General Condition for Polymer Cononsolvency in Binary Mixed Solvents

Starting from a generic model based on the thermodynamics of mixing and abstracted from the chemistry and microscopic details of solution components, three consistent and complementary computational approaches are deployed to investigate the general condition for polymer cononsolvency in binary mixed solvents at the zeroth order. The study reveals χPS – χPC + χSC as the underlying universal parameter that regulates cononsolvency, where χαβ is the immiscibility parameter between the α- and β-component. Two disparate cononsolvency regimes are identified for χPS – χPC + χSC < 0 and χPS – χPC + χSC > 2, respectively, based on the behavior of the second osmotic virial coefficient at varying solvent mixture composition xC. The predicted condition is verified using self-consistent field calculations by directly examining the polymer conformational transition as a function of xC. It is further shown that in the regime χPS – χPC + χSC < 0, the reentrant polymer conformation transition is driven by maximizing the solvent-cosolvent contact, but in the regime χPS – χPC + χSC > 2, it is driven by promoting polymer and cosolvent contact. In-between the two regimes when neither effect is dominant, a monotonic response of polymer conformation to xC is observed. Effects of the mean-field approximation on the predicted condition are evaluated by comparing the mean-field calculations with computer simulations. It shows that the fluctuation effects lead to a higher threshold value of χPS – χPC + χSC in the second regime, where local enrichment of cosolvent in polymer proximity plays a critical role.


INTRODUCTION
Solution processing is a fundamental process in polymer science in which solvent mixtures are often in use.If two good solvents become poor for a polymer when mixed, then the solvent mixture is called a cononsolvent pair for the polymer.The cononsolvency effect had been observed for many types of polymers in a variety of solvent mixtures, characterized by either reentrant coil−globule−coil (C-G-C) conformational transition 1−3 and/or reentrant mixing-demixing-mixing phase transition with respect to the solvent mixture composition. 4,5omputer simulations incorporating atomistic details had revealed a multitude of system-dependent molecular mechanisms that can give rise to the phenomenon, ranging from competitive/cooperative/frustrated hydrogen bonding of polymer molecules with polymer/solvent/cosolvent molecules 4,6−9 to formation of solvent/cosolvent complexation cluster, 10−14 polymer-mediated S−C interactions, and alteration of the solvent structuring by the presence of cosolvent, 15,16 etc.−23 The explanation however struggles with polymer cononsolvency observed in water− alcohol mixtures, which are known to have positive excess free energy of mixing. 24Recent studies by Mukherji's et al. claim that preferential adsorption of cosolvent by polymer is responsible for cononsolvency of PNIPAM in water−methanol mixtures. 25The authors demonstrate the generic nature of the preferential-adsorption effect by showing that atomistic simulations of PNIPAM in water−methanol mixtures are in excellent agreements with the coarse-grained simulations that are deprived of all chemistry details. 26 A generic adsorption model is further proposed to show that competition of two adsorption modes of cosolvents by polymer toward minimizing the adsorption free energy leads to a reentrant polymer conformation transition with respect to solvent mixture composition.−29 Recently, understanding of the two types of polymer cononsolvency under one unified generic formulation had been discussed by us. 30Early investigations have reinforced the plausibility that polymer cononsolvency is governed by general principles that are chemistry nonspecific, although mechanisms at the molecular level may vary.
Despite the progress, the general condition for polymer cononsolvency that applies to a broad variety of systems has not yet been clearly presented.This leads to doubts and confusion about the validity of the generic considerations.For example, numerous experiment studies had reported that preferential adsorption of cosolvents by polymer does not necessarily give rise to polymer cononsolvency. 31,32Using coarse-grained simulations, a recent study by Bharadwaj et al.  shows that polymer C-G-C conformation transition can occur under a myriad of different settings, including strong solventcosolvent energetic interactions, solvent-cosolvent size difference, and strong solvent−solvent energetic interactions. 33hey proceed to conclude that neither solvent-cosolvent energetic attractions nor preferential adsorption of cosolvent by polymer warrants or are prerequisites for polymer cononsolvency.These seemingly contradictory findings highlight two outstanding needs toward an improved understanding of the phenomena based on generic considerations.First, current generic understandings seem to suggest separate cononsolvency conditions depending on the driving force (e.g., "solvent-cosolvent attractions" vs "preferential adsorption").There needs to be a unified formulation of the condition that naturally governs polymer cononsolvency of seemingly distinct origins.In order to do this, the relevant underlying parameter that regulates polymer.cononsolvencyneeds to be identified.For it to apply to a broad variety of systems, the parameter would likely be of a thermodynamic nature.Microscopic interactions (of either enthalpic or entropic origins) are certainly not suitable choices.On the other hand, "preferential adsorption" as a thermodynamic phenomenon refers to the corresponding excess cosolvent density in the solvation shell.However, it is premature to expect the onset of preferential adsorption to commence the reentrant polymer conformation transition.
In view of the needs, in this study, we make the attempt to determine the underlying governing parameter and the corresponding condition for polymer cononsolvency in binary mixed solvents.A generic formulation based on the thermodynamics of mixing was used to keep the results independent of the microscopic details.Three separate but complementary approaches will be combined to achieve the task.An analytical theory offers a clear mathematical path to derive the general condition and identify the relevant parameters.Using the same model, field theoretical calculations provide direct access to polymer conformation transitions for the verification of the analytical predictions.Effects of the mean-field approximations on the theoretical predictions will be assessed and examined in computer simulations.The manuscript is organized as the following.Section 2 gives an account of the development of the model and methods employed in this work.In Section 3, the conditions for polymer cononsolvency are first predicted analytically based on the behavior of the second osmotic virial coefficient, followed by a study of polymer conformation transitions under the corresponding conditions to verify and rationalize the predictions.We then compare the results with computer simulations to estimate the effects of mean-field approximations on the predicted conditions.In Section 4, we conclude with a discussion of the implications of the current study for rationalizing available experiments and simulation results and then touch on the possible improvements that can be made to the current approach.

MODEL AND METHODS
We aim at a generic model that contains the essential ingredients for producing the gross character of polymer cononsolvency.Based on the model, three separate but complementary approaches will be developed and combined to derive and verify the general conditions for polymer cononsolvency.Toward this, we develop the model from a thermodynamic perspective that involves less hypothesis concerning the structure and laws of interactions between molecular species.The starting point is a generalized, however, exact expression of the Helmholtz free energy per unit volume of a ternary system containing polymer (P), solvent (S), and cosolvent (C).
The first sum accounts for the contribution from the translational entropy of the three components, with ρ i being the number density of the i-type monomer in the mixture, N i is the degree of polymerization (N S = N C = 1) and Λ i is the thermal de Broglie wavelength for the degrees of freedom of center of mass of an i-type molecule.The second term arises from the conformation entropy due to the internal degree of freedom of molecules, with Z i being the reference ideal-state conformation partition function of an i-type molecule.The last term f ̃mixture ex is the excess free energy per unit volume due to all of the interaction effects in the mixture.By assuming no volume change upon mixing (i.e., V = ∑ i V i (0) with V i (0) being the volume of the icomponent in its pure state), a formally exact expansion of the excess free energy of mixing per unit volume can be obtained (see Section 5 for details) as where χ ij and χ ijk are expansion coefficients of free-energy nature that are in general ϕ-dependent.
The models for this study are based on eq 2. We restrict our considerations to the zero-order case; that is, the χ-parameters are assumed to be ϕ-independent.For simplicity, we also neglect the nonaddititive effects χ PSC .Considering a homogeneous system where ϕ i is location-independent, inclusion of change in translational entropy of molecular center of mass into eq 2 (change in polymer conformational entropy due to mixing is contained in χ-parameters) yields the free energy of mixing which has the form of the Flory−Huggins (F−H) theory.Considering an inhomogeneous system with ϕ i (r) being the volume fraction of component i at location r, the effective Hamiltonian of a particlebased model can be constructed based on eq 2. Specifically, the local excess free energy of mixing in volume element dr at location r can be written as drρ 0 (r)Δf ex , where ρ 0 (r) is the total packing fraction of the ef fective particles (referred to hereafter as segments) at location r.
Dropping the χ ijk -term in eq 2 for consistence, the effective Hamiltonian becomes Upon introducing the segmental microscopic densities ρ̂i ,c (r′) ≡ ∑ k δ(r′ − r i,k ) where r i,k is the center location of the k-th segment of type i, ρ 0 (r) and ϕ i (r) in eq 4 can be written as ρ 0 (r) = ∑ i=P,S,C ρ̂i(r) , respectively, where ρî(r) = ∫ dr′S i (r − r′)ρî ,c (r′) and

Macromolecules
S i (r − r′) is the shape function of the i-type segment that satisfies ∫ drS i (r − r′) = v i .For a strictly incompressible system, ρ 0 (r) = ρ 0 and eq 4 becomes By defining u ij (r′ − r″) ≡ ∫ drS i (r − r′ i )S j (r − r″ j ), a pairwise form is arrived in which u r r ( ) acts as the effective interaction between segment i and j (i ≠ j) due to immiscibility.In the particle-based model, the incompressibility condition = = r ( ) for convenience, we explicitly specify the function form for u ij (r − r′) while keeping S i (r − r′) implicit.For all ij-pair we set The center locations of effective segments introduced in eq 5 are also used to construct the effective Hamiltonian arising from polymer chain connectivity, i.e.
where r k,s is the center location of the s-th segment on the k-th polymer chain.Equations 7−9 define an effective particle-based model that is consistent with eq 2, and in which polymer conformational degree of freedom beyond the segment scale in an inhomogeneous environment can be explicitly accounted for.High κ values (κ = 16 throughout this study) are employed in implementations of the particle-based models to ensure near incompressibility so that the χ-parameters are comparable to that in the F−H model.For simplicity, we also assume the same segmental volume for all types of segments (v i = v 0 , i = P, S, C) in the calculations.The F−H theory will be used to derive an analytical expression of the second osmotic virial coefficient from which the conditions for polymer cononsolvency are determined.Using the effective particle model, self-consistent field (SCF) calculations will be conducted to examine the polymer conformational transition as a function of solvent mixture composition to verify the predicted condition.Monte Carlo (MC) simulations will be used to assess the effects of mean-field approximation on the predictions with the inclusion of fluctuation effects, which prove to be important in dilute polymer solutions.Details of the implementations can be found in Section 5.

Second Osmotic Virial Coefficient
2 .For a sufficiently dilute polymer/solvent/cosolvent solution (ϕ P ≪ 1), the second osmotic virial coefficient can be derived as C (see Section 5 for derivation), where PS corresponds to the second osmotic virial coefficient in binary polymer/solvent mixtures, and the term accounts for the cosolvent effects with The dependence of 2 on x C derives solely from which is controlled by two parameters: the solvent/cosolvent immiscibility parameter χ SC and Δχ ≡ χ PS − χ PC that characterizes the immiscibility mismatch between polymer/ cosolvent and polymer/solvent.With no loss of generality, we let Δχ ≥ 0, i.e., cosolvent is defined as the species that is more miscible with polymer.Equation 10 gives = x ( ) 0 shows both a minimum and a maximum.Since 2 manifests the strength of effective repulsions between polymer segments, an inverse response of 2 in regions I and II with increasing x C implies a reentrant type polymer conformational transition.For this reason, we define regions I and II as the two cononsolvency regimes in this study, and their exact boundaries are determined as respectively.Contrastingly,

2
C exhibits an overshoot in response to increasing x C in region III which can be accordingly defined as the cosolvency region. 22We defer the discussion of cosolvency to a future study and will focus on cononsolvency in this work.It is worth mentioning that the boundaries in Figure 1 depend on Δχ = χ PS − χ PC but not on PS .In comparison, the monomer chemical potential μ P can be derived from eq 3 as ) ( ) where the monomer chemical potential in binary P/S mixtures, and the second term arises from the cosolvent effects.It can be shown that in Regime I (χ SC + Δχ < 0), μ P changes synchronously with 2 , possessing a maximum with respect to where 2 is at its minimum.In Regime II (χ SC + Δχ > 2), however, μ P decreases monotonically with respect to x C ∈ [0, 1] whereas 2 possesses a minimum, indicating a decoupled behavior of μ P and 2 in this region.We now seek further insights into the role of Δχ and χ SC in inducing reentrant polymer conformational transitions.Toward that, we examine the local solution structure in polymer proximity which can be characterized by the overall polymer-(co)solvent contact and the local cosolvent composition.The contact between the polymer and the α-solvent can be quantified as m Pα ≡ ρ 0 E Pα /χ Pα (α = S and C) with E Pα being the polymer-α interaction energy.The total polymer-(co)solvent contact and the local solvent mixture composition can then be accordingly calculated as m total = m PC + m PS and x C ′ = m PC /m total , respectively.Figure 3(a,b) shows m total and x C ′ /x C as the function of x C at a set of Δχ and χ SC values, respectively.With Δχ = 0 and χ SC = −3 (inside Regime I), Figure 3(a) shows that m total (x C ) exhibits a reentrant behavior symmetric about x C = 0.5 that tracks the transition of x ( ) ee C .On the other hand, the local cosolvent composition x C ′ shown in Figure 3(b) indicates a depletion of the minor solvent component (i.e., x C ′ < x C when x C < 0.5 and x C ′ > x C when x C > 0.5).This depletion signals that solvent-cosolvent contact is being promoted in the solution by moving the minor solvent species away from polymer proximity and replacing them with the primary solvent.

Polymer Conformation
As x C → 0.5, this tactic becomes increasingly less effective, as there are not enough primary solvents to keep both the minor solvent and polymer in accompany.As the result, promoting S−C contact relies increasingly on polymer size contraction, which explains x ( ) ee C approaching minimum at x C = 0.5.Therefore, the polymer conformation transition in Regime I can be understood as being driven by system maximizing solvent-cosolvent contact.In consequence, greater χ SC or Δχ values will act to reduce this driving force, leading to lesser to none depletion by the minor solvent specie as shown in Figure 3(b).This results in the eventual disappearance of reentrant transition by ee as the system moves out of Regime I.As comparison, with Δχ = 3 and χ SC = 1 (inside Regime II), Figure 3(d) shows that the local cosolvent composition x C ′ exhibits a pronounced peak at x C ∼ 0.28 corresponding to a > 20% increase from the bulk cosolvent composition.The peak compares well with a similar extent of contraction in x ( ) ee C shown in Figure 2, but contrasting to only ∼3% change in m total shown in Figure 3(c).This suggests that unlike in Regime I where the overall contact between polymer and (co)solvent is reduced in favor of solvent-cosolvent contact, polymer contraction in Region II is driven by promoting the polymercosolvent contact through partitioning of cosolvent molecules in favor of polymer proximity.The extent of cosolvent localization, characterized by x C ′ /x C , depends on the competitive interplay of entropy of mixing and favorable energetics.Strongly favorable energetics are required to overcome the entropic penalty to generate local cosolventenriched regions at low x C , which induces polymer conformational contraction.Effectively favorable energetics for polymercosolvent contact can derive from excess affinity of the cosolvent to the polymer and/or greater solvent-cosolvent immiscibility.This means that greater Δχ and χ SC values will lead to more pronounced peak in x C ′ (Figure 3(d)) and drive the system toward Regime II.The above analysis rationalizes the emergence of Δχ + χ SC as the relevant parameter in the general condition predicted in eq 11.The two cononsolvency regimes are located on opposite ends of the Δχ + χ SC scale, separated by a monotonic region as the driving force for polymer conformation transition changes from maximizing S/ C contact to promoting P/C contact.
The effect of the mean-field approximation (employed in the Flory−Huggins and SCF calculations) on the predicted general condition can be assessed by comparison to MC simulations.Using the same model, such comparisons will reveal unambiguously the effects of fluctuating local solvent mixture compositions on the reentrant polymer conformation transition.Figure 4  contraction comparing to the MC simulations.Figure 5 compares the radial distribution functions obtained from the SCF calculations and MC simulations for the two cases in Figure 4(a−b) at x C = 0.5 and 0.2 where ee reaches their respective minimum.Again, good agreements are seen in all three pairs of radial distributions at χ SC = −2 and Δχ = 0.But at Δχ = 4 and χ SC = 1, a much higher degree of cosolvent enrichment around polymer is predicted by the SCF calculations.The results suggest that the influence of fluctuation effects on the reentrant polymer conformation transition is much stronger in Regime II than in Regime I.This is probably because, as discussed above, the reentrant transition inside Regime II relies on the local cosolvent enrichment in polymer proximity, which is more subject to the effects of composition fluctuations, whereas the reentrant transition inside Regime I is driven by promoting solventcosolvent contact, which maximizes at x C = 0.5 and takes place dominantly in the bulk region far from the polymer.We therefore conclude that the mean-field approximation leads to an underestimate of the threshold value of χ SC + Δχ for Regime II, but has much less effects on that for Regime I.

SUMMARY AND DISCUSSION
Starting from a generic framework based on the thermodynamics of mixing and abstract from the chemistry details of solution components, a model is developed and studied using three complementary computational approaches to investigate the general condition for polymer cononsolvency in binary solvent systems.Our study shows Δχ + χ SC as the underlying universal parameter that regulates cononsolvency.Two disparate cononsolvency regimes are identified for Δχ + χ SC < 0 and Δχ + χ SC > 2, respectively, based on the behavior of the second osmotic virial coefficient 2 .These conditions are further verified in the self-consistent field calculations by directly examining the polymer conformation transitions.It is further shown that in the regime Δχ + χ SC < 0, reentrant polymer conformational transition is driven by maximizing the solvent and cosolvent contact, but in the regime Δχ + χ SC > 2, it is driven by promoting polymer and cosolvent contact.Inbetween, when neither effect is dominant, a monotonic response of polymer conformation to cosolvent composition is observed.Results from the mean-field calculations are in qualitative agreement with the Monte Carlo simulations.Effects of compositional fluctuations are found to play a more significant role in the Δχ + χ SC > 2 regime, suppressing local enrichment of the cosolvent in polymer proximity.
Results of this study rationalize trends observed in several experiments and computer simulations.In particular, connections can be made with reference to the three microscopic cononsolvency models discussed in ref 33.In the model that features dominant solvent-cosolvent energetic attraction, the settings of the microscopic simulations corresponds to having χ SC < 0 and Δχ ≡ χ PS − χ PC > 0 in the current formulation.Given enough solvent-cosolvent attraction strength (χ SC being more negative), χ SC + Δχ turns negative and places the system inside Regime I, where reentrant conformation transition is accompanied by depletion of the minority solvent specie (similar to what is shown in Figure 2(b)).Increasing monomer-cosolvent attraction strength corresponds to decreasing χ PC and as the result increasing Δχ.According to eq 11, this will drive the system from Regime I toward the monotonic regime, leading to a reduced extent of reentrant conformation transition by the polymer (Figure 2(a)), exactly as observed in ref 33.The "entropic" model in ref 33 features a size difference between the monomer and solvent as the cosolvent size is varied in-between.This microscopic setting corresponds to having χ SC > 0 and Δχ > 0 (i.e., χ PS > χ PC because of a smaller size difference between monomer and cosolvent).Increasing monomer−solvent size difference will increase both χ SC and Δχ and eventually bring the system inside Regime II according to eq 11.In the meanwhile, reducing cosolvent size toward the size of solvent corresponds to reducing both χ SC and Δχ toward zero.According to eq 11, this acts to drive the system from Regime II toward the monotonic regime and reduce the extent of reentrant transition, as observed in ref 33.Notably, in Regime II, the onset of preferential adsorption does not commence the reentrant conformation transition, which can be seen in Figures 2(d) and 3(d) for the case (Δχ = 3, χ SC = −1).At last, in the model that features strong solvent−solvent energetic attractions, this setting corresponds to having χ SC > 0 and Δχ > 0. Increasing solvent−solvent attraction strength results in greater χ SC and Δχ, which will eventually bring the system inside Regime II.Keeping everything else unchanged, increasing solvent-cosolvent energetic attraction strength will act to reduce χ SC and as the result χ SC + Δχ.According to eq 11, this will drive the system toward the monotonic regime and reduce the extent of reentrant transition, again in agreement with ref 33.The above comparisons demonstrate χ SC + Δχ acting as the underlying universal parameter in models with differing microscopic details in regulating polymer cononsolvency.
Although this study is based on the Gaussian chain model, we expect that the qualitative conclusions still apply when considering real polymers with finite chain length and excluded volume effects.This is based on two considerations: (1) As long as the polymer chain length exceeds several persistence lengths, the bending rigidity of real polymers can be renormalized into an effective Kuhn length in the Gaussian chain model; and (2) polymer cononsolvency concerns mostly with the long-length-scale properties of polymers, i.e., meansquare end-to-end distance as the function of solution composition.At this length scale, the averaged effect of microscopic interactions between monomers, responsible for the excluded volume effects in real polymers, can be modeled using effective pseudo potentials such as the one shown in eq 8. Furthermore, calculations in this study are conducted at the zeroth order with the intention of showcasing the viability of formulating general cononsolvency conditions using generic considerations.Improvements can be systematically conceived toward more accurate descriptions by incorporating higherorder effects into the calculations.One possible improvement is to still assume concentration-independent χ but retain the term χ PSC ϕ P ϕ S ϕ C in eq 2. By adsorbing χ PSC ϕ P ϕ S ϕ C into χ PC ϕ P ϕ C , the effective parameter χ PC eff ≡ χ PC + χ PSC ϕ S contains the first-order correction to χ PC due to the solvent-mediated effects.This may help address the atypical high threshold values in Δχ obtained by the current calculations.A full treatment of concentration-dependent χ will have to rely on the microscopic details for determining the exact form of the dependence.In its current form, the generic model can be readily applied to studies of cosolvent effects in more complex systems, such as the solution-phase polymer self-assemblies, which are of practical importance but difficult to address directly using the chemistry-specific atomistic models.Studies along these lines will be reported in future publications.

Expansion of Δf ex and χ-Parameters.
Assuming no volume change upon mixing (i.e., V = ∑ i V i (0) with V i (0) being the volume of the i-component in its pure state), the free energy of mixing per unit volume is given by where v i is the volume of an i-type monomer, ϕ i = V i (0) /V is the volume fraction of the i-type monomer in the mixture = = ( 1) , and βΔf ex is the excess free energy of mixing given as with f ̃i ex being the excess free energy per unit volume of the pure-i state.As a mixing property, βΔf ex must vanish when ϕ i = 1 and reduce to the excess free energy of mixing of a binary mixture when ϕ i = 0 (i = P or S or C).For that, one can argue that βΔf ex can be expanded into series of binary ϕ i m ϕ j n and ternary ϕ i m ϕ j n ϕ k p terms, with m, n, p ≥ 1 being integers. 34The high-order terms in the expansion can be adsorbed into the leading-order terms by introducing ϕ-dependent coefficients χ ij and χ ijk , which leads to the concise expression for βΔf ex shown in eq 2.
The nature of the χ-parameters in eq 2 can be inferred by a similar expansion of f ̃mixture Considering that , the double sum can be interpreted as the additive effects of mixing, with the expansion coefficients f ̃ij ex (i ≠ j) and f ̃ii ex = f ̃i ex carrying the meaning of per unit volume excess free energy due to i−j cross interactions and i−i interactions, respectively.The f ̃PSC ex term corrects for the "non-additive" effects such as the free volume and contact surface dissimilarities, 35 or the formation of coordinated polymer−solvent-cosolvent complexes. 36By substituting eq 15 into eq 14 and comparing with eq 2, one obtains χ ij = 2f ̃ij ex − f ̃i ex − f ̃j ex and χ ijk = f ̃ijk ex from which their freeenergy nature becomes evident.More specifically, the χparameters encode all (both enthalpic and entropic) effects of mixing due to, e.g., dispersion interactions, dissimilarities in the excluded volume, and chain conformation changes, etc., and are generally composition dependent.
5.2. 2 from the Flory−Huggins Theory.Consider a binary solution of solvent (S) and cosolvent (C) in osmotic equilibrium with a ternary solution consisting of S, C, and polymer (P), the osmotic pressure is then defined through equating the chemical potentials of the diffusible components in the two systems, i.e.
In principle, Π be solved in terms of ϕ S , ϕ C , and ϕ P from eqs 24 and 25 together with ϕ S 0 + ϕ C 0 = 1.To the best of our efforts, solutions are obtainable only via density expansions.For ϕ P → 0, substitute into eqs 24 and 25 and apply Taylor series for ln(x); the expansion coefficients A 1 , A 2 are determined (by matching the corresponding coefficients of the two resulting polynomials) as Following this, the osmotic second virial coefficient 2 can then be identified as It is noteworthy that using a similar approach in ref 8 obtains an expression for 2 that from eq 28.Here, we explain the cause of this disparity.Following the definitions in eqs 16 and 17, but express μ S 0 and μ S (similarly for μ C 0 and μ C ) using t h e H e l m h o l t z f r e e e n e r g y , i .e ., Changing variables from ΔF(n S , n C , n P , T) to ΔF(V, ϕ P , x C , T) by applying chain rule to the second term, i.e., , and substituting into eq 29, one obtains where f x T ( , , ) The first two terms in eq 31 do not contribute to .In the special case of a binary P + S solution (i.e., x C = 0), eq 31 yields = P 0 being the excluded volume parameter.This corresponds to the result reported in ref 8.However, for ternary solutions, the last term in eq 31 does not vanish and will also contribute to , which was neglected in ref 8.This results in conclusions that are different from those in the current study.

Polymer End-to-End Distance and Radial Distributions from the Self-Consistent Field Calculations.
The extended Percus test-particle method 38 is applied in the SCF calculations to obtain the polymer end-to-end distance and the radial distributions.We consider a system of volume V consisting of a single polymer (P) chain tethered to the origin by the t-th segment mixed with solvent (S) and cosolvent (C).The partition function of the system is given by with nb and b being given in eqs 7 and 9.By labeling the chain section 1 ≤ s < t as "L" and the section t < s ≤ N P as "R", and defining ρL ,c (r) ≡ ∑ s=1 t−1 δ (r − r P,s ), ρR ,c (r) ≡ ∑ s=t+1 N P δ (r − r P,s ), after integrating out the δ-function, the partition function of the system becomes where ϵ α α′ = χ α α′ + κ for α ≠ α′ and κ for α = α′.Following the standard Hubburd-Stratonovich transformation, the SCF equations are obtained under saddle-point approximation as In calculations, a cutoff radius L r is used to carry out the integration, i.e., ∫ 0 ∞ ≈ ∫ 0 L r , with L r being large enough such that the solution composition at r = L r approaches that of a binary S + C solvent mixture.A similar expression can also be derived for q † (r, s) starting from eq 39b.
To calculate the end-to-end distance of polymer chain ee , we set t = 1.After solving the SCF equations, the end-to-end distance can be calculated as .The trial move is accepted or rejected according to the Metropolis acceptance criterion

2 C
the constraints of x C,1 * ∈ [0, 1] and x C,2 * ∈ [0, 1].Depending on the presence and nature of the extrema in , the χ SC − Δχ plane can be divided into four regions as shown in Figure 1. 2 C possesses a minimum (<0) in region I and II, a maximum in region III, but is monotonic in region IV.In the overlapping region between II and III, 2 C

Figure 1 .
Figure 1.Respective regions in the χ SC − Δχ plane in which 2 C , derived from the F−H theory, varies monotonically (IV), possesses minimum (I and II), or maximum (III) with respect to x C .Symbols are estimated boundary points separating the monotonic and reentrant polymer conformation transitions in the SCF calculations.

Figure 3 .
Figure 3. (a) and (c): Total polymer−solvents contact m total ; (b) and (d): the local cosolvent enrichment x C ′ /x C as the function of x C at a set of Δχ and χ SC values.See the text for details.
(a,b) compares x ( ) ee C obtained from the SCF calculations and MC simulations.At χ SC = −2 and Δχ = 0 (χ PS

ex.
Recognizing that f ̃mixture ex must recover the excess free energy of the pure-i state when ϕ i = 1 (i = P or S or C), f ̃mixture ex can be expanded into the following form34

,
and G 0 and G being the Gibbs free energy of the binary and ternary systems, respectively.Assuming no volume change upon mixing and segmental volume v 0 = 1, G 0 and G can be written asMacromolecules ΔF are the Helmholtz free energy of mixing of S + C and P + S + C mixtures, respectively.Plugging μ S 0 and μ S into eq 16, one obtains Meng − Biomaterials Division, Department of Molecular Pathobiology, New York University, New York, New York 10010, United States; orcid.org/0000-0003-1763-6411;Email: dm173@nyu.edu