Solvation shell thermodynamics of extended hydrophobic solutes in mixed solvents

Interfacial solvent density fluctuations play an important role in hydrophobic effects, which regulate the functional characteristics of a wide variety of soft matter systems, including bio-molecules.^1–4 It has been observed that small hydrophobic solutes are hydrated without disrupting the hydrogen bonded water network around them, whereas the inability of water molecules to maintain their hydrogen bonded network around the extended interface leads to enhanced density fluctuations near large non-polar solutes.^5–7 These enhanced fluctuations make the hydration shell of large solutes susceptible to small perturbations,^6,8 which, in case of a hydrophobic polymer with a large solvent accessible surface area (SASA), can let monomers easily shed their hydration water and subsequently lead to hydrophobic collapse. The crossover between the small and large length scale regimes is found to further depend on the solute–solvent attraction⁹ and also has crucial implications in the folding/unfolding of protein.^2,6,10

Cosolutes (but also cosolvents, hereafter referred to as cosolutes as the third species in the solute–water system) can either enhance or quench interfacial solvent density fluctuations and thereby regulate the hydration behavior of non-polar solutes. This modulation results from the preferential accumulation or depletion of the cosolutes on the solute surface. However, the effect of preferential cosolute accumulation/depletion on the solvent density fluctuations and, subsequently, on the thermodynamics of the solvation shell is not fully understood. In this regard, Nayar et al. have shown that urea preferentially adsorbs onto poly(N,N-diethylacrylamide) (PDEA) and leads to polymer swelling.¹¹ This observation has been attributed to the quenching of interfacial solvent density fluctuations by the adsorbing urea molecules.¹² Interestingly, urea also preferentially adsorbs onto the chemically similar poly(N-isopropylacrylamide) (PNIPAM), but instead induces polymer collapse. This difference in the mechanism of urea action has been attributed to the length scale effects, as PDEA has a larger surface area compared to PNIPAM.¹² Note that this phenomenon of preferential cosolvent adsorption accompanying both polymer collapse and swelling has also been observed for aqueous polymer solutions with amphiphilic cosolutes such as alcohols and acetone.^13–17 Here, it is important to note that preferentially accumulating cosolutes can be broadly classified into two categories. The first one includes cosolutes such as urea, which preferentially adsorb due to the attractive solute–urea interactions.^11,18–23 The second type includes amphiphilic ones such as alcohols, which preferentially adsorb onto surfaces irrespective of the solute–cosolvent attractive interactions.^13,16,24,25 Therefore, the effect of the cosolute on interfacial solvent density fluctuations and, thereby, the hydration behavior has to depend on the interplay between the solute–cosolute and cosolute–solvent attractive interactions and the solute size.

In our previous work,²⁶ we investigated the effect of solute size and solute–water interaction on the density fluctuations in the hydration shell (HS) of a model extended hydrophobic solute in pure water. By systematically strengthening the repulsive interaction between the solute and water, we modeled the system between the two limiting cases of an attractive solute with a small SASA and a repulsive solute with a large SASA. We observed a transition from a “stable” HS with lower density fluctuations and compressibility at weak solute–water repulsion to that with significantly larger density fluctuations and compressibility at stronger solute–water repulsion, relative to shells of the same size in pure water. Given the distinct nature of solute–solvent–cosolute interactions, it is intriguing to investigate their effect on the density fluctuation in the solvation shell (SS) of the hydrophobic solute and the SS thermodynamics. In this paper, we probed the effect of urea and methanol on the SS thermodynamics of the model hydrophobic solute by systematically strengthening the solute–urea, solute–water, and solute–methanol repulsion. The choice of the systems here was motivated by the distinct hydration behavior and solute interaction of the two adsorbing cosolutes. Urea forms a near homogeneous mixture with water (at low concentration),²⁷ which is mostly governed by its favorable hydrogen bonding with water,²⁸ and the preferential accumulation of urea on a non-polar solute is governed by attractive solute–urea van der Waals interactions.¹² On the contrary, aqueous methanol is nano-scale segregated²⁹ owing to inadequate water hydrogen bonding capability,³⁰ and methanol can preferentially accumulate on a non-polar solute surface even in the absence of attractive interactions owing to its amphiphilic nature.^24,25,31

We used the Small System Method (SSM)^32,33 to accurately determine sample density fluctuations in these finite size SS. This method is based on the “thermodynamics of small systems” presented by Hill^34,35 and provides a route to estimate, using finite size simulation systems, properties in the thermodynamic limit (TL) by means of scaling laws. We computed the Kirkwood–Buff integrals (KBIs)³⁶ via the density fluctuation route and used them to estimate various thermodynamic properties of the SS.³⁷ We observed distinctly different effects of urea and methanol on the thermodynamics of the SS of the extended hydrophobic solute assessed in terms of the KBIs, excess/deficit of various species, activity derivatives, molar volumes, and SS compressibilities. For a fixed urea concentration, the preferential accumulation of urea on the solute diminished as the solute–urea and solute–water repulsive interactions were progressively strengthened. Such a trend is expected as the preferential accumulation of urea is driven solely by attractive solute–urea van der Waals interactions.¹² This decrease in the preferential accumulation of urea was found to enhance the compressibility of the SS. In contrast, strengthening the solute–methanol and solute–water repulsive interactions, at a fixed methanol concentration, enhanced the preferential accumulation of methanol in the solute SS. As opposed to the case in urea, the increase in the preferential accumulation of methanol led to an increase in the SS compressibility. The distinct modes of action of the two types of preferentially accumulating cosolutes can be attributed to the difference in their hydration behavior: water–urea mixtures are nearly ideal,^27,28 while water–methanol mixtures are nano-scale segregated.^29,30 As discussed later, these results may have implications in the cosolute induced swelling and collapse of polymers in aqueous solutions.

The rest of this paper is organized as follows: In Sec. II, we discuss the details of the methodology, including the simulation details and the methods used to estimate thermodynamic properties of the SS using KBIs. In Sec. III, we present our main results. We conclude in Sec. IV by summarizing our main findings and discussing their implication in the context of cosolute induced polymer conformational transition.

We considered a model extended hydrophobic solute, which interacted with the water–cosolute mixture via Lennard-Jones (LJ) interactions. The extended surface was created by periodically replicating a stretched uncharged 80 bead polymer chain in the Z-direction.³⁸ The end to end distance along the Z-direction, R_ee, was 12.2 nm. Additional position restraints were applied on the beads (force constant k = 10⁵ kJ mol⁻¹ nm⁻²) to keep the chain conformation fixed. The LJ parameters for the beads were σ_p = 0.4 nm and ϵ_p = 1.0 kJ mol⁻¹. The bonds between the beads were kept rigid at 0.153 nm. The solvent was either an aqueous urea mixture (with the urea concentration between 1.526 and 9.014M) or an aqueous methanol mixture (with the methanol concentration between 9.000 and 17.241M corresponding to the methanol mole fraction between 0.2 and 0.5). The concentration of urea and methanol was so chosen based on their concentrations in previous studies on polymer conformational equilibria.^13,15,19,39 Readers are referred to the supplementary material for additional details on system composition (see Tables S1 and S2). The Extended Simple Point Charge (SPC/E) model⁴⁰ was used for water, the Kirkwood–Buff force-field for urea,²³ and the Optimized Potentials for Liquid Simulations - United Atom (OPLS-UA) force-field for methanol.⁴¹ The repulsive part of the LJ interaction, between the solute and the molecules in the solvent–cosolute mixture, was systematically scaled using a multiplicative parameter α,

where U_sx was the LJ potential between the solute and species “x” (solvent or cosolute). The Lorenz–Bertholet mixing rule was used to calculate the corresponding LJ parameters. As shown in Fig. S1, an increase in α led to a decrease in the attractive solute–water/cosolute interactions and an increase in the effective bead size. Thus, a systematic increase in α modeled the solute between the two limiting cases: an attractive extended solute with a small SASA and a repulsive extended solute with a larger SASA, as discussed in our previous work.²⁶ Note that α simultaneously strengthened the repulsion interaction between both the solute–solvent and solute–cosolute. The model solute in water–urea mixtures was simulated at α values 1, 2, and 4 and in water–methanol mixtures at α values 4, 8, and 15. These α values were chosen such that the cosolute molecules remain preferentially accumulated on the model solute.

MD simulations were performed using the Gromacs 2019.3⁴² package using a time step of 2 fs. A cutoff of 1.4 nm was used for the van der Waals and short range Coulomb interactions. The particle mesh Ewald method was used for the long range electrostatic interactions. All bonds were constrained using the LINCS algorithm.⁴³ Following an initial energy minimization and an NVT run using the velocity-rescale thermostat,⁴⁴ an NPT run using the Nośe–Hoover thermostat⁴⁵ and the Berendsen barostat⁴⁶ was performed. The Parinello–Rahman barostat⁴⁷ was used in a subsequent production NPT run, beyond which, repeated NPT–NVT cycles were used to properly equilibrate the systems at 1 bar pressure and 300 K temperature. These repeated runs used the Nośe–Hoover thermostat and the Parinello–Rahman barostat with time constants 1 and 2 ps, respectively. The last NVT run was extended as production trajectory for subsequent analyses using system snapshots collected at every 1 ps. The production trajectories for the solute–urea–water and solute–methanol–water systems were both 100 ns long. We also simulated pure urea–water and methanol–water mixtures at all the concentrations without the solute (see Tables S1 and S2 in the supplementary material for further details). Readers are referred to our previous work²⁶ for further details on the simulation protocol.

The preferential adsorption coefficient Γ_sc was calculated to quantify the adsorption behavior of urea and methanol, which indicated their relative adsorption/depletion on the solute, as compared to the far away bulk,

where n_x(r) denotes the number of molecules of species “x” (s: solute, c: urea/methanol, w: water) within a radial distance r around the solute and N_x denotes the total number of molecules of species “x.” At each concentration of urea/methanol and at each α value, the radial extension of the solute SS was identified by the position of the first peak in the corresponding Γ_sc curve (see Figs. S2 and S3). The density fluctuations were sampled within these limits in the XY plane around the solute.

The KBIs are calculated either as integrals of the local structure or using the density fluctuation in an open system and, in the thermodynamic limit (TL), are related to the thermodynamic quantities such as isothermal compressibility, derivatives of the chemical potential, activity derivatives, and molar volumes,³⁶ 

where i and j are the two species in the binary mixture, V is the integration volume, $gijo$ is the radial distribution functions (RDFs) between i and j calculated in the open system, N_x denotes the number of particles of species “x,” δ_ij is the Kronecker delta, and $⋯$ denotes an ensemble average in an open system. The KBIs, G_ij, can be understood as the relative affinity between the participating species i and j.³⁶ 

We calculated the thermodynamic properties of water/urea/methanol in the SS of the solute, based on the well-known relations derived for binary mixtures.³⁷ The excess/deficit of particles of species j with and without particle i at the center of a small observation volume was calculated as Δn_ij = ρ_jG_ij,⁴⁸ where ρ_i denotes the number density of species i in the SS. The activity derivative γ_cc, where c = {u, m}, was calculated as

where a_x is the activity of species “x.” The activity derivative γ_wc was calculated as

The molar volumes $ṽw$ and $ṽx$⁠, where x = {u, m} for urea–water and methanol–water systems, were calculated as

where

and

The isothermal compressibility was calculated as

where R is the gas constant and T is the temperature.

Characterizing the thermodynamic properties of interfaces is non-trivial. For example, the compressibility of the system, with and without the solute, can be computed by analyzing volume fluctuations in the NPT ensemble to estimate the net effect of the solute. However, in such a case, the compressibility will indicate the cumulative effect of all the SS of the solute. In order to quantify the subtle changes that take place in the immediate vicinity of the solute, one has to establish a method to accurately compute the thermodynamic properties within the first SS, which is a finite size system. Toward this, Trinh et al. extended the small system method (SSM) to characterize the thermodynamics and kinetics of CO₂ and CH₄ adsorption on graphite surfaces,^49,50 where the properties of interest were calculated in mono-layers and the TL corresponded to infinite planes. In the same line, we employed the SSM to quantify the fluctuations and the resulting compressibility of the hydration shell of a model hydrophobic solute.²⁶ Therein, we calculated these properties in concentric cylinder shaped hydration shells around the extended solute and the TL corresponded to infinite cylinders. In this work, we followed this framework to quantify the thermodynamics of the solvation shell of the solute in mixed solvents.

For a finite size system, the KBIs can be calculated from the number density fluctuations in an open system,

where V_L is the volume of the subsystem with dimension L. Note that KBIs can be calculated in two different ways. The first method involves direct analysis of particle number fluctuations in small open sub-volumes embedded in a larger reservoir,⁵¹ whereas the second method is based on the computation of the running integrals over the RDFs. For bulk liquid mixtures, results from both the methods are in quantitative agreement.⁵² In this work, the SS of the solute was infinite in only one dimension (Z) and finite in the other two dimensions (see Fig. 1). Converging the RDF between the solvent and cosolute molecules in the SS of the solute was, therefore, extremely difficult. In such a case, KBIs could be better estimated via the direct analysis of particle number fluctuations.

The finite size KBIs, G_ij(L), are known to scale with the system size L^51,53–55 as

where C is a constant. This provides a route to compute the corresponding KBIs in the TL, $Gij∞$⁠, using the above scaling law. Here, we calculated G_ij(L) (i, j: {u, m, w} indicating water/urea/methanol), within concentric cylindrical shaped sub-volumes V_L of height L²⁶ following the SSM^32,33 (Fig. 1). The height of the subsystem, L, was varied from 1 Å to the box size in the Z-direction (R_ee = 12.2 nm) in units of 1 Å. The linear regime in LG_ij(L) vs L plots was fitted to straight lines, where the slope indicated the corresponding $Gij∞$ values (see Figs. S4–S7). Further details on using SSM to calculate SS specific properties can be found in our previous work.²⁶ 

The $Gij∞$ values calculated in the above mentioned manner corresponded to KBIs in an infinitely long SS of the solute (Fig. 1). We denoted this quantity as $GijSS$ and used it to calculate various thermodynamic properties of the SS as discussed earlier. To interpret the computed thermodynamic quantities of the SS, we also computed $GijSS,•$⁠, i.e., the KBIs of concentric shells in water–urea and water–methanol mixtures without the solute, at the corresponding concentration of urea/methanol (see the supplementary material for details). The SS extensions were fixed based on the concentration and α values. For every thermodynamic quantity that we calculated for the solute SS using $GijSS$ values, we calculated the corresponding quantity in shells without the solute and analyzed the resulting excess quantities,

Thus, the excess thermodynamic quantities reported here captured the sole effect of the solute on the interactions and thermodynamic properties of the molecular species in the solute SS relative to shells with the same dimensions in the corresponding bulk mixtures. We also included the bulk $Gij∞$ values and the corresponding thermodynamic quantities in our results wherever possible to indicate the agreement/difference between the SS specific quantities and the corresponding bulk ones.

The preferential accumulation of urea is driven by the attractive solute–urea van der Waals interactions. Therefore, strengthening the solute–urea (and solute–water) repulsive interactions reduced the preferential accumulation of urea in the solute SS. As shown in Fig. S2, for a fixed urea concentration, the preferential accumulation of urea diminished with increasing α, i.e., α↑, Γ_sc↓. In contrast, for a fixed methanol concentration, the preferential accumulation of methanol got enhanced with increasing α, i.e., α↑, Γ_sc↑ (Fig. S3). Here, it is important to emphasize that the preferential accumulation of methanol is driven by a combination of two factors: the solute–methanol attractive interactions and the surfactant-like effect originating from the amphiphilic nature of methanol.^4,25 This behavior can be inferred from the observation that such molecules preferentially adsorb at the air–water interface and reduce the surface tension of water. The surfactant-like effect is present under all conditions and may be reinforced by the solute–methanol attractive interactions. In contrast, the accumulation of water on the solute is driven only through solute–water attractive interactions. As one can expect, at fixed α, the preferential accumulation of both urea and methanol increased as a function of their concentrations (Figs. S2 and S3). As we discuss next, the interplay of these effects determined the relative population of urea/water/methanol in the solute SS.

In Fig. 2, we compare the excess number density $Δρx=ρx−ρx•$ (x: u/m/w) in the solute SS for various α values. As the preferential accumulation of urea in the solute SS increased as a function of the urea concentration (Fig. S2), for C_u < 2M, this led to an increase in Δρ_u and Δρ_w decreased to accommodate the excess urea [Fig. 2(a)]. Between a concentration of 2 and 4M (depending on the α value), urea saturated the solute SS, leading to a subsequent decrease in Δρ_u as a function of the urea concentration and a corresponding increase in Δρ_w. With increasing α, preferential accumulation of urea decreased resulting in a decrease in Δρ_u and an increase in Δρ_w. Interestingly, Δρ_u and Δρ_w almost compensated each other at all α values. Furthermore, at α = 4, both Δρ_u and Δρ_w approached zero, indicating their composition in the solute SS to be almost similar to their respective bulk compositions.

Figure 2(b) shows the corresponding trends in methanol–water mixtures. At low methanol concentrations (C_m < 10M), strengthening the solute–methanol and solute–water repulsive interactions (i.e., increasing α) led to a decrease in Δρ_w, whereas Δρ_m remained almost constant. At low α values, the preferential accumulation of methanol was driven by a combination of attractive solute–methanol interaction and surfactant-like effect and the water molecules competed with the methanol molecules to interact with the solute. At high α, the preferential accumulation of methanol was driven predominantly by the surfactant-like effect and there was no competition from the water molecules due to the enhanced solute–water repulsion. As the SS was expected to be saturated with methanol at C_m = 10M [similar to Fig. 2(a) for urea], Δρ_m decreased and Δρ_w increased with a further increase in C_m. Moreover, it could be seen that Δρ_m and Δρ_w did not compensate each other as in the case of water–urea mixtures [Fig. 2(a)]. This could be attributed to the accumulation of methanol at the solute interface with increasing α, which could push the water away from the solute surface in the absence of favorable solute–water van der Waals interaction. Unlike the case of water–urea mixtures, the compositions of either methanol or water in the SS did not approach their respective bulk compositions at large α values (except for Δρ_m at α = 15 and C_MeOH > 17M).

In Fig. 3, we show the $GijSS$⁠, {i, j} = {w, u} or {w, m}, for the SS of the solute and shells without the solute for urea–water and methanol–water mixtures. These quantities were calculated using the SSM using the scaling law in Eq. (12) (see Figs. S4–S7). Interestingly, the values of $GuuSS,•$ (for shells without the solute) were comparable to $Guu∞$ values for bulk water–urea mixtures observed in experiments and simulation.^27,56 As mentioned earlier, $GijSS$ indicates the mutual affinity between the species i and j. Clearly, the urea–urea and methanol–methanol mutual affinities decreased in the presence of the solute, while the water–urea and water–methanol affinities increased. The KBIs, $GijSS$⁠, play a very important role in the thermodynamic characterization of solvated systems and can be used to calculate various thermodynamic quantities such as partial molar volumes, compressibility, and activity derivatives [see Eqs. (4)–(10)]. We discuss these properties for the SS of the extended solute in Sec. III C.

To understand the distinct ways in which urea and methanol are structured locally in the solute SS, we computed the excess/deficit of these molecules using the corresponding KBIs, $Δnij=ρjGijSS$ (see Sec. II), and looked at these quantities in the SS of the solute relative to those computed for shells in bulk mixtures without the solute, i.e., $ΔΔnij=Δnij−Δnij•$⁠. As evident in Fig. 4, the presence of the solute led to a relative deficit of urea around urea and methanol around methanol molecules, while both urea and methanol were better hydrated in the SS of the solute. In all these cases, the concentration of urea/methanol had a non-monotonic effect on ΔΔn_ij, where the overall trend in each case remained comparable for all the α values.

The excess of urea in the solute SS led to significant excluded volume repulsion resulting in large negative ΔΔn_uu, while higher concentrations of water kept the urea molecules well hydrated, resulting in large positive ΔΔn_uw at α = 1. With increasing α, at a fixed urea concentration, the amount of urea in the SS decreased due to reduced solute–urea attraction. Thus, a lower number of urea molecules competed to maximize their interaction with both the solute and water, and the urea–water mixture in the solute SS was very similar to the bulk water–urea mixture as indicated by both ΔΔn_uu and ΔΔn_uw approaching zero.

On the contrary, methanol accumulation in the solute SS increased with increasing α (see Fig. S2) and ΔΔn_mm became increasingly negative due to the excluded volume repulsion. Interestingly, both ΔΔn_ww and ΔΔn_mw became increasingly positive with α. One possible explanation of increasing hydration behavior of methanol with α could be the following: as the solute–water and solute–methanol repulsive interactions were scaled, methanol molecules, via a surfactant-like mechanism, preferentially accumulated around the solute and reduced the number of unfavorable water–solute contacts. Such a screening effect increased the hydration of methanol, while simultaneously enhancing the segregation of water molecules. Such segregation is well known for bulk water–methanol mixtures under methanol rich conditions, which is similar to the case in the SS of the solute.^30,57,58

In Figs. 5 and 6, we compare the density maps of urea, water, and methanol in the solute SS for the two extreme α values in each case, with α = 4 as the reference. We calculated these maps by binning the positions of urea/methanol/water in the solvation shells, with and without the solute, within a height L = 1 nm, and performed an average over a 10 ns long trajectory. In Figs. 5 and 6, the bottom left corner represents the center of mass of the solute and only a quarter of the SS (top view) is shown. We show these density maps for the highest urea concentration and the lowest methanol concentration and compare them to the corresponding density maps in bulk mixtures without the solute. In shells without the solute, water and urea/methanol were always homogeneously distributed (the right column in Figs. 5 and 6). The presence of the solute led to preferential accumulation of species in the SS, thereby rearranging their spatial distribution. The preferential adsorption of urea increased from α = 4 to α = 1 with an increase in solute–urea attraction (Fig. S2). This was accompanied by accumulation of water near the solute surface owing to attractive solute–water interaction. Urea and water remained relatively well mixed in the solute SS at both α values. On the contrary, preferential adsorption of methanol increased with increasing α (see Fig. S2) and due to a decrease in water–solute attraction, water simultaneously depleted out of the SS. With α increasing from 4 to 15, as more methanol molecules got adsorbed on the solute interface, they pushed the water away and the two species were inhomogeneously distributed within the SS of the solute.

Given the stark contrast between the structure of the SS of the solute in urea–water and methanol–water mixtures, it was worthwhile to investigate how this distinct structuring affects the thermodynamic properties of the SS. The activity derivatives of urea and methanol in the solute SS and shells in the corresponding pure mixtures without the solute are shown in Fig. 7 along with the resulting excess quantities Δγ = γ − γ^•. The values of γ_uu as a function of concentration for shells without the solute (open symbols) were comparable to that for bulk urea–water mixtures observed experimentally^23,59 (see the black line in Fig. 7) and also in previous simulations,^27,55 indicating almost ideal behavior (γ_uu = 1) until close to 5M concentration. The stronger urea–water affinity in the solute SS, as opposed to shells without the solute, led to γ_uu values much larger than 1. With increasing α, the urea–water affinity decreased (see Figs. 3 and 4), which led to a decrease in γ_uu in the solute SS. The resulting excess quantity, Δγ_uu, was, therefore, positive for all α values analyzed. The concentration dependence, while being non-monotonous, was rather weak. Owing to the insufficient sampling for such low urea concentrations in narrow SS, the overall concentration dependence could be considered to be weakly linear.

For the case of methanol, γ_mm for both SS and shells in the pure methanol–water mixture showed a monotonous increase with concentration and was always $>$1, indicating strong methanol–water affinity. This affinity, however, was stronger in the solute SS and increased with increasing α (see Fig. 4). The resulting Δγ_mm was always positive and clearly followed a monotonous increase with methanol mole fraction. As discussed previously, with increasing repulsion (α), more and more methanol accumulated in the SS and competed to interact with the solute. The resulting excluded volume repulsion and subsequent inhomogeneous distribution of methanol and water within the SS (Fig. 6) resulted in stronger methanol–water affinity (Fig. 4), leading to an increase in γ_mm.

In Fig. 8, we show the preferential interaction parameter $Δij=GiiSS+GjjSS−2GijSS$ for the two mixtures, where {i, j} = {w, u} or {w, m}. This quantity indicates the combined self (solvent–solvent and cosolute–cosolute) affinities of the solvent and cosolute relative to their mutual affinity. As evident, Δ_uw was lower (and negative) in the presence of the solute as compared to shells without the solute, indicating that urea–water affinity was stronger in the solute SS. This was in line with the observations from Figs. 4 and 5 that urea remained well mixed with water and was better hydrated in the solute SS. With increasing α, as urea accumulation decreased, Δ_uw of the solute SS approached that of shells without the solute. In contrast, Δ_mw for the solute SS was much higher than that in shells without the solute for methanol concentrations less than 15M, indicating weaker methanol–water mutual affinity. With increasing α, as methanol accumulation in the SS increased, Δ_mw increased further indicating enhanced segregation of methanol and water within the SS. This again is in line with the observations from Fig. 6 that increasing methanol accumulation near the solute interface pushed the water molecules away. These observations were further consolidated by the corresponding partial molar volumes of urea, water, and methanol and are shown in Fig. 9. The increased mutual affinities between urea and water led to a decrease in $ṽu$ for solute SS as compared to shells in the pure urea–water mixture without the solute. This meant that urea was better hydrated in the solute SS, which could be attributed to its favorable hydrogen bonding with water.²⁸ Together, these two observations suggested that urea and water were very well mixed in the solute SS, which should be a result of a good hydrogen bonded network. The smaller molar volume of urea in the solute solvation shell may have implications in pressure induced unfolding of small proteins, where it has been observed that the volume change upon unfolding is larger in the urea–water mixture as compared to pure water.⁶⁰ Here, the data for $ṽu$ was noisy due to the limited sampling in narrow shells at such low concentrations. In contrast, $ṽm$ was much larger in the presence of the solute and increased with increasing α, which could be attributed to the inhomogeneous distribution of methanol and water within the SS. The water molecules remained segregated resulting in lower $ṽw$ in the solute SS. Interestingly, the values of partial molar volumes in the SS of the solute (except $ṽu$⁠) and shells in pure water–urea/methanol mixtures were comparable to that for bulk water–urea/methanol mixtures as observed in experiments and simulations (see black lines in Fig. 9).^23,61

In our earlier study on the extended solute in pure water, we observed that the tetrahedral order parameter in the HS of the solute was lower in comparison to that for shells in pure water, indicating that the solute disrupted the water hydrogen bonding network around it, i.e., Δq_tet < 0.²⁶ Furthermore, the tetrahedral order parameter monotonically decreased as a function of α indicating that the hydrogen bonding network was progressively distorted around extended solutes with larger SASA. In the same study, we quantified the density fluctuations (in terms of the inverse thermodynamic correction factor 1/Γ) in the HS, which was directly related to the isothermal compressibility. We observed that the excess thermodynamic correction factor and, thus, the excess compressibility (relative to shells in pure water without the solute) exhibited a negative to positive transition at intermediate α values (between 8 and 12),²⁶ indicating the HS to be more “stable” as compared to shells in pure water at lower α and less “stable” at higher α. Herein, the stability of the HS indicated lower density fluctuations and compressibility. The emerging picture from the study indicated that the broken water hydrogen bonds near the extended solute lead to enhanced fluctuations in the HS and resulted in large HS compressibility.

These lines of thought could be extended to the present study, where we investigated the SS thermodynamics in mixed solvents. The isothermal compressibility is now governed by the KBIs [Eq. (11)], which are related to the number density fluctuations in the SS [Eq. (10)]. Therefore, the compressibility of the SS could be thought of as an indirect measure of the density fluctuations within the SS. Here, the mutual affinities (Fig. 8) and partial molar volumes (Fig. 9), together with the density distribution of the various species (Figs. 5 and 6), indicated well mixed urea and water in the solute SS in one case, while inhomogeneously distributed methanol and water in the other. This could be attributed to the distinct hydration behavior of urea and methanol and their hydrogen bonding capability in aqueous solutions.^28,30 To understand their effect on the density fluctuations, we computed the compressibility of the SS. In Fig. 10, we present the compressibility of the solute SS and of shells in pure mixtures and the resulting excess compressibility, Δκ_T. Δκ_T exhibited a non-monotonic dependence, decrease and then increase, on the urea concentration for α = 1, 2 whereas it monotonically increased with urea concentration for α = 4. Furthermore, for a fixed urea concentration, Δκ_T increased as a function of α. For water–methanol, the excess compressibility increased as a function of both α and methanol concentration. Moreover, it appeared that both urea and methanol showed the same trend in Δκ_T as a function of α (increasing repulsion), even though the trends in all the thermodynamic quantities reported so far had been opposite as a function of α. As urea preferential adsorption decreased with increasing α and that for methanol increased, the mechanisms by which urea and methanol enhanced the SS compressibility with α should be rather distinct.

At low urea concentrations at α = 1, 2, the preferential accumulation of urea led to a decrease in the excess compressibility of the SS, as the adsorbed urea molecules could engage in hydrogen bonding with the water molecules, which in turn compensated for the broken water hydrogen bonds around the extended solute interface. With increasing urea concentration, the excess compressibility decreased, indicating reduced solvent density fluctuations with an increase in the preferential accumulation of urea. This is also in line with an earlier study that showed that urea can quench the enhanced water density fluctuations near extended hydrophobic surfaces.⁶² A further increase in urea concentration led to excess urea in the SS, beyond what was required to compensate the water hydrogen bond network, and led to an increase in Δκ_T. With an increase in α, urea preferential adsorption decreased and, therefore, Δκ_T increased.

In contrast, it could be seen that an increase in the preferential accumulation of methanol enhanced solvent density fluctuations and thereby Δκ_T under all conditions. This occurred as the accumulated methanol pushed away the adsorbed water leading to inhomogeneous distribution of the two phases. Given that liquid methanol itself has a large compressibility than water,^63,64 Δκ_T for the SS became much larger than that for shells in the pure methanol–water mixture indicating enhanced density fluctuations in the SS.

The solvation of hydrophobic solutes exhibits a well-known length scale crossover: small hydrophobic solutes are hydrated without disrupting the hydrogen bonded water network around them, while water density fluctuations are enhanced around large hydrophobic solutes as the water molecules can no longer maintain their hydrogen bonded network around an extended interface.^5,6 In this line, a recent study has shown that the tetrahedral order parameter of water is larger in the solvation shell of monomers in comparison to polymers.⁶⁵ Enhanced fluctuations around extended solutes can drive the system toward the onset of a drying transition and make it sensitive to small perturbations. Beyond idealized hydrophobes, this mechanism also has crucial implications in the functional dynamics of proteins.^2,6,10 In our previous study,²⁶ we observed a similar length scale transition for a model extended hydrophobic solute in pure water when the repulsive interaction between the solute and water was systematically scaled by a parameter α that modeled the solute between the two extremes of an attractive solute with lower SASA and a repulsive solute with higher SASA. With increasing α, we observed a negative to positive change in the excess fluctuations (quantified in terms of the inverse thermodynamic correction factor) and excess compressibility of the solute hydration shell relative to shells in pure water without the solute. Additionally, we observed a concurrent decrease in the tetrahedral order parameter of water in the hydration shell. The comprehensive picture that emerged from these observations suggested that water molecules in the solute hydration shell with broken hydrogen bonds led to enhanced density fluctuation in the hydration shell and, thus, larger compressibility.

Interestingly, it has been shown that enhanced fluctuations near extended hydrophobic surfaces could be quenched by the addition of cosolutes such as urea and guanidinium chloride.⁶² This study demonstrated a de-wetting transition when two extended hydrophobic plates were brought closer together in pure water, while urea was found to stabilize the interfacial water. In this line, our study aims at investigating how two adsorbing cosolvents, urea and methanol, affect the thermodynamics of the solvation shell of an extended hydrophobic solute and understand the mechanisms behind their distinct actions.

We observed that both urea and methanol exhibited significant excluded volume repulsion and enhanced hydration in the SS of the solute as compared to shells in pure mixtures without the solute. The relative affinities between these participating species got either enhanced or diminished as a function of increasing repulsion, based on the nature of preferential accumulation of urea and methanol in the SS. While urea and water are known to form ideal mixtures at low urea concentrations²⁷ owing to favorable hydrogen bonding,²⁸ their mutual affinity was further enhanced in the presence of the solute, indicating optimal hydrogen bonding in the solute SS. On the contrary, the low affinity for hydrogen bonding between methanol and water leads to nano-scale segregation.^29,30 The mutual affinity between methanol and water was further reduced in the solute SS, indicating enhanced segregation between the two. Interestingly, in both these cases, strengthening the solute–water/cosolute repulsion led to an increase in the compressibility. Note that the increase in compressibility as a function of solute–urea (solute–methanol) repulsion was accompanied by a decrease (an increase) in the preferential accumulation of urea (methanol). While urea is capable of decreasing water density fluctuation near the extended solute surfaces via engaging in hydrogen bonding with SS water,²⁸ strengthening of the solute–water/urea repulsion led to a decrease in the preferential accumulation of urea and, therefore, enhanced density fluctuation. On the other hand, strengthening of the solute–water/methanol repulsion led to an increase in methanol accumulation and water depletion in the SS. This led to further enhancement in the solvent density fluctuations within the SS, thus increasing the compressibility. Compressibility, herein, is calculated using the KBIs, which are computed using the density fluctuation route. Thus, SS compressibility can be considered as an indirect measure of density fluctuations in the SS.

The observations from this simple model may provide us some useful insights into the effect of cosolute on interfacial solvent density fluctuations, which play an important role in regulating hydrophobic hydration and hydrophobic interactions.^2,6,11,62 It is important to point out that, in real systems such as polymer solutions, the contribution from cosolute modulated solvent density fluctuations will be compensated or reinforced by other contributions such as conformational entropy and solvent excluded volume. It is the interplay of these various contributions that governs the polymer conformational equilibria.⁶⁶ A wide range of studies on different polymeric systems have shown that the preferential accumulation of cosolutes can drive polymer swelling and collapse.^11,18,25,67 Recently, experimental and simulation studies have indicated that the accumulation of cosolutes such as Trimethylamine N-oxide (TMAO) and thiocyanate ions results in a length scale crossover behavior, where they deplete from small solutes and preferentially adsorb onto larger ones.^65,68 This preferential accumulation of the cosolute onto extended solutes has been shown to be related to the enhanced solvent density fluctuations resulting from the loss in tetrahedral order of water near these solutes.⁶⁵ Therefore, understanding the effect of cosolutes on interfacial solvent density fluctuations is crucial toward unraveling their mode of action.

In this line, our observations herein indicate that at low bulk urea concentrations, urea accumulation in the SS, with favorable urea–water hydrogen bonding,²⁸ makes the solute SS less compressible as compared to shells in pure urea–water mixtures. This quenching of density fluctuations by urea may be an important factor in the urea induced swelling of PDEA in water–urea mixtures.¹¹ Note that this effect of urea may also play a similar role in PNIPAM–water–urea mixtures, but is overcompensated by contributions from solvent excluded volume interactions, which lead to polymer collapse.¹¹ 

It appears that the hydrogen bonding capability of the cosolute plays a major role in its ability to quench interfacial solvent density fluctuation. Unlike urea, methanol has a lower tendency to hydrogen bond with water and together with large solute–water repulsion, this leads to a segregation of the two species within the SS with methanol saturating the solute interface.^28,30 This leads to further enhancement of interfacial density fluctuation as indicated by an increase in the SS compressibility. This might be a generic characteristic of the cosolute that exhibits a surfactant like mechanism, such as alcohols and acetone,^24,25 and might have implication in alcohol induced polymer collapse. However, as mentioned earlier, polymer conformational entropy and excluded volume effects will also play crucial roles in this collapse transition. Studies, such as ours, which are aimed at isolating each of these factors and investigating the effect of cosolutes can improve our overall understanding of the complex phenomena of cosolute induced polymer conformational transition.

See the supplementary material for more details on the system composition, solute–solvent/cosolute interaction potentials, preferential adsorption coefficients, and scaling behavior of Kirkwood–Buff integrals.

This research study was funded by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center Transregio Grant No. TRR146 Multiscale Simulation Methods for Soft Matter Systems. Computations for this work were performed on the Lichtenberg High Performance Computer of Technische Universität Darmstadt, Germany.

The authors have no conflicts to disclose.

M.T. and S.B. contributed equally to this work.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.