No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Solvation of a sponge-like geometry
Pure and Applied Chemistry
Abstract
Periodic entanglements of filaments and networks, which resemble sponge-like materials, are often found as self-assembled materials. The interaction between the geometry of the assembly and a solvent in its interstices can dictate the geometric configuration of the structure as well as influence macroscopic properties such as swelling and mechanics. In this paper, we show the calculation of the solvation free energy as a function of the solute–solvent interaction from hydrophilic to hydrophobic, for a candidate entanglement of filaments. We do this using the morphometric approach to solvation free energy, a method that disentangles geometric properties from thermodynamic coefficients, which we compute via density functional theory.
... The theoretical study of fluids confined in porous media is still carried out essentially in a case-by-case way. The morphological thermodynamics proposed and advocated by K. Mecke, R. Roth and co-workers [1][2][3][4][5][6][7][8] provides a framework for a general thermodynamic description of complex interfacial systems. Starting from Hadwiger's theorem in integral geometry, morphological thermodynamics postulates that four geometrical measures are enough to characterize the thermodynamic potential of a complex interfacial system, i.e., volume, surface area, integrated mean curvature and integrated Gauss curvature [1]. ...
... Moreover, Mecke et al assume that the thermodynamic variables conjugated to these geometrical measures, i.e. pressure, surface tension, as well as two surface bending rigidities, can be determined for a simple system then be used to describe systems with more complex morphology [1][2][3]. Although morphological thermodynamics gives promising results for some systems [1][2][3][4][5][6][7][8], the validity of one of its fundamental postulates has been questioned. Theoretical and simulation investigations have provided evidence for the existence of non-Hadwiger terms, i.e., high-order curvature contributions to surface tension, which are not taken into account in the morphological thermodynamics [9][10][11][12]. ...
We propose a general approach based on morphological thermodynamics for determining adsorption isotherms, i.e., the chemical potential of a confined fluid as a function of its density. The validity of this approach and its versatility are established by its remarkable accuracy compared to Monte-Carlo simulation results and its capability of accounting for a quite large variety of porous media, ranging from a simple slit pore to a random sponge matrix. It is also revealed that the contribution of the curvature terms to the chemical potential of the confined fluid is negligibly small when the interface curvature is not too large. This finding is of a particular importance for simplifying the treatment of experimental results of adsorption isotherms since no experimental technique is currently available for determining the curvatures of the pore surface inside a porous material.
The stratum corneum, the outer layer of mammalian skin, provides a remarkable barrier to the external environment, yet it has highly variable permeability properties where it actively mediates between inside and out. On prolonged exposure to water, swelling of the corneocytes (skin cells composed of keratin intermediate filaments) is the key process by which the stratum corneum controls permeability and mechanics. As for many biological systems with intricate function, the mesoscale geometry is optimized to provide functionality from basic physical principles. Here we show that a key mechanism of corneocyte swelling is the interplay of mesoscale geometry and thermodynamics: given helical tubes with woven geometry equivalent to the keratin intermediate filament arrangement, the balance of solvation free energy and elasticity induces swelling of the system, importantly with complete reversibility. Our result remarkably replicates macroscopic experimental data of native through to fully hydrated corneocytes. This finding not only highlights the importance of patterns and morphology in nature but also gives valuable insight into the functionality of skin.
Significance
Chirality and hierarchical ordering are two fundamental properties found in many of nature’s most complex self-assembled structures such as living cells. Simultaneous control over these properties in synthetic systems is vital to mimic or even surpass nature’s designs. Via numerical simulations, we describe a class of complex morphologies that afford radically new architectures for self-assembled shapes. Specifically, a mixture of two star block-copolymers are shown to form multiple interwoven 2D and 3D labyrinths—all chiral—and hierarchically ordered on two different length scales. Furthermore, we show that such intricate network morphologies forming at a confined, hyperbolic interface can be classified and modeled in terms of a much simpler isotropic model of packing based on tilings of the hyperbolic plane.
Ordered arrays of cylinders, known as rod packings, are now widely used in descriptions of crystalline structures. These are generalized to include crystallographic packed arrays of filaments with circular cross sections, including curvilinear cylinders whose central axes are generic helices. A suite of the simplest such general rod packings is constructed by projecting line patterns in the hyperbolic plane () onto cubic genus-3 triply periodic minimal surfaces in Euclidean space (): the primitive, diamond and gyroid surfaces. The simplest designs correspond to ‘classical’ rod packings containing conventional cylindrical filaments. More complex packings contain three-dimensional arrays of mutually entangled filaments that can be infinitely extended or finite loops forming three-dimensional weavings. The concept of a canonical ‘ideal’ embedding of these weavings is introduced, generalized from that of knot embeddings and found algorithmically by tightening the weaving to minimize the filament length to volume ratio. The tightening algorithm builds on the SONO algorithm for finding ideal conformations of knots. Three distinct classes of weavings are described.
High-symmetry free tilings of the two-dimensional hyperbolic plane () can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.
We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S (2)), Euclidean (E (2)) and hyperbolic (H (2)) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.
Recent theoretical work on the microscopic structure and surface tension of the liquid-vapour interface of simple (argon-like) fluids is critically reviewed. In particular, the form of pairwise intermolecular correlations in the liquid surface and the capillary wave treatment of the interface are examined in some detail. It is argued that conventional capillary wave theory, which leads to divergences in the width of the density profile, is unsatisfactory for describing all the equilibrium aspects of the interface. The density functional formalism which has been developed to study the liquid-vapour interface can also be profitably applied to other problems in the statistical mechanics of non-uniform fluids; here a new generalization of the `linear' theory of spinodal decomposition is formulated and by considering a `nearly uniform' fluid, some useful results for the long-wavelength behaviour of the liquid structure factor of various monatomic liquids are obtained. Some other topics of current interest in this area are briefly discussed.
A novel technique to generate three-dimensional Euclidean weavings, composed of close-packed, periodic arrays of one-dimensional fibres, is described. Some of these weavings are shown to dilate by simple shape changes of the constituent fibres (such as fibre straightening). The free volume within a chiral cubic example of a dilatant weaving, the ideal conformation of the G(129) weaving related to the Σ(+) rod packing, expands more than fivefold on filament straightening. This remarkable three-dimensional weaving, therefore, allows an unprecedented variation of packing density without loss of structural rigidity and is an attractive design target for materials. We propose that the G(129) weaving (ideal Σ(+) weaving) is formed by keratin fibres in the outermost layer of mammalian skin, probably templated by a folded membrane.
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences.
Six packings of symmetry-related cylinders, with cylinder axes in invariant positions (coordinates completely determined by symmetry), are described. Two have axes along 〈100〉 and four have axes along 〈111〉. It is shown that there can be no cubic cylinder packing with axes along 〈110〉. Earlier errors concerning the numbers of such packings and their symmetries are corrected.
Cubic membranes represent highly curved, three-dimensional nanoperiodic structures that correspond to mathematically well defined triply periodic minimal surfaces. Although they have been observed in numerous cell types and under different conditions, particularly in stressed, diseased, or virally infected cells, knowledge about the formation and function of nonlamellar, cubic structures in biological systems is scarce, and research so far is restricted to the descriptive level. We show that the "organized smooth endoplasmic reticulum" (OSER; Snapp, E.L., R.S. Hegde, M. Francolini, F. Lombardo, S. Colombo, E. Pedrazzini, N. Borgese, and J. Lippincott-Schwartz. 2003. J. Cell Biol. 163:257-269), which is formed in response to elevated levels of specific membrane-resident proteins, is actually the two-dimensional representation of two subtypes of cubic membrane morphology. Controlled OSER induction may thus provide, for the first time, a valuable tool to study cubic membrane formation and function at the molecular level.
We present a systematic study of the cuticular structure in the butterfly wing scales of some papilionids (Parides sesostris and Teinopalpus imperialis) and lycaenids (Callophrys rubi, Cyanophrys remus, Mitoura gryneus and Callophrys dumetorum). Using published scanning and transmission electron microscopy (TEM) images, analytical modelling and computer-generated TEM micrographs, we find that the three-dimensional cuticular structures can be modelled by gyroid structures with various filling fractions and lattice parameters. We give a brief discussion of the formation of cubic gyroid membranes from the smooth endoplasmic reticulum in the scale's cell, which dry and harden to leave the cuticular structure behind when the cell dies. The scales of C. rubi are a potentially attractive biotemplate for producing three-dimensional optical photonic crystals since for these scales the cuticle-filling fraction is nearly optimal for obtaining the largest photonic band gap in a gyroid structure.
We suggest that bubbles are the bistable hydrophobic gates responsible for the on-off transitions of single channel currents. In this view, many types of channels gate by the same physical mechanism-dewetting by capillary evaporation-but different types of channels use different sensors to modulate hydrophobic properties of the channel wall and thereby trigger and control bubbles and gating. Spontaneous emptying of channels has been seen in many simulations. Because of the physics involved, such phase transitions are inherently sensitive, unstable threshold phenomena that are difficult to simulate reproducibly and thus convincingly. We present a thermodynamic analysis of a bubble gate using morphometric density functional theory of classical (not quantum) mechanics. Thermodynamic analysis of phase transitions is generally more reproducible and less sensitive to details than simulations. Anesthetic actions of inert gases-and their interactions with hydrostatic pressure (e.g., nitrogen narcosis)-can be easily understood by actions on bubbles. A general theory of gas anesthesia may involve bubbles in channels. Only experiments can show whether, or when, or which channels actually use bubbles as hydrophobic gates: direct observation of bubbles in channels is needed. Existing experiments show thin gas layers on hydrophobic surfaces in water and suggest that bubbles nearly exist in bulk water.
The transitions lamellar → cubic → hexagonal in the aqueous system of sunflower oil monoglycerides are analysed. X-Ray diffraction data show linear relationships between the lattices of the three phases, which are discussed on the basis of structures formed by lipid bilayer units. The cubic structure is related to ‘Schwarz's primitive cubic minimal surface’ and consists of a three-dimensional continuous bilayer system separating two separate water channel systems.It is also pointed out that the three-dimensional membrane system in plant plastids, the prolamellar body, which is involved in the formation of thylakoid membranes of chloroplasts, has a structure which is closely related to or identical with that of the cubic phase of monoglyceride-water systems.
It is known that the monoolein-water system exhibits one cubic phase with a body-centered lattice at low water content and at higher water content another cubic phase, with a primitive lattice. Evidence is given for a structure of the body-centered phase with the lipid bilayer forming an infinite periodic minimal surface of the gyroid type. The transition from this structure to the primitive cubic structure with a lipid bilayer forming the diamond type of infinite periodic minimal surface, as proposed by Longley and Mclntosh, can take place without change in curvature of the lipid bilayer. The transformation between these two minimal surfaces is described. The lattice parameters of the two cubic phases, the transition interval and an extremely low transition enthalpy are features consistent with such a transformation, which occur without change in curvature.
Recent studies of cubic lipid-water phases have shown that there are three basic structure alternatives, each consisting of an infinite periodically curved lipid bilayer separating two continuous water regions. The geometric factors that determine these structures are discussed. The biological relevance of cubic phases is related to the possibility of two-dimensional analogues of these structures, which are consistent with earlier observations of so-called nonbilayer conformations in membranes. Evidence is given for phase transitions in membranes involving such intrinsically curved bilayers.
A new model is proposed for the study of porous media and complex fluids using morphological measures to describe homogeneous spatial domains of the constituents. Under rather natural assumptions a general expression for the Hamiltonian can be given extending the model of Widom and Rowlinson for penetrable spheres. The Hamiltonian includes energy contributions related to the volume, surface area, mean curvature, and the Euler characteristic of the configuration generated by overlapping sets of arbitrary shapes. Phase diagrams of the model are calculated and discussed. In particular, we find that the Euler characteristic in the Hamiltonian stabilizes a highly connected bicontinuous structure, resembling the middle phase in oil - water microemulsions.
A set of four morphological measures, so-called Minkowski functionals, was defined which allows to quantitatively characterize the shape of spatial structures, to optimally reconstruct porous media, and to accurately predict material properties. The method is based on integral geometry and Kac's theorem which relates the spectrum of the Laplace operator to the four Minkoski functionals. The integral geometric measures, i.e., Minkowski functionals of the spatial structure proved to be structural quantities which are important for many physical properties of heterogeneous materials. It was shown that completely porous media exhibit very similar conductivities and elastic moduli as long as the structures have the same Minkowski functionals.
By means of a density functional approach the phase equilibria of a simple fluid confined by two adsorbing walls have been investigated as a function of wall separation H and chemical potential μ for temperature T corresponding to both partial and complete wetting situations. For large values of H and small undersaturations Δμ ≡ μsat−μ, we recover the macroscopic formulas for the undersaturation at which a first- order phase transition (capillary condensation) from dilute ‘‘gas’’ to a dense ‘‘liquid’’ occurs in a single, infinitely long slit. For smaller H we compute the lines of coexistence between gas and liquid in the (Δμ, 1/H) plane at fixed values of T. The adsorption Γ(Δμ), at fixed T and H, is characterized by a loop. At the first order transition Γ jumps discontinuously by a finite amount; however metastable states exist and these could give rise to hysteresis of the adsorption isotherms obtained for the single slit. The loop disappears at a capillary critical point (Δμc, 1/Hc) at each T. For HΔμc, condensation can no longer occur and no metastable states are present. The location of the critical points is described and for a complete wetting situation we find that these lie outside the bulk two phase region. Our theory provides a simple explanation of phase transitions observed in earlier computer simulations and mean-field lattice gas calculations for confined fluids and suggests that measurements of the forces between plates, either by simulation or in real fluids, should provide rather direct information about capillary condensation and, possibly, capillary critical points. The relevance of our results for adsorption experiments on mesoporous solids is discussed briefly.
Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.
The identification of evolutionary conserved membrane morphologies whose architecture is governed by cubic symmetry--cubic membranes--adds a new dimension to cell membrane functions and, perspicuously, to their role in subcellular space organization. Through analysis of electron micrographs, three families of cubic membranes have been unequivocally identified in which one or more (parallel) membranes, described by periodic cubic surfaces, partition space into two or more independent, albeit convoluted, subspaces of membrane potential determined dimensions. The choice of a particular cubic symmetry is suggested to be due to its activity. Here the architecture and function of multiple (> or = 3) subspace organization in classical membrane bound organelles is addressed. As it can be precisely determined with cubic membranes suggests that they can be employed as a reference morphology.
We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are satisfied, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term extensive and have as a consequence that the surface tension and other thermodynamic quantities contain, aside from a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.
The helix is a ubiquitous motif for biopolymers. We propose a heuristic, entropically based model that predicts helix formation
in a system of hard spheres and semiflexible tubes. We find that the entropy of the spheres is maximized when short stretches
of the tube form a helix with a geometry close to that found in natural helices. Our model could be directly tested with wormlike
micelles as the tubes, and the effect could be used to self-assemble supramolecular helices.
We show that the solvation free energy of a complex molecule such as a protein can be calculated using only four geometrical measures of the molecular structure and corresponding thermodynamical coefficients. We compare results from this morphometric approach to those obtained by an elaborate statistical-mechanical theory in liquid state physics for a large variety of different structures of protein G and find excellent agreement. Since the computational time is drastically reduced, the new approach provides a practical and efficient way for calculating the solvation free energy which can be employed when this quantity has to be calculated for a large number of structures, as in a simulation study of protein folding.
We calculate the solvation free energy of proteins in the tube model of Banavar and Maritan [Rev. Mod. Phys. 75, 23 (2003)10.1103/RevModPhys.75.23] using morphological thermodynamics which is based on Hadwiger's theorem of integral geometry. Thereby we extend recent results by Snir and Kamien [Science 307, 1067 (2005)10.1126/science.1106243] to hard-sphere solvents at finite packing fractions and obtain new conclusions. Depending on the solvent properties, parameter regions are identified where the beta sheet, the optimal helix, or neither is favored.
de Accepted for publication
- Kirkensgaard
The Language of Shape The Role of Curvature in Condensed Matter Chemistry Biology
- Hyde
The Language of Shape: The Role of Curvature in Condensed Matter: Physics
- S T Hyde
- S Andersson
- K Larsson
- Z Blum
- T Landh
- S Lidin
- B W Ninham