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

fig. 2 A spherical solute with radius *R*_{sp} submerged in a solvent. This geometry is well suited for a calculation of the solvation free energy *F*_{sol}(*R*_{sp}) or the excess surface free energy per area *f*_{surf} (*R*_{sp}), which we employ in order to determine the thermodynamic coefficients *γ, κ*, and

using DFT.

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 . If Θ is smaller than 90° the solute is hydrophilic.

Table 1 Thermodynamic coeffcients as determined by classical DFT for a spherical solute with different values of the radius *r*_{w}.

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