## Introduction

Geometry plays an important role in soft biological systems, from function to mechanics. In particular, spongy, space-filling geometry is readily observed in soft matter systems [1–8], with a variety of physical ramifications. The idea that solvents and surface interactions can dictate the geometry of a deformable interface is worth exploring. Complex geometries, such as ordered and disordered space-filling structures, are ubiquitous in biological and physical systems. figure 1 shows the Σ^{+} periodic arrangement of helical filaments in a high symmetry entanglement [9, 10]. We examine here the solvation free energy calculation of a solvent in this weaving of filaments using the morphometric approach [11–14], varying the solute–solvent interaction by how hydrophobic or hydrophilic we make the surface. What happens when we solvate such a complicated geometry? How do the solvent properties change? And how does the solute change geometry to minimize free energy of the system? [14–16].

In this paper, we examine the solvation free energy balance for the helical Σ^{+} rod packing, whose unit cell with length *L* is shown in fig. 1. This is a well-known arrangement in structural chemistry [9], as well as in soft matter physics [10, 17, 18]. In the present study, this is considered a test case to demonstrate how solvation can dictate geometry of complicated structures. We study deformations of the filaments restricted to helical trajectories, scanning from straight components through to a large helical radius *R*, and from small to large unit cell volume (edge length *L*). The helical tubes make exactly two helical turns from corner to corner of the unit cell, and we require that the helical tubes never intersect. We alter the variables *L* and *R* until the tubes just touch. We study the solvation free energy within a single unit cell as a representative of the material as a whole.

## Morphometric approach and density functional theory

In general, the calculation of the solvation free energy (*F*_{sol}) of a complex sponge-shaped solute is challenging and can be computationally demanding. The so-called morphometric approach [9–12] is a convenient shortcut to the computation of *F*_{sol} for solutes [12, 14], because it allows one to transfer the experience gained with simple solutes to complicated ones. The simplicity of the computation enables us to compute *F*_{sol} for very complex geometry, such as the spongy weaving we consider here. *F*_{sol} (per unit cell) can be computed as a function of only four purely geometric quantities, pairing thermodynamic prefactors with geometric measures of the solute. The prefactors are independent of the specific geometry and are determined by the solvent–solvent and solvent–solute interactions. It is precisely this property that allows us to determine the thermodynamic coeffcients for a simple geometry with high symmetry and employ them for complex solutes. The morphometric form of *F*_{sol} is given by

where the geometrical measures are the volume *V*, the surface area *A*, the integrated (over the surface) mean (*C*), and Gaussian (*X*) curvatures of the filaments per unit cell. It is convenient to calculate these geometrical measures for the solvent accessible surface, i.e., the surface that is accessible for the centers of solvent particles. The corresponding thermodynamic coeffcients in eq. 1 are the solvent pressure *p*, the planar wall surface tension *γ*, and the bending rigidities *κ* and [12]. The void space of the geometry is filled with a solvent and gives rise to the grand potential of the total system per unit cell of Ω = −*pV*_{cell} + *F*_{sol}, with volume *V*_{cell} = *L*^{3}.

We vary the interaction between the helical tubes and the solvent from hydrophobic to hydrophilic, to see how this affects the minimal energy configuration. The thermodynamic coeffcients (for a given solute–solvent interaction strength as measured by the contact angle Θ) *p, γ, κ*, and which are independent of the solute geometry, can be computed in a simple geometry via classical density functional theory (DFT) [19]. To this end, we calculate the solvation free energy of a spherical particle with radius *R*_{sp}, as shown in fig. 2, for which the geometrical measures are particularly simple. We calculate the solvation free energy for the sphere in two steps. first, we fix the position of the solute sphere (big red particle in fig. 2) and thereby make it into an external potential for the solvent. In this external field we minimize a DFT for a water-like square-well fluid in order to obtain the inhomogeneous equilibrium density distribution *ρ*(*r*) of the solvent (*r* is the radial distance from the center of the solute sphere). In a second step, we calculate the total grand potential of the system Ω, where we make use of the fact that in equilibrium the density functional reduces to the grand potential Ω = Ω[*ρ*(*r*)] [19]. From Ω we can extract the solvation free energy *F*_{sol}(*R*_{sp}) of the sphere and *f*_{surf} (*R*_{sp}), the excess surface free energy per area, via

For a sphere (and our definition of *X*) we obtain *C/A* = 1/*R*_{sp} and so that the thermodynamic coefficients *γ, κ*, and can be extracted from a quadratic fit in 1/*R*_{sp} to numerical DFT calculations for various values of the radius *R*_{sp}. The solvent used in our calculation is a square-well fluid that models the properties of water at ambient conditions [20]. Due to the high symmetry of the solute, the calculation of the density distribution *ρ*(*r*) of the solvent around a spherical solute is simple and can be done with a high numerical accuracy. The coefficients are obtained by fitting numerical DFT data of *f*_{surf} (*R*_{sp}) for various values of *R*_{sp} by eq. 2.

The thermodynamic coeffcients for a range of contact angle values Θ, which are determined by the solute–solvent interaction strength, are given in Table 1. If Θ is smaller than 90° the solute is hydrophilic.

Thermodynamic coeffcients as determined by classical DFT for a spherical solute with different values of the radius *r*_{w}.

Contact angle Θ | βκr_{w} | ||
---|---|---|---|

49.6° | –0.0991 | –0.835 | –17.5 |

63.7° | –0.0687 | –0.703 | –15.9 |

76.2° | –0.0391 | –0.572 | –14.4 |

82.1° | –0.0247 | –0.506 | –13.6 |

87.9° | –0.0105 | –0.441 | –12.8 |

93.6° | 0.00340 | –0.376 | –12.1 |

99.3° | 0.0170 | –0.311 | –11.3 |

104.9° | 0.0303 | –0.246 | –10.6 |

110.6° | 0.0433 | –0.181 | –9.9 |

116.3° | 0.0559 | –0.117 | –9.2 |

## Computation of geometric properties

We now discuss the computation of the geometric properties of the complicated sponge-like geometry. Each unit cell of the structure contains four distinct helical filaments of length and helical radius *R*. We fix the thickness of the filaments, *t*, to be 3.7 nm in radius, taken from the radius of keratin intermediate filaments in application to the structure of corneocytes in the skin [10].

The excluded volume encompassed by the solvent accessible surface of the fibers (*t* + *r*_{w}, the radius of water) is the length of the central spine of the helix times the cross-sectional area of the filament

We can define the surface of each helical cylinder by the parametric equations, with each pitch and *t*_{w} = *t* + *r*_{w} by the parametric equations

With *X* = {*x*(*u, v), y*(*u, v), z*(*u, v*)}, we can compute the surface area as follows:

Where the coefficients of the first and second fundamental forms are as follows:

the mean curvature at each point is given by

and the integrated mean curvature is calculated by

The helical filaments are forbidden from overlapping, but where the filaments are very close to each other, the excluded volume bounded by the solvent accessible surface does overlap. In this case, each of these quantities must be adjusted for the intersections. Here, the intersection was computed numerically using the program Karambola [21, 22].

The integrated Gaussian curvature is an integer invariant of the body and is prescribed by its topology. This can be simply computed by hand.

## Results

We combine the geometric data over a large range of helical pitch and unit cell edge lengths with each set of prefactors for different contact angles given in Table 1. We examine the grand potential of the system as a function of geometry. figure 3 shows the grand canonical potential landscape for three very hydrophilic examples (a–c) and three very hydrophobic examples (d–f). The qualitative behavior is opposite for the opposing regimes: with hydrophilic filaments, Ω minimizes with increasing *R* and *L*, however with hydrophobic filaments, Ω minimizes for the most compact state, essentially squeezing out as much water as possible until the system jams.

There is an additional driving force when hydrophobic fibers are sufficiently close together, where a liquid–gas interface can form [23]. This is a small-scale analogue of the so-called capillary evaporation phase transition. A full treatment of the interface formation is beyond the scope of this study and would require one to compute the thermodynamic coefficients for the meta-stable gas phase and a minimization of the shape of the liquid–gas interface. Neglecting the effects of curvature, which are of importance to the present problem, one can estimate the distance between two fibers *d*(Θ < 90°), for which capillary evaporation becomes important [23]:

where *γ*_{lg} is the liquid–gas interfacial tension and Δ*p* = *p*_{g} – *p*_{l} is the pressure difference between the meta-stable gas (g) and the stable liquid (l) phases, respectively.

An image of the compacted state of the structure, with minimal solvent in the channels, is shown in fig. 4. The compacted state of this particular geometric arrangement is interesting: the ordered geometry dictates that the system will jam at a packing fraction of approximately 0.44, which is still very porous. Interest in the structural chemistry community has been in relation to this very low packing fraction [9].

In addition to the behavior for sternly hydrophobic or hydrophilic filaments, we wish to ascertain how Ω minimizes as we change from slightly hydrophilic to slightly hydrophobic. Plots of Ω as a function of geometry are shown in fig. 5 for contact angles ranging from 82°to 99°. We see that the change from expansion to contraction is interesting: we start for slightly hydrophilic filaments by favoring swelling with a large helical radius, and then change to swelling with straight filaments almost in contact, and finally once we pass to hydrophobic filaments, the structure compacts. This transformation from expansion to contraction is gradual, and changes somewhere slightly less than a contact angle of 99°. This is where the water pressure term slightly favors expansion, but this is balanced by the slightly hydrophilic surface of the filaments favoring contraction.

## Conclusions

The computation of configurations that minimizes Ω suggests that changing the surface properties of the filaments can have a drastic effect. For hydrophilic filaments, the structure will absorb the solvent, and for hydrophilic filaments, the structure will squeeze out the solvent until it reaches a jamming point.

These calculations give some insight into the solvation properties of ordered sponge-like materials (most relevant to biomaterials), however, the approach is more general. In addition to finding the minimal Ω conformation of the structure, this method might provide a way to probe the mechanics of filamentous and wet materials, where external forces can lead to rearrangements and sliding inside the structure, which should respond nonlinearly to external mechanical forces. This would be a nice way to probe the mechanics of highly tangled structures, such as tangled polymer melts.

In combination with the elastic free energy of the system, these methods have been applied to the study of swelling of the dead cells in the outer layer of mammalian skin. The structure detailed in this article is precisely the arrangement of keratin intermediate filaments in these dead cells, called corneocytes [17]. The minimization of the solvation free energy, which expands the structure as the filaments are hydrophilic, is balanced by the elastic energy stored in the elastic filaments themselves. This dictates precisely the swelling behavior of the corneocytes, most readily observed as wrinkling skin after too long in the bath [24].

Exploring solvation in ordered complex geometry could give significant insight into self-assembly processes in nature, most obviously in mesoscale biological systems. Very complicated ordered patterns, sometimes with local helical geometry, readily self-assemble in a variety of systems [25, 26]. Solvation and the geometry of a solute in the solvent might play a role in this complex and poorly understood phenomenon. These processes might be considered via systematic studies of the solvation of enumerated ordered periodic patterns, such as those given in [16, 18, 27, 28].

## Footnotes

A collection of invited papers based on presentations at the 33^{rd} International Conference on Solution Chemistry (ICSC-33), Kyoto, Japan, 7–12 July 2013.

Myfanwy Evans thanks the Humboldt foundation and the DFG Forschergruppe “Geometry and Physics of Spatial Random Systems” for support.

## References

- [1]
K. Larsson, K. Fontell, N. Krog. Chem. Phys. Lipids 27, 321 (1980).

- [2]
S. T. Hyde, S. Andersson, B. Ericsson, K. Larsson. Z. Kristallogr. 168, 2139 (1984).

- [3]
K. Larsson. J. Phys. Chem. 93, 7304 (1989).

- [4]
S. T. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B. W. Ninham. The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology, Elsevier Science B.V. (1997).

- [5]
V. Luzzati. Curr. Opin. Struct. Biol. 7, 661 (1997).

- [6]
Z. A. Almsherqi, S. D. Kohlwein, Y. Deng. J. Cell Biol. 173, 839 (2006).

- [7]
K. Michielsen, D. G. Stavenga. J. R. Soc. Interface 5, 85 (2008).

- [8]
T. Landh. FEBS Lett. 369, 13 (1995).

- [9]
K. R. Mecke. J. Phys.: Condens. Matter 8, 9663 (1996).

- [10]
K. Mecke, C. H. Arns. J. Phys.: Condens. Matter 17, S503 (2005).

- [11]
P.-M. König, R. Roth, K. Mecke. Phys. Rev. Lett. 93, 160601 (2004).

- [12]
H. Hansen-Goos, R. Roth, K. Mecke, S. Dietrich. Phys. Rev. Lett. 99, 128101 (2007).

- [13]
Y. Snir, R. D. Kamien. Science 307, 1067 (2005).

- [14]
R. Roth, Y. Harano, M. Kinoshita. Phys. Rev. Lett. 97, 078101 (2006).

- [15]
M. O’Keeffe, J. Plevert, Y. Teshima, Y. Watanabe, T. Ogama. Acta Crystallogr., A 57, 110 (2001).

- [16]
M. E. Evans, V. Robins, S. Hyde. Acta Crystallogr., A 69, 262 (2013).

- [17]
M. E. Evans, S. T. Hyde. J. R. Soc. Interface 8, 1274 (2011).

- [18]
M. E. Evans, V. Robins, S. Hyde. Acta Crystallogr., A 69, 241 (2013).

- [19]
R. Evans. Adv. Phys. 28, 143 (1979).

- [20]
R. Roth, D. Gillespie, W. Nonner, R. E. Eisenberg. Biophys. J. 94, 4282 (2008).

- [21]
G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reicheldorfer, W. Peukert, W. Schwieger, K. Mecke. Adv. Mater. 23, 2535 (2011).

- [22]
G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, K. Mecke. New J. Phys. 15, 083028 (2013).

- [23]
R. Evans, U. M. B. Marconi. J. Chem. Phys. 84, 2376 (1985).

- [24]
M. E. Evans, R. Roth. Phys. Rev. Lett. 112, 038102 (2014).

- [25]
J. J. K. Kirkensgaard, M. E. Evans, L. de Campo, S. T. Hyde. Proc. Natl. Acad. Sci. USA (2013). Accepted for publication. Doi: 10.1073/pnas.1316348111.

- [26]
G. E. Schröder-Turk, S. Wickham, H. Averdunk, F. Brink, J. fitz-Gerald, L. Poladian, M. C. Large, S. T. Hyde. J. Struct. Biol. 174, 290 (2011).

- [27]
S. J. Ramsden, V. Robins, S. T. Hyde. Acta Crystallogr. 39, 365 (2009).

- [28]
T. Castle, M. E. Evans, S. T. Hyde, S. J. Ramsden, V. Robins. Interface Focus 2, 555 (2012).