Modelling of auxetic networked polymers built from calix[4]arene building blocks

Auxetic materials (i.e. materials with a negative Poisson's ratio) expand laterally when stretched and become thinner when compressed. This unusual yet very useful property arises from the way by which the nano or microstructure of the material deforms when subjected to uniaxial mechanical loads. This paper discusses a novel class of molecular-level auxetic (networked polymers) built from calix[4]arene building blocks. These calix[4]arene subunits are connected in such a way that they mimic the shape of a “folded macrostructure” which is known to exhibit auxetic behaviour. We confirm through force-field based simulations that these newly proposed networked polymers exhibit negative Poisson's ratios, the magnitudes of which can be changed by introducing slight variations in the molecular structure of these polymers. We also develop simple geometry-based models which explain the values of the Poisson's ratios obtained through the force-field based simulations, and which give an insight into the features of the molecular structure that are responsible for the auxetic effect.     

Modelling of the mechanical and mass transport properties of auxetic molecular sieves: an idealised inorganic (zeolitic) host–guest system

Force field based simulations have been employed to model the structure, and mechanical and mass transport properties of the all-silica zeolite MFI (ZSM5—Si₉₆O₁₉₂). Undeformed and deformed MFI subject to uniaxial loading in each of the three principal directions were investigated. The mechanical properties are predicted to include negative on-axis Poisson's ratios (auxetic behaviour) in the x ₁–x ₃ plane of the undeformed structure, and are strain-dependent. Transformation from positive-to-negative Poisson's ratio behaviour, and vice versa, is predicted for most on-axis Poisson's ratios at critical loading strains. Simulations of the simultaneous sorption of neopentane and benzene guest molecules onto the undeformed host MFI framework indicate a low neopentane-to-benzene loading ratio, consistent with experimental observation. The sorption of these two molecular species onto deformed MFI is Poisson's ratio- and strain-dependent. Uniaxial tensile loading along a direction containing a negative on-axis Poisson's ratio leads to an increase in the loading of the larger neopentane molecules with respect to benzene, strongly correlated with the increase in volume associated with auxetic behaviour.     

Poisson's ratio of bovine meniscus determined combining unconfined and confined compression

Poisson's ratio has not been experimentally measured earlier for meniscus in compression. It is however an important intrinsic material property needed in biomechanical analysis and computational models. In this study, equilibrium Poisson's ratio of bovine meniscus (n = 6) was determined experimentally by combining stress-relaxation measurements in unconfined and confined compression geometries. The average Young's modulus, aggregate modulus and Poisson's ratio were 0.182 ± 0.086 MPa, 0.252 ± 0.089 MPa and 0.316 ± 0.040, respectively. These moduli are consistent with previously determined values, but the Poisson’s ratio is higher than determined earlier for meniscus in compression through biomechanical modelling analysis. This new experimentally determined Poisson’s ratio value could be used in the analysis of biomechanical data as well as in computational finite element analysis when the Poisson’s ratio is needed as an input for the analysis.     

The role of particle softness in determining the value of Poisson's ratio for soft sphere solids

The influence of particle softness on the Poisson's ratio of model solids has been investigated. We have used the repulsive inverse power potential (～r ^? n for particle separations, r) between the particles, which is conveniently characterised by one adjustable parameter, ? = 1/n. For large ?, the interaction is soft whereas in the ? → 0 limit the particles approach hard spheres. The pressure and elastic constants of the solid phase have been calculated at various densities with constant temperature molecular dynamics (MD) simulation for a range of the softness parameter in the range, n>12. Density-softness surfaces of these quantities were determined which revealed hitherto unrecorded trends in the behaviour of the elastic moduli and Poisson's ratio. It was found that the pressure and some elastic properties, e.g. the C₁₂ elastic constant and the bulk modulus, manifest a maximum value or ‘ridge’ on this surface. The height of the maximum increases with density and interaction steepness (small ?). The Poisson's ratio varies essentially linearly with softness and is relatively insensitive to density. However, at higher densities and for larger steepness a considerable lowering of the Poisson's ratio is observed. In order to identify possible mechanisms for reducing the value of Poisson's ratio, ν, the fluctuation and Born-Green contributions were analysed. Changes in the Poisson's ratio are mainly determined by the fluctuation contribution which can cause a considerable decrease as well as increase of its value.     

Micromechanical models of helical superstructures in ligament and tendon fibers predict large Poisson's ratios

Experimental measurements of the Poisson's ratio in tendon and ligament tissue greatly exceed the isotropic limit of 0.5. This is indicative of volume loss during tensile loading. The microstructural origin of the large Poisson's ratios is unknown. It was hypothesized that a helical organization of fibrils within a fiber would result in a large Poisson's ratio in ligaments and tendons, and that this helical organization would be compatible with the crimped nature of these tissues, thus modeling their classic nonlinear stress–strain behavior. Micromechanical finite element models were constructed to represent crimped fibers with a super-helical organization, composed of fibrils embedded within a matrix material. A homogenization procedure was performed to determine both the effective Poisson's ratio and the Poisson function. The results showed that helical fibril organization within a crimped fiber was capable of simultaneously predicting large Poisson's ratios and the nonlinear stress–strain behavior seen experimentally. Parametric studies revealed that the predicted Poisson's ratio was strongly dependent on the helical pitch, crimp angle and the material coefficients. The results indicated that, for physiologically relevant parameters, the models were capable of predicting the large Poisson's ratios seen experimentally. It was concluded that helical organization within a crimped fiber can produce both the characteristic nonlinear stress–strain behavior and large Poisson's ratios, while fiber crimp alone could only account for the nonlinear stress–strain behavior.     

Characterization of molecular structures and properties of polyurethanes using molecular dynamics simulations

Thermotropic polyurethanes with mesogenic groups in side chains were prepared from two diisocyanates and four diols with stoichiometric ratios of reactive isocyanate (NCO) and hydroxy (OH) groups. Their thermal behavior was determined by differential scanning calorimetry. The effect of structure modifications of the diisocyanates and diols, in particular changes in the mesogen, were investigated. Introduction of mesogenic segments into the polymers suppresses the ordering. Stiff end substituents (phenyl and alkoxy groups) of the mesogens stabilize the mesophases to such an extent that the negative influence of long polymer chains is compensated and the liquid-crystalline properties are recovered. All-atom molecular dynamics simulations in the Cerius² modeling environment were carried out to characterize the structures of the polymers. Analysis of the dynamic trajectories at 20, 100, 120 and 170 °C revealed changes in conformation of macromolecules, which correlate with DSC measurements.Figure Example of structure relaxation of D4/TDI molecule at indicated simulation times (temperature 20 °C): a complete structure; b backbone structure; c top view of molecule     

Multi-Platform Compatible Software for Analysis of Polymer Bending Mechanics

Cytoskeletal polymers play a fundamental role in the responses of cells to both external and internal stresses. Quantitative knowledge of the mechanical properties of those polymers is essential for developing predictive models of cell mechanics and mechano-sensing. Linear cytoskeletal polymers, such as actin filaments and microtubules, can grow to cellular length scales at which they behave as semiflexible polymers that undergo thermally-driven shape deformations. Bending deformations are often modeled using the wormlike chain model. A quantitative metric of a polymer''s resistance to bending is the persistence length, the fundamental parameter of that model. A polymer''s bending persistence length is extracted from its shape as visualized using various imaging techniques. However, the analysis methodologies required for determining the persistence length are often not readily within reach of most biological researchers or educators. Motivated by that limitation, we developed user-friendly, multi-platform compatible software to determine the bending persistence length from images of surface-adsorbed or freely fluctuating polymers. Three different types of analysis are available (cosine correlation, end-to-end and bending-mode analyses), allowing for rigorous cross-checking of analysis results. The software is freely available and we provide sample data of adsorbed and fluctuating filaments and expected analysis results for educational and tutorial purposes.     

A unit-cell-based three-dimensional molecular mechanics analysis for buckling load,effective elasticity and Poisson's ratio determination of the nanosheets

By using a three-dimensional (3D) space-frame-like model, a molecular mechanics (MM) approach is proposed for determination of the buckling loads, effective Young's modulus and Poisson's ratio of the nanosheets, using a proper unit cell. The governing equations are derived based on the 3D kinematics of deformations and the principle of minimum total potential energy. The unit-cell-based results are employed for the space-frame-like finite element model of the nanosheet. The nonlinear MM equations are solved by representing bonds of the boron nitride nanosheet (BNNS) by beam elements to extract the local characteristics. These properties are employed in modelling of the nanosheet, as a space-frame-like finite element structure. The force field constants are chosen according to the Morse, AMBER, UFF and DREIDING models to determine the buckling strength, and effective Poisson's ratio and in-plane rigidity of the whole graphene and BNNSs. Silicon Carbide nanosheets are analysed based on different force constants. These results are concordant with the results available in the literature. The comparisons reveal that the DREIDING force field usually gives the most accurate predictions.     

GPU-Accelerated Molecular Dynamics Simulation to Study Liquid Crystal Phase Transition Using Coarse-Grained Gay-Berne Anisotropic Potential

Gay-Berne (GB) potential is regarded as an accurate model in the simulation of anisotropic particles, especially for liquid crystal (LC) mesogens. However, its computational complexity leads to an extremely time-consuming process for large systems. Here, we developed a GPU-accelerated molecular dynamics (MD) simulation with coarse-grained GB potential implemented in GALAMOST package to investigate the LC phase transitions for mesogens in small molecules, main-chain or side-chain polymers. For identical mesogens in three different molecules, on cooling from fully isotropic melts, the small molecules form a single-domain smectic-B phase, while the main-chain LC polymers prefer a single-domain nematic phase as a result of connective restraints in neighboring mesogens. The phase transition of side-chain LC polymers undergoes a two-step process: nucleation of nematic islands and formation of multi-domain nematic texture. The particular behavior originates in the fact that the rotational orientation of the mesogenes is hindered by the polymer backbones. Both the global distribution and the local orientation of mesogens are critical for the phase transition of anisotropic particles. Furthermore, compared with the MD simulation in LAMMPS, our GPU-accelerated code is about 4 times faster than the GPU version of LAMMPS and at least 200 times faster than the CPU version of LAMMPS. This study clearly shows that GPU-accelerated MD simulation with GB potential in GALAMOST can efficiently handle systems with anisotropic particles and interactions, and accurately explore phase differences originated from molecular structures.     

The three-for-one model for Gram-negative wall growth: a problem and a possible solution

The murein wall in Gram-negative bacteria is so thin that the mechanism of growth is necessarily complicated. From analytical data of murein components, Höltje suggested a model for the growth mechanism that would lead to safe wall enlargement. The model depended on the formation of trimers of peptidoglycan disaccharides linked via their pentapeptides. In the `three-for-one' model three oligopeptidoglycan chains are linked to each other in the usual linkages between the carboxyl group of d-alanine residues and the ϵ-amino group of diaminopimelic acid residues; these are designated `tail-to-tail' linkages. This three-chained raft is then linked to the stress-bearing wall via the formation of trimers, defined as three peptide chains linked together by tail-to-tail linkages. Then by autolyzing the oldest bonds in each trimer, the old chain is excised and the raft becomes part of the stress-bearing wall and the wall is enlarged. There is a problem with the three-for-one model in that it demands a precise fitting of the prefabricated raft of three crosslinked chains to a stress-bearing chain in the wall fabric to allow the series of trimer linkages to form. Because the wall, when bearing stress, must be pulled into a `honeycomb' structure, the end-to-end distance would be shortened. The possibility is raised here that the glycan chains in the stress-bearing wall are stretched to a sufficient degree by the cell's turgor pressure to compensate for its zig-zag structure; this could allow the model to function. A calculation is presented that assumes that the area of the pores in the fabric, called tessera, is maximized by the cell's turgor pressure. In this case the glycan chain must stretch 10% (and the end-to-end distance of peptide strands stretch 28%) so that the end-to-end distance of a glycan chain in the stress-bearing wall and the unstretched nascent wall can be the same and permit indefinite stable growth.     

New resource for the computation of cartilage biphasic material properties with the interpolant response surface method

Cartilage material properties are important for understanding joint function and diseases, but can be challenging to obtain. Three biphasic material properties (aggregate modulus, Poisson's ratio and permeability) can be determined using an analytical or finite element model combined with optimisation to find the material properties values that best reproduce an experimental creep curve. The purpose of this study was to develop an easy-to-use resource to determine biphasic cartilage material properties. A Cartilage Interpolant Response Surface was generated from interpolation of finite element simulations of creep indentation tests. Creep indentation tests were performed on five sites across a tibial plateau. A least-squares residual search of the Cartilage Interpolant Response Surface resulted in a best-fit curve for each experimental condition with corresponding material properties. These sites provided a representative range of aggregate moduli (0.48–1.58 MPa), Poisson's ratio (0.00–0.05) and permeability (1.7 × 10^? 15–5.4 × 10^? 15 m⁴/N s) values found in human cartilage. The resource is freely available from https://simtk.org/home/va-squish.     

Modelling of the mechanical and mass transport properties of auxetic molecular sieves: an idealised organic (polymeric honeycomb) host–guest system

Force field-based simulations have been employed to model the mechanical properties of a range of undeformed molecular polymeric honeycombs having conventional and re-entrant hexagon pores. The conventional and re-entrant hexagon honeycombs are predicted to display positive and negative in-plane Poisson's ratios, respectively, confirming previous simulations. The structure, and mechanical and mass transport properties of a layered re-entrant honeycomb ((2,8)-reflexyne) were studied in detail for a uniaxial load applied along the x ₂ direction. The mechanical properties are predicted to be stress- (strain-) dependent and the trends can be interpreted using analytical expressions from honeycomb theory. Transformation from negative to positive Poisson's ratio behaviour is predicted at an applied stress of σ₂ = 2 GPa. Simulations of the loading of C₆₀ and C₇₀ guest molecules into the deformed layered (2,8)-reflexyne host framework demonstrate the potential for tunable size selectivity within the host framework. The entrapment and release of guest molecules is attributed to changes in the size and shape of the pores in this host–guest system.     

A Monte Carlo Simulation of Nematic and Discotic Ordering in a Polymeric Liquid Crystal

A cyclic polymeric liquid crystal system is simulated using the Metropolis Monte Carlo method in the NVT ensemble. The polymeric system consists of mesogenic moieties attached by alkyl chain spacers to siloxane ring polymers. In the model, the mesogenic moieties are represented individually by an anisotropic Lennard-Jones potential and the polymer ring is represented solely as a constraint on the relative motions of the attached mesogens. A transition from calamitic ordering to discotic ordering is observed as the ring-mesogen bond is varied from full flexible to rigid.     

Dilute liquid crystals used to enhance residual dipolar couplings may alter conformational equilibrium in oligosaccharides

The solution structures of a trisaccharide and a pentasaccharide containing the Lewis(x) motif were determined by two independent approaches using either dipolar cross-relaxation (NOE) or residual dipolar coupling (RDC) data. For the latter, one-bond 13C[bond](1)H RDC enhanced by two different mineral liquid crystals were used alone. Home-written programs were employed firstly for measuring accurately the coupling constants in the direct dimension of non-decoupled HSQC experiments, secondly for transforming each RDC data set into geometrical restraints. In this second program, the complete molecular structure was expressed in a unique frame where the alignment tensor is diagonal. Assuming that the pyranose rings are rigid, their relative orientation is defined by optimizing the glycosidic torsion angles. For the trisaccharide, a good agreement was observed between the results of both approaches (NOE and RDC). In contrast, for the pentasaccharide, strong discrepancies appeared, which seem to result from interactions between the pentasaccharide and the mesogens, affecting conformational equilibrium. This observation is of importance, as it reveals that using simultaneously NOE and RDC can be hazardous as the former represent 99% of the molecules free in solution, whereas the latter correspond to less than 1% of the structure bound to the mesogen.     

Nanomechanical properties of mineralised collagen microfibrils based on finite elements method: biomechanical role of cross-links

Hierarchical structures in bio-composites such as bone tissue have many scales or levels and synergic interactions between the different levels. They also have a highly complex architecture in order to fulfil their biological and mechanical functions. In this study, a new three-dimensional (3D) model based on the finite elements (FEs) method was used to model the relationship between the hierarchical structure and the properties of the constituents at the sub-structure scale (mineralised collagen microfibrils) and to investigate their apparent nanomechanical properties. The results of the proposed FE simulations show that the elastic properties of microfibrils depend on different factors such as the number of cross-links, the mechanical properties and the volume fraction of phases. The results obtained under compression loading at a small deformation < 2% show that the microfibrils have a Young's modulus (E_f) ranging from 0.4 to 1.16 GPa and a Poisson's ratio ranging from 0.26 to 0.3. These results are in excellent agreement with experimental data (X-ray, AFM and MEMS) and molecular simulations.     

Hydraulic permeability and compressive properties of porcine and human synovium

The synovium is a multilayer connective tissue separating the intra-articular spaces of the diarthrodial joint from the extra-synovial vascular and lymphatic supply. Synovium regulates drug transport into and out of the joint, yet its material properties remain poorly characterized. Here, we measured the compressive properties (aggregate modulus, Young's modulus, and Poisson's ratio) and hydraulic permeability of synovium with a combined experimental-computational approach. A compressive aggregate modulus and Young's modulus for the solid phase of synovium were quantified from linear regression of the equilibrium confined and unconfined compressive stress upon strain, respectively (H_A = 4.3 ± 2.0 kPa, E_s = 2.1 ± 0.75, porcine; H_A = 3.1 ± 2.0 kPa, E_s = 2.8 ± 1.7, human). Poisson's ratio was estimated to be 0.39 and 0.40 for porcine and human tissue, respectively, from moduli values in a Monte Carlo simulation. To calculate hydraulic permeability, a biphasic finite element model's predictions were numerically matched to experimental data for the time-varying ramp and hold phase of a single increment of applied strain (k = 7.4 ± 4.1 × 10^?15 m⁴/N.s, porcine; k = 7.4 ± 4.3 × 10^?15 m⁴/N.s, human). We can use these newly measured properties to predict fluid flow gradients across the tissue in response to previously reported intra-articular pressures. These values for material constants are to our knowledge the first available measurements in synovium that are necessary to better understand drug transport in both healthy and pathological joints.     

Mechanical properties of the porcine growth plate and its three zones from unconfined compression tests

The aim of the study was to determine intrinsic mechanical properties of the complete growth plate and its reserve, proliferative and hypertrophic zones. Growth plate disk samples from newborn swine's ulnae were tested using stress relaxation tests under unconfined compression. The Transversely Isotropic Biphasic Model (TIBPE) derived by [Cohen, B., Lai, W. M., Mow, V. C., 1998. A transversely isotropic biphasic model for unconfined compression of growth plate and chondroepiphysis. Journal of Biomechanical Engineering, 120, pp. 491–496] was used to extract intrinsic mechanical properties using a four-parameter optimization procedure. Significant differences were found for the transverse permeability k₁, the Poisson's ratio in the transverse plane ν₂₁, the out-of-plane Poisson's ratio ν₃₁ and the out-of-plane Young's modulus E₃ between the reserve zone and the proliferative zone as well as between the reserve zone and the hypertrophic zone. The same trends were obtained for the Young's modulus in the transverse plane E₁, but significant differences were also found between the reserve zone and the complete growth plate. The proliferative and hypertrophic zones are half as stiff as the reserve zone along the compression axis and about three times less stiff than the reserve zone in the transverse plane. These two zones are also three times as permeable as the reserve zone in the radial direction. The mechanical behavior of the newborn porcine distal ulna growth plate is non-uniform along its thickness. The reserve zone, with its greater zonal component at that development stage, has noteworthy effects on the complete growth plate intrinsic mechanical properties. This study provides, for the very first time, an investigation of the intrinsic mechanical properties of the reserve, proliferative and hypertrophic zones of the growth plate.     

Mean-square amplitudes of the longitudinal vibrational of helical polymers

A simple model for the longitudinal acoustical vibrations of helical polymers has been developed. This model consists of a linear string of masses, M, connected by springs. The mass must be taken as that of one helical turn of the true helix. It has been found that all of the previously observed or calculated frequencies for the longitudinal acoustical modes of helical polymers can be calculated from this model using only one adjustable parameter, the force constant f, between the turns, for each type of helix. Using this simplified model it has been possible to obtain the rms amplitudes of the changes in the overall end-to-end length of helical polymers as a function of their chain length and temperature. At room temperatures the rms end-to-end length is found to vary from a few tenths of an angstrom for hydrocarbon chains of the length found in phospholipid membrane bilayers to several angstroms in a rigid α-helix of 100 peptide residues. The damping of these α-helical modes is not considered but may be appreciable in solution.     

Molecular dynamics simulation of melting and crystallization processes of polyethylene clusters confined in armchair single-walled carbon nanotubes

The confined interaction is important to understand the melting and crystallization of polymers within single-wall carbon tube (SWNT). However, it is difficult for us to observe this interaction. In the current work, the structures and behaviors of melting and crystallization for polyethylene (PE) clusters confined in armchair single-walled carbon nanotubes ((n,n)-SWNTs) are investigated and examined based on molecular dynamics (MD) simulations. The nonbonded energies, structures, Lindemman indices, radial density distributions, and diffusion coefficients are used to demonstrate the features of melting phase transition for PE clusters confined in (n,n)-SWNTs. The chain end-to-end distance (R _n) and chain end-to-end distribution are used to examine the flexibility of the PE chain confined in SWNT. The global orientational order parameter (P₂) is employed to reveal the order degree of whole PE polymer. The effect of polymerization degree on melting temperature and the influence of SWNT chirality on structure of PE cluster are examined and discussed. Results demonstrate that within the confined environment of SWNT, PE clusters adopt novel co-axial crystalline layer structure, in which parallel chains of each layer are approximately vertical to tube axis. The disordered-ordered transformation of PE chains in each layer is an important structural feature for crystallization of confined PE clusters. SWNTs have a considerable effect on the structures and stabilities of the confined PE clusters.