Spontaneously Forming Dendritic Voids in Liquid Water Can Host

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Biophysical Chemistry, Biomolecules, and Biomaterials; Surfactants and Membranes

Spontaneously Forming Dendritic Voids in Liquid Water Can Host Small Polymers Narjes Ansari, Alessandro Laio, and Ali A. Hassanali J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b02052 • Publication Date (Web): 30 Aug 2019 Downloaded from pubs.acs.org on August 30, 2019

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The Journal of Physical Chemistry Letters

Spontaneously Forming Dendritic Voids in Liquid Water Can Host Small Polymers Narjes Ansari,† Alessandro Laio,‡,† and Ali Hassanali∗,† †The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy ‡SISSA, Via Bonomea 265, I-34136 Trieste, Italy E-mail: [email protected]

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Abstract Some liquids are characterized by the presence of large voids with dendritic shapes and are for this reason are dubbed transiently porous. By using a battery of data analysis tools, we demonstrate that liquid water and methane both characterized by transient porosity. We show that the thermodynamics of porosity is distinct from that associated with cavitation a´ la classical nucleation theory. The shapes of dendritic voids in both liquids with very different chemistries, resemble those of small polymers. We further show, using free energy calculations, that the cost of solvating small hydrophobic polymers in water is consistent with the work associated with creating dendritic voids. The entropic and enthalpic contributions associated with hosting these polymers can thus be rationalized by the thermodynamics of fluctuations in bulk water.

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Graphical TOC

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Introduction Developing a molecular understanding of liquids has been an active area of study for over a century. 1–8 In this context, water has been the most studied liquid due to its crucial role in biochemistry. Its anomalous properties compared to other simple liquids are well known 9–12 and determine the solvation of ions as well as the physical chemistry of hydrophobicity, a cornerstone of numerous biophysical processes. 13–21 One aspect that has received particular attention in the literature is the origin and behavior of density fluctuations. 10,22 At the molecular scale, variations in density are tied to the presence of cavities and the correlations between them. 23–25 Most studies focus on spherical cavities 6,26–29 which are extremely important since they are related with the nucleation of the gas phase. In this contribution, we concentrate on cavities which are, instead, highly non-spherical and study their implications on hydrophobic solvation. As we will see, non-spherical voids are rather common not only in water, but also in liquids with a totally different nature. Their presence gives rise to what has been called liquid porosity. Although the concept of porosity is rather well established in the solid-state, the porosity of common liquids is much less appreciated, and porous liquids are considered an oddity. 30,31 We will show here how fertile and important this concept is: underpinning the transient porosity of a liquid is its ability to sustain cavities with varying morphology that can host guest molecules. Here, we employ atomistic simulations to study cavities of two liquids namely, water at ambient conditions and methane at the estimated pressure and temperature conditions of the oceans of Titan. Despite the difference in their chemistries, both these liquids are characterized by large-dendritic voids with highly complex morphologies. Spherical voids are also present in both liquids, but are much smaller than the dendritic voids. In short, both the polar liquid, water and non-polar methane can be considered as transiently porous in the thermodynamic conditions we considered. In order to characterize the voids, we employ the Smooth Overlap of Atomic Positions (SOAP) 32 to construct a descriptor for the 4


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regions of empty space. This provides a natural way to compare different voids within a framework that complies rotational, translational and permutational invariance. With this metric in hand by applying state-of-the-art data analysis techniques 33–35 we characterize the free energy landscape associated with the creation of the voids, that turns out to be high dimensional, and characterized by the presence of several local free energy minima. We demonstrate further, that the shapes of the dendritic voids created by thermal fluctuations in these liquids have a rather uncanny similarity to those that surround short hydrophobic polymers. Using free energy calculations, we confirm that cost of solvating these hydrophobic polymers is consistent with the work required to create a dendritic shaped void in liquid water. The total solvation free energy involves a large positive entropic contribution which confirms the important role of dendritic voids in hydrophobic solvation.

Computational Methods MD Simulations All-atom molecular dynamics simulations (MD) of water and methane were performed using the GROMACS 5.0 package. 36 For both liquids, the time step used for the simulations was 2 fs, equilibration runs were 2 ns and production runs were carried out for 40 ns using the Verlet leap-frog algorithm. 104 configurations were used for analysis, sampled every 4 ps. All simulations with a total number of 4096 molecules were conducted in the NPT ensemble. Water molecules have been modeled using the TIP4P-Ew rigid water model 37 at ambient pressure and temperature of 270 K, because the melting temperature for the TIP4P-Ew water model is Tm ∼ 244 K. The OPLS-AA model was used for liquid methane at 111 K and ambient pressure, corresponding to a density of 0.47 of Titan (0.45

g ). cm3

g , cm3

which is close to that on the surface

A Nose-Hoover thermostat 38,39 and Parrinello-Rahman barostat 40 was

used to maintain constant temperature and pressure with coupling time constants of 2 ps and 1 ps respectively. The Ewald 41 summation was used for the electrostatic interactions, 5



and a cutoff of 10 ˚ A was used for the van-der- Waals interactions. A cubic box was used with an average side length 49.68 ˚ A and 61.14 ˚ A, for water and methane, respectively. In order to investigate the ability of the liquids to accommodate polymer of certain sizes, we did three 40 ns MD simulations of a polymer in a large water box (4000 molecules) using the NVT ensemble. The three polymer molecules are modeled as freely jointed chains of 6 and 10 Lennard-Jones (LJ) particles and one consisting of 10 particles with a small branch. The parameters of the Lennard-Jones potential describing the polymerwater interactions were determined by the Lorentz-Berthelot combining rules using a sigma=3.4 ˚ A and epsilon= 0.46 kJ/mol for polymer atoms. As seen from our free energy calculations, these interaction parameters mimic the hydrophobic character of decane. In addition, we calculate the solvation free energies using thermodynamic integration 42 method as implemented in GROMACS 5.0. To connect the two end states, which correspond to the polymer in vacuum and in solution, we use a path including 20 intermediate states with different values of the alchemical parameter λ, which couples to the polymer-solvent interaction potential. For each state, we run 1 ns of Langevin dynamics with a 2 fs time step at 300 K. In the intermediate steps (λ 6= 1) a soft-core potential is used for all the non-bonded interactions. 43 In order to extract the entropic and enthalpic contributions to the free energy, the thermodynamic integration calculations are performed at a set of temperature values in the range 280 K to 315 K with a step of 5 K.

Voronoi Voids To characterize the geometry of the empty space between molecules in studied liquids, we compute the Voronoi-Delaunay (VD) voids 44 using the VNP code 45 . In order to construct the Voronoi voids, one needs both a probe radius (RB ) and a bottleneck radius (RP ) in order to identify the cavities and merge connected regions or channels in the hydrogen bond network. VNP code is used in this report to generate the VD voids with (RP and (RB equal 6


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to 1.2 ˚ A and 1.1 ˚ A, respectively. The interested reader is referred to the original paper (see Ref. 46) for more details, and Refs. 24,25, where we perform a complete analysis of the effect of different parameters on the morphology of voids in liquid water.

SOAP Distance In order to evaluate the similarity between the voids in each of the liquids or between the voids of different liquids, we need to define an appropriate descriptor. Among the many descriptors of local environments, the Smooth Overlap of Atomic Positions (SOAP) descriptor 32 has recently received a lot attention in being able to describe different chemical environments in complex molecular systems. 47 As the voids are by definition the empty space, in order to use SOAP, we fill the voids (see in SI Figure S6) with ghost particles on grids with a uniform spacing of 0.5 ˚ A between them. Using the ghost particles (j) within the void, we determine a local density of geometric center of the void (i) defined as a sum of Gaussian functions.

ρi (r) =

X

exp

j

− |r − rij |2 2σ 2

! (1)

The SOAP kernel is constructed using the overlap of two local atomic neighbour densities integrated over all rotations. The similarity kernel of two voids is defined then as the inner product of the densities of the ghost particle on the void’s geometrical center within a cutoff rc = 10 ˚ A. The SOAP kernel can be expanded in a basis of spherical harmonics yielding a power spectrum. It can be shown that by building the elements of the power spectrum into a vector of unit length which subsequently defines the SOAP kernel κ. The distance between √ two voids can then be defined as d = 2 − 2κ.

Free energies and clustering of void shapes By using the SOAP distances we first estimated the intrinsic dimension (ID) of the space of the void shapes. We used TWO-NN, 34 which infers the ID d from the probability distribution 7



of the ratio between the distances to the second and first nearest neighbors of each data point. Using this value of d as a parameter, we then computed the free energy of each data point by following the approach in Ref. 35. In this approach the free energy is estimated implicitly as a function of the d variables charting the manifold embedding the data points. Therefore, the free energy is a computed without defining explicitly any collective variable. It can be interpreted as a measure of the logarithm of the probability of observing a specific void shape, without projecting this probability in any specific coordinate. We finally find the independent minima in this free energy landscape (the clusters) by using the approach in Ref. 48. This approach is an unsupervised extension of Density Peak clustering 33 in which the number of clusters is determined automatically, retaining only those which are statistically significant up to a confidence level z, here fixed to 1. The program implementing all this pipeline is available at bluehttps://github.com/alexdepremia/Advanced-Density-Peaks/tree/master.

Results and discussion We considered liquid water at ambient conditions, and methane at a pressure of 1 atm and temperature of 111 K where it is in a liquid state. These two liquids were chosen because they widely differ in their chemical properties, and secondly because they are considered putative solvent environments for pre-biotic chemistry. 49 The molecular volume of water is the smallest. The liquid is characterized by an averagely tetrahedral network of hydrogen bonds. Methane on the other hand, is a non-polar solvent held together by weak van-derWaals interactions. A total of 40 ns of molecular dynamics simulations was performed for both liquid water and methane from which Voronoi voids are extracted and analyzed. After cataloging the voids based on their morphology, the SOAP metric is used to cluster the voids and subsequently examine whether shapes of voids surrounding small hydrophobic polymers bear any similarity to those found in the liquid. To confirm that small polymers can be docked

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into dendritic voids, we also determined the solvation free energy of three polymers using thermodynamic integration. 42 We begin by showing in Figure 1 the distribution of the volume of the voids normalized by the molar volume of each molecule shown on a log-log scale. The numbers on the x-axis essentially say how many molecules of each liquid can fill up the void. In a recent work, 25 we used an asphericity parameter to distinguish between spherical and non-spherical voids. We here build on these result to characterize the properties of these two different voids. The distribution of voids for both liquids has an asymmetric structure characterized by a single peak and a fat tail corresponding to spherical and non-spherical voids respectively (see SI Figure S1). The typical volume of the spherical voids is about ≤25 ˚ A3 while the non-spherical ˚3 . The spherical voids formed in our simulation, as indicated ones range between >25-400 A by the scaling behavior between volume and surface area, is consistent with the previous cavitation studies examining density fluctuations in liquid water. 6,26,27 Surrounding Figure 1 we illustrate some snapshots of the voids that are obtained from the water and methane color coded as: red for water and orange for methane. It is clear that both the liquids feature large dendritic shaped voids. These large voids are a manifestation of transient porosity that occurs on a nano-meter length scale and exist on a sub-picosecond to picosecond timescale. In some of the cases (marked by star), we observe voids that can form closed loop-like channels. The two liquids differ in the extent of the tails corresponding to the large dendritic voids. Relative to its small molecular size, water appears to produce larger voids than methane. The presence of dendritic shaped voids (seen in the surrounding panel of Figure 1) in both a polar and non-polar solvent, indicates that transient porosity may be a rather generic property of many liquids. However, it appears to be finely tuned by specific thermodynamic conditions. For example, an increase in the temperature of water up to the boiling point (400 K) results in an increase in the transient porosity of voids, while an increase in the pressure up to 1 GPa completely eliminates the dendritic shaped voids (see SI Figure S2).

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Figure 1: The distribution of the volume of spherical and non-spherical cavities in liquid water and methane is shown averaged over all voids in the system (solid colored lines red and orange for water and methane respectively) and then constrained to only those that are spherical shaped (dashed colored lines gray and light-orange for water and methane respectively). The volumes of the voids are normalized to the the molecular volumes associated with the respective liquids - 30 ˚ A3 for water and 55.80 ˚ A3 for methane. Surrounding the figure we show examples of the non-spherical dendritic shaped voids and associated volumes in water (red) and methane (orange). The volume of voids are in ˚ A3 unit.

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Although we observe dendritic voids both in water and in methane, the mechanism underlying their formation is very different: methane is an apolar liquid, which can host cavities without paying a significant energy cost. Water on the other hand is a complex liquid, characterized by the presence of a network of hydrogen bonds. Dendridic voids there form due to the fluctuations of a percolating hydrogen bond network. This feature can be seen by examining the rings around the voids. Figure S3 in SI shows the closed ring-statistic around spherical and dentritic voids at three different thermodynamic conditions. The long tail of distribution around the dendritic voids at ambient and high temperature reveals the formation of extended networks of hydrogen bond connections between the water molecules surrounding the voids. In order to understand the fluctuations associated with the voids, we turn next to characterizing the thermodynamic landscape underlying their formation. The complex shapes adopted by the voids shown in Figure 1 indicate that they can grow or nucleate in many different directions. As a reference, a sphere would grow in only one dominant direction. We determine the intrinsic dimension (ID) associated with the nucleation of our voids by TWO-NN, 34 an estimator which allows computing the ID in data sets harvested from high dimensional probability distributions. More details can be found in the Methods section. ˚3 were extracted To determine the ID, 104 voids with the volume range of 90 - 120 A from the simulations after which pairwise SOAP distances were computed and the TWONN methodology applied. The ID of the voids for both water and methane are remarkably similar: 9.11 and 9.32 ± 0.09 for water and methane respectively (see SI Figure S4). Firstly, the large ID values indicate that the number of directions in which the dendritic shapes grow is much larger than that of a sphere. The ID of a perfect sphere is 3. Since the spherical voids are not perfect sphere, their ID is approximately 4.5 (for volumes less than 25 ˚ A3 ). Secondly, the fact that the ID values are very similar for water and methane shows that the shape of the voids is very similar even though the chemistry associated with the interactions is very different.

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The preceding results indicate that the free energy landscape in which the nucleation of the dendritic voids occurs, is intrinsically high dimensional. We reconstruct this landscape using the approach in Ref. 35. Figure 2 shows the free energy surface (FES) for liquid water illustrated as a dendrogram for voids constrained to have volumes between the range of 90 -120 ˚ A3 . The vertical axis of the dendrogram shows the free energy of each basin in the unit of kB T. The horizontal lines and the radius of circles are proportional to the population of each basin. A saddle point between basins is indicated by an horizontal line joining the two vertical lines corresponding to the basins.

Figure 2: The free energy surface associated with the voids in liquid water is shown in the form of a dendrogram. The various circles correspond to cluster basins or free energy minima. Moving from the bottom left to the top right, we go from more spherical looking voids to larger cavities with more branching. The snapshots correspond to the cluster centers and essentially involve voids surrounded by water networks resembling thermally excited clathrate-like structures. The FES shown is characterized by roughly 13 different minima. Towards the left bottom side, the most thermodynamically stable voids correspond to those that are spherically shaped. As one moves higher up in free energy (right top side) at approximately 10-15 kB T, we observe the presence of larger dendritic shaped voids. Furthermore, it is clear that the FES associated with the nucleation of our channel-like voids is very rough. This observation is akin to an effect noticed earlier in our group where we examined the high-dimensionality 12


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of the FES associated with water wires around a model peptide. 50 The thermodynamics associated with the fluctuations of liquid methane is also characterized by a similar FES as seen in the corresponding dendrograms - there are several local minima resulting in a rough landscape (see in SI Figure S5). In the case of water, the basins are analogous to thermally excited clathrate-like structures 51 stabilized by hydrogen bond networks. For liquid methane, the origin of the basins is distinct from water since they essentially originate from the weak van-der-Waals forces needed to separate two methane molecules. Figure 1 shows that in the largest non-spherical voids, one can potentially fit up to 8 water molecules. A visual inspection of these voids, indicates that they are shaped like dendrimers. Although, these voids form more rarely than small spherical voids, it is natural to ask whether they are large enough to accommodate polymers. In this regard, there are two related aspects that we focus on: firstly, the extent of the similarity in the shapes of the voids between the two ensembles (with and without the polymer) and secondly, the thermodynamics associated with docking the hydrophobic polymers in water. In order to assess the applicability of these notions, we performed different simulations of three model polymers in liquid water. We focused on neutral polymers modeled as LennardJones particles: two linear polymer consisting of 6 and 10 atoms respectively, and a branched polymer, made up of 10 atoms. To confirm whether there are voids spontaneously forming in the liquid that have a similar shape to those that surround the polymer, we searched for the closest voids across the two systems (bulk liquid vs liquid+polymer). A total of 1000 configurations for each of the three model polymers were sampled and then the SOAP distance between the voids surrounding the polymer and the closest one found in the pure liquid, was determined. Figure 3 shows the distribution of this SOAP distance for our three simulated systems. The results of Figure 3 are very striking. For all the three simulated polymer systems, there is a remarkable similarity between the shape of the voids encapsulating the polymer

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and those that can be found in the liquid without it. As one might expect, for the shorter chain made of 6 atoms the similarity is smaller compared to that obtained with the larger polymer. One of the three simulated systems, includes a short branch in order to build in some complexity resembling a side chain in a bio-polymer. Again, even a more complex branched-like polymer (middle panel of Figure 3) has a remarkable similarity in shape to spontaneously occurring voids created by density fluctuations in the water.

Figure 3: The SOAP distance distribution for the three LJ polymers described in the main text. The x-axis in all the three panels quantifies the distribution of the most similar void that surrounds the polymer that can be found in the water or methane. In each panel we also show snapshots obtained from the simulations visually depicting the similarity between the voids surrounding the polymer and those found in water. The distributions shown in dashed lines correspond to methane. Since liquid water and methane seem to be very similar in their tendencies to produce large dendritic voids (see Figure 1), we sieved through the empty space in liquid methane searching for a void that looks closest to the ones surrounding the polymer in water. Shown 14


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in dashed-lines in Figure 3 are the corresponding SOAP distance distributions for the three polymers with respect to the liquid methane simulations. For all three polymers, we observe that the shapes of the empty space in methane are also quite similar to those voids associated with the polymer. The preceding analysis does not give any quantitative insights into the relationship between the thermodynamics of solvation and the fluctuations creating the dendritic voids. In order to understand this, we turn next to our free energy calculations using thermodynamic integration (TI). TI simulations were performed for all three polymers in liquid water at room temperature. The total solvation free energy as well as the enthalpic and entropic contributions are shown in Table 1. The free energy of solvation (∆Gsol ) for all three polymers ranges between 15-20 kJ/mol consistent with the fact that the polymers are hydrophobic. As a reference, we also determined the solvation free energy for decane which works out to ∼14 kJ/mol consistent with experimental results. 52–54 Our polymers are thus slightly more hydrophobic compared to decane. As expected, the 6-MEM chain has a lower solvation free energy compared to the 10-MEM ones. In order to understand if the solvation free energies for the polymers are consistent with the intrinsic density fluctuations in liquid water, we determined the work required to create a dendritic void of similar size and shape. For each of the three systems, scanning a total of 5 ×105 voids, we determined the probability of finding a dendritic void in liquid water with a size and shape corresponding to a volume of ∼65 ˚ A3 and ∼140 ˚ A3 for the 6-MEM and 10-MEM respectively. The volume is the one enclosed within a van-der-Waals surface of the polymer. From this, the cavitation free energy ∆Gcav for the void formation also reported in Table 1, ranges between 12-17 kJ/mol. ∆Gcav are in close agreement with ∆Gsol confirming one of the central points of our findings: the free energy cost for solvating a hydrophobic polymer in water practically coincides with the free energy cost of producing the dendritic cavity hosting it. This cost, in liquid water at ambient conditions, happens to be small.

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By performing the TI at a series of temperatures (see SI for details), we also extracted the enthalpic and entropic contributions to the solvation energy. As we can see in Table 1, ∆Gsol is dominated by large positive entropic contributions consistent with the hydrophobic effect. 16,27 Interestingly, enthalpy also plays a role in the solvation of the hydrophobic polymer. Part of the origin of this effect comes from the fact that the void creation results in the formation of slightly more tetrahedral-like water in close vicinity to it. 25 Table 1: Free energy, enthalpy and entropy of solvation for 6-member (6-MEM), 10-member branched- (10-MEM(B)) and chair-like (10-MEM(C)) polymers. ∆Gcav is calculated from dendritic void’s statistics. 6-MEM 10-MEM(B) 10-MEM(C) ∆Gcav (kJ/mol) ≈12 ≈17 ≈ 17 ∆Gsol (kJ/mol) 15.55 ± 0.21 20.33 ± 0.33 19.71 ± 0.17 ∆Hsol (kJ/mol) -21.01 ± 0.81 -30.89 ± 1.23 -31.09 ±1.06 T ∆Ssol (kJ/mol) 36.36 ± 0.81 52.17 ± 1.23 50.52 ± 1.06 Indeed the cavity free energy determined by the void statistics takes into account multiple conformations of the polymer. It is estimated by taking into account all the cavities with the same volume, which are all dendritic in the volume ranges relevant for our estimate. To clarify this point, for the case of the 10-MEM in Table 1, we computed the ∆Gcav by considering only one reference configuration (the bottom most shape in the middle panel of Figure 3 with SOAP distance 0.025). When we do this, we obtain a value of approximately 26 kJ/mol which is 9 kJ/mol more than the value reported in Table 1, which takes into account multiple configurations. There have been several previous studies examining the effect of polymer conformations on the solvation free energy. Pettit and co-workers used similar TI type calculations and showed that the van-der-Waals contribution to the solvation free energy of alanine peptides is very sensitive to the peptide flexibility. 55 In some more recent work, Asthagiri and co-workers have shown that for explicit atom deca-alanine structures, the protein-solvent interactions play an important role in solution thermodynamics. 56 In future work, we intend to invesigate how the the thermodynamics of dendritic void creation couples with strong polymer-water 16


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interactions.

Conclusions The thermodynamics of hydrophobic solvation associated with cavity formation and its implications on polymer solvation has been the subject of several important studies in the literature. 57,58 Chandler, s original theory regarding the 2 length scale cross over for hydrophobic solutes posits that for small molecules like methane, the hydrogen bond network of water is not significantly perturbed. 6 In this case, it is the entropic cost of creating some excluded volume that dominates the solvation energy of a hydrophobic solute. Our analysis is akin to this idea, but now extended to larger molecules - the formation of relatively large dendritic shaped voids does not incur a significant disruption of the hydrogen bond network. Furthermore, the solvation free energy of small polymers is rooted in the entropic dominated penalty of the creation of dendritic voids. We have shown that density fluctuations in liquids appear to be able to create regions of empty space with exotic shapes bearing close similarity to those surrounding small polymers. The surprise is that although small dendritic shaped voids are more rare compared to their spherical counterparts, the situation is reversed for large voids, which are dominantly dendtritic in the thermodynamic conditions we considered. This feature is observed in both water and methane. Rather than being controlled by specific chemical details, the formation of the voids is tied to the effective volume of the molecule. 59 The thermodynamics associated with the formation of large voids does not follow the canonical cavitation associated with classical nucleation theory. Indeed the large voids we observe are all dendritic and the volume scales almost linearly with the surface which is at variance with the spherical voids. The free energy landscape on which dendritic voids grow is high dimensional and highly potholed by local minima. Rather than reflecting the growth of the vapor phase in the liquid, we suggest that the formation of these voids provides a measure of the transient porosity of a liquid. In this regard, both water and methane are 17



equally transiently porous liquids The cavities in transiently porous liquids have previously been considered as putative sites for absorbing only small gas molecules. Here we find that the large dendritic shaped voids can actually host larger molecules such as polymers. By performing molecular dynamics simulations of small polymers, we corroborate this proposal and indeed find that thermal fluctuations in both water and methane can produce voids that have a strikingly similar shape to those that surround small polymers. Using free energy calculations we confirm that the free energy of solvation of three hydrophobic polymers is consistent with the statistics of finding dendritic voids of similar size and shape generated by fluctuations in bulk water. Consistent with hydrophobic solvation, we elucidate the dominant role of entropic effects in the thermodynamics of creation of dendritic voids. Patel and co-workers have recently been developing some novel methods to study the importance of solvent fluctuations with both static and dynamically evolving complex shapes. 60,61 It would be interesting in the future to couple these ideas to the sampling of larger dendritic voids in different systems.

Supporting Information Available SI consist of 7 Figures (S1-S7).

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