous demixing transition for positive coupling.

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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 acknowledges fruitful discussions with André Galuschko, Huaisong Yong, and Holger Merlitz. Financial support by the German Research Foundation (DFG) within the project SO-277/17 is gratefully acknowledged.

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ADDITIONAL NOTE Please note that f is given in monomer units and not in volume units. An interconversion between both representations is given by f = f V/c, where f V denotes the free energy in volume units. a

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REFERENCES

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DOI: 10.1021/acs.macromol.8b00370 Macromolecules XXXX, XXX, XXX−XXX