Boltzmann Inversion Research Articles

Mechanical behaviour of cis-1,4-polyisoprene under uniaxial tension with silica and POSS filling: A coarse-grained molecular dynamics simulation

Abstract In this paper, the iterative Boltzmann inversion (IBI) method has been employed to develop the coarse-grained (CG) force field of cis-1,4-polyisoprene (PI) rubber and its silica-filled and POSS-filled composites. Uniaxial tension simulations have been conducted for the pure PI model, the silica-filled PI model, and the POSS-filled PI model, as well as their particle grafting and matrix cross-linking models. By examining the evolution of molecular chains, filled particles, and the entire matrix in terms of mechanical behaviour, it was found that particle-induced molecular chain straightening is the primary factor contributing to the elevated macroscopic stress level observed in the particle-filled PI model. However, the smooth silica particles do not result in an increase in the stress level of the system due to the weaker interactions. The grafting and matrix cross-linking processes serve to enhance this induced process, with the former increasing particle roughness and the latter increasing molecular chain roughness. In terms of the entire condensed system, the macroscopic stress level of the system is strongly correlated with the inhomogeneity of the microscopic strain. The inhomogeneity of the strain distribution in the system demonstrates the existence of a region of strong strain, which results in the loosening of the structure and a decrease in the load-carrying capacity. The nucleation of microvoids in the system cannot be attributed to microscopic stress concentrations, which are more likely to occur in the weak interaction region. In POSS-filled systems, the weak interaction region is located in the unfilled region, which explains the formation of microvoids in the matrix. In contrast, in silica-filled systems, the weak interaction region becomes particle-filled, which results in microvoids forming at the particle-filled region.

Elasticity of Swollen and Folded Polyacrylamide Hydrogel Using the MARTINI Coarse-Grained Model.

One of the key advantages of using a hydrogel is its superb control over elasticity obtained through variations of constituent polymer and water. The underlying molecular nature of a hydrogel is a fundamental origin of hydrogel mechanics. In this article, we report a Polyacrylamide (PAAm)-based hydrogel model using the MARTINI coarse-grained (CG) force field. The MARTINI hydrogel is molecularly developed through Iterative Boltzmann inversion (IBI) using all-atom molecular dynamics (AAMD), and its quality is evaluated through the experimental realization of the target hydrogel. The developed model offers a mechanically high-fidelity CG hydrogel that can access large-scale water-containing hydrogel behavior, which is difficult to explore through AAMD in practical time. With the modeled hydrogel, we reveal that the polymer conformation modulates the elasticity of the hydrogel from a folded state to a swollen state, confirmed by the Panyukov model. The results provide a robust bridge for linking the polymer conformations and alignment to their bulk deformation, enabling the multifaceted and material-specific predictions required for hydrogel applications.

Representing Structural Isomer Effects in a Coarse-Grain Model of Poly(Ether Ketone Ketone).

Carbon-fiber composites with thermoplastic matrices offer many processing and performance benefits in aerospace applications, but the long relaxation times of polymers make it difficult to predict how the structure of the matrix depends on its chemistry and how it was processed. Coarse-grained models of polymers can enable access to these long-time dynamics, but can have limited applicability outside the systems and state points that they are validated against. Here we develop and validate a minimal coarse-grained model of the aerospace thermoplastic poly(etherketoneketone) (PEKK). We use multistate iterative Boltzmann inversion to learn potentials with transferability across thermodynamic states relevant to PEKK processing. We introduce tabulated EKK angle potentials to represent the ratio of terephthalic (T) and isophthalic (I) acid precursor amounts, and validate against rheological experiments: The glass transition temperature is independent to T/I, but chain relaxation and melting temperature is. In sum we demonstrate a simple, validated model of PEKK that offers 15× performance speedups over united atom representations that enables studying thermoplastic processing-structure-property-performance relationships.

Continuum-Particle Coupling for Polymer Simulations.

We report a concurrent hybrid multiscale simulation method, in which a particle domain is coupled with a surrounding continuum domain. The particle domain consists of a coarse-grained model of poly(lactic acid) and the continuum domain is treated using the finite element method. The coarse-grained model is derived from an atomistic model, using the iterative Boltzmann inversion scheme. The particle- and the finite element-domains overlap in a bridging domain through anchor points. In this coupling, the information passes back and forth between the high- and the low-resolution domains, effectively bridging the gap between the nano and macro-scales. This scheme is employed to simulate the coupled particle-continuum domains under both stochastic and semistochastic boundary conditions. While the anchor points keep the volume of the particle domain fixed in the former case, there is no anchor point in the planes normal to the periodic direction, in the latter case. The stress-strain behavior of polymer under both stochastic and semistochastic boundary conditions is investigated and the results are compared with those calculated from pure finite element reference simulations. Furthermore, the stress-strain relationship for the coupled system under the semistochastic boundary conditions is examined under plane stress and plane strain conditions, and the results are compared with those of pure finite element reference simulations. The hybrid particle-continuum method reproduces the pure finite element simulation results well.

Adaptive resolution force probe simulations: Coarse graining in the ideal gas approximation.

The unfolding of molecular complexes or biomolecules under the influence of external mechanical forces can routinely be simulated with atomistic resolution. To obtain a match of the characteristic time scales with those of experimental force spectroscopy, often coarse graining procedures are employed. Here, building on a previous study, we apply the adaptive resolution scheme (AdResS) to force probe molecular dynamics (FPMD) simulations using two model systems as examples: One system is the previously investigated calix[4]arene dimer that shows reversible one-step unfolding, and the other example is provided by a small peptide, a β-alanine octamer in methanol solvent. The mechanical unfolding of this peptide proceeds via a metastable intermediate and, therefore, represents a first step toward a complex unfolding pathway. We show that the average number of native contacts serves as a robust order parameter for the forced unfolding of this small peptide. In addition to increasing the complexity of the relevant conformational changes, we study the impact of the methodology used for coarse graining. Apart from the iterative Boltzmann inversion method, we apply an ideal gas approximation, and therefore, we replace the solvent by a non-interacting system of spherical particles. In all cases, we find excellent agreement between the results of FPMD simulations performed fully atomistically and those of the AdResS simulations also in the case of fast pulling. This holds for all details of the unfolding pathways, such as the distributions of the characteristic forces and also the sequence of hydrogen-bond opening in case of the β-alanine octamer. Therefore, the methodology is very well suited to simulate the mechanical unfolding of systems of experimental relevance.

The acquisition and calibration of coarse-grained force field parameters for graphene/nitrile rubber nanocomposites

Graphene (GNS) is an ideal reinforcing material for developing high-performance nitrile rubber (NBR) nanocomposites to withstand complex and harsh working conditions. The mesoscale molecular simulation is significant in exploring the microscopic mechanism of nanocomposites. Initially, a coarse-grained (CG) scheme for NBR was established, and the potential parameter files were obtained through the iterative Boltzmann inversion method. Then, the bond and angle parameters were fitted by harmonic potential, the non-bonded interactions were fitted by LJ 12–6 potential. The CG force field parameters were coefficients of the potential function. The parameters such as the stiffness values, potential well parameter, etc., were calibrated by stretching the NBR molecular chains, and comparing the density and stretching performances of the NBR all-atomic and CG models. A rapid method for obtaining the non-bonded parameters between NBR and GNS was proposed based on the radial distribution function curves. The non-bonded parameters were calibrated by comparing GNS pull-out simulations, and the density and tensile properties of the GNS/NBR models. Finally, the CG force field parameters applicable to the GNS/NBR nanocomposites were obtained. The study is expected to be significant for subsequent mesoscale studies on GNS/NBR nanocomposites.

A Temperature-Transferable Coarse-Grained Model for Poly(lactic Acid) Melts.

We present a temperature-transferable coarse-grained (CG) model for poly(lactic acid) (PLA), specifically designed to replicate its volumetric properties and solubility parameter in the molten state. The CG-bonded potentials were derived by using the iterative Boltzmann inversion (IBI) optimization method to match structural properties from detailed atomistic models. A parametrization workflow was employed to determine nonbonded interaction parameters with temperature-dependent corrections that provide agreement with the target properties across the melting temperature range. The CG model successfully replicates key features of the PLA melt. It satisfactory reproduces the density and solubility parameter, maintains the dependence of chain conformation on molecular weight, and captures the dynamic behavior through agreement in scaled mean squared displacement and diffusion coefficients with the atomistic model. Additionally, the CG model offers much faster equilibration compared with the atomistic model. The proposed model is expected to be particularly useful for investigating the miscibility characteristics of PLA in various blends and composites that remain challenging to explore using fully atomistic simulations or experiments.

Martini-Based Coarse-Grained Soil Organic Matter Model Derived from Atomistic Simulations.

The significance of soil organic matter (SOM) in environmental contexts, particularly its role in pollutant adsorption, has prompted an increased utilization of molecular simulations to understand microscopic interactions. This study introduces a coarse-grained SOM model, parametrized within the framework of the versatile Martini 3 force field. Utilizing models generated by the Vienna Soil Organic Matter Modeler 2, which constructs humic substance systems from a fragment database, we employed Swarm-CG to parametrize the fragments and subsequently assembled them into macromolecules. Direct Boltzmann inversion (DBI) facilitated the determination of bonded parameters between fragments. The parametrization yielded favorable agreement in the radius of gyration and solvent-accessible surface area. Transfer free energies exhibited a strong correlation with hexadecane-water and chloroform-water values, albeit deviations were noted for octanol-water values. Comparing densities of modeled Leonardite humic acid systems at coarse-grained and atomistic levels revealed promising agreement, particularly at higher water concentrations. The DBI approach effectively reproduced average values of bonded interactions between fragments. Radial distribution functions between carboxylate groups and calcium ions showed partial agreement, however, reproducing certain peaks was challenging due to fixed bead sizes. Detailed analysis of atomistic systems revealed different configurations between the groups, explaining discrepancies. The present contribution provides a comprehensive insight into the properties, strengths, and weaknesses of the coarse-grained SOM model, serving as a foundation for future investigations encompassing pollutant interactions and varied SOM compositions.

Predicting the artificial dynamical acceleration of binary hydrocarbon mixtures upon coarse-graining with roughness volumes and simple averaging rules.

Coarse-grained (CG) molecular models greatly reduce the computational cost of simulations allowing for longer and larger simulations, but come with an artificially increased acceleration of the dynamics when compared to the parent atomistic (AA) simulation. This impedes their use for the quantitative study of dynamical properties. During coarse-graining, grouping several atoms into one CG bead not only reduces the number of degrees of freedom but also reduces the roughness on the molecular surfaces, leading to the acceleration of dynamics. The RoughMob approach [M. K. Meinel and F. Müller-Plathe, J. Phys. Chem. B 126(20), 3737-3747 (2022)] quantifies this change in geometry and correlates it to the acceleration by making use of four so-called roughness volumes. This method was developed using simple one-bead CG models of a set of hydrocarbon liquids. Potentials for pure components are derived by the structure-based iterative Boltzmann inversion. In this paper, we find that, for binary mixtures of simple hydrocarbons, it is sufficient to use simple averaging rules to calculate the roughness volumes in mixtures from the roughness volumes of pure components and add a correction term quadratic in the concentration without the need to perform any calculation on AA or CG trajectories of the mixtures themselves. The acceleration factors of binary diffusion coefficients and both self-diffusion coefficients show a large dependence on the overall acceleration of the system and can be predicted a priori without the need for any AA simulations within a percentage error margin, which is comparable to routine measurement accuracies. Only if a qualitatively accurate description of the concentration dependence of the binary diffusion coefficient is desired, very few additional simulations of the pure components and the equimolar mixture are required.

Coarse-grained simulation of thermal conductivity of boron nitride/epoxy composites based on DPD and SPH method

In this study, thermal conductivity (TC) of BN/polymer composites is investigated utilizing dissipative particle dynamics (DPD) and smoothed-particle hydrodynamics (SPH) methods by establishing the coarse-grained (CG) models of boron nitride nanotube (BNNT) and boron nitride nanosheet (BNNS). The Iterative Boltzmann inversion (IBI) approach is employed to calculate the parameters of the potential function of the hexagonal boron nitride (h-BN) coarse grain model for DPD method. Mesoscopic simulations on the TC of BNNT/EP, BNNS/EP, and BNNT/BNNS/EP composites are performed using SPH and DPD simulations to investigate the effects of nanofiller ratio, aspect ratio, size, and orientation on TC. The CG models established for the BNNT and BNNS structures are proven to be efficient in DPD and SPH simulations for predicting the TC of h-BN/polymer composites. Furthermore, it is demonstrated that the TC of BNNT/BNNS/EP composites is superior to that of BNNT/EP and BNNS/EP composites at the same filler ratio due to the synergistic effect of BNNT and BNNS.

Synthetic Force-Field Database for Training Machine Learning Models to Predict Mobility-Preserving Coarse-Grained Molecular-Simulation Potentials.

Balancing accuracy and efficiency is a common problem in molecular simulation. This tradeoff is evident in coarse-grained molecular dynamics simulation, which prioritizes efficiency, and all-atom molecular simulation, which prioritizes accuracy. Despite continuous efforts, creating a coarse-grained model that accurately captures both the system's structure and dynamics remains elusive. In this article, we present a data-driven approach for constructing coarse-grained models that aim to describe both the structure and dynamics of the system equally well. While the development of machine learning models is well-received in the scientific community, the significance of dataset creation for these models is often overlooked. However, data-driven approaches cannot progress without a robust dataset. To address this, we construct a database of synthetic coarse-grained potentials generated from unphysical all-atom models. A neural network is trained with the generated database to predict the coarse-grained potentials of real liquids. We evaluate their quality by calculating the combined loss of structural and dynamical accuracy upon coarse-graining. When we compare our machine learning-based coarse-grained potential with the one from iterative Boltzmann inversion, the machine learning prediction turns out better for all eight hydrocarbon liquids we studied. As all-atom surfaces turn more nonspherical, both ways of coarse-graining degrade. Still, the neural network outperforms iterative Boltzmann inversion in constructing good quality coarse-grained models for such cases. The synthetic database and the developed machine learning models are freely available to the community, and we believe that our approach will generate interest in efficiently deriving accurate coarse-grained models for liquids.

Chemically Specific Systematic Coarse-Grained Polymer Model with Both Consistently Structural and Dynamical Properties.

The coarse-grained (CG) model serves as a powerful tool for the simulation of polymer systems; its reliability depends on the accurate representation of both structural and dynamical properties. However, strong correlations between structural and dynamical properties on different scales and also a strong memory effect, enforced by chain connectivity between monomers in polymer systems, render developing a chemically specific systematic CG model a formidable task. In this study, we report a systematic CG approach that combines the iterative Boltzmann inversion (IBI) method and the generalized Langevin equation (GLE) dynamics. Structural properties are ensured by using conservative CG potentials derived from the IBI method. To retrieve the correct dynamical properties in the system, we demonstrate that using a combination of a Rouse-type delta function and a time-dependent short-time kernel in the GLE simulation is practically efficient. The former can be used to adjust the long-time diffusion dynamics, and the latter can be reconstructed from an iterative procedure according to the velocity autocorrelation function (ACF) from all-atomistic (AA) simulations. Taking the polystyrene as an example, we show that not only structural properties of radial distribution function, intramolecular bond, and angle distributions can be reproduced but also dynamical properties of mean-square displacement, velocity ACF, and force ACF resulted from our CG model have quantitative agreement with the reference AA model. In addition, reasonable agreements are observed in other collective properties between our GLE-CG model and the AA simulations as well.

Force matching and iterative Boltzmann inversion coarse grained force fields for ZIF-8.

Despite the intense activity at electronic and atomistic resolutions, coarse grained (CG) modeling of metal-organic frameworks remains largely unexplored. One of the main reasons for this is the lack of adequate CG force fields. In this work, we present iterative Boltzmann inversion and force matching (FM) force fields for modeling ZIF-8 at three different coarse grained resolutions. Their ability to reproduce structure, elastic tensor, and thermal expansion is evaluated and compared with that of MARTINI force fields considered in previous work [Alvares et al., J. Chem. Phys. 158, 194107 (2023)]. Moreover, MARTINI and FM are evaluated for their ability to depict the swing effect, a subtle phase transition ZIF-8 undergoes when loaded with guest molecules. Overall, we found that all our force fields reproduce structure reasonably well. Elastic constants and volume expansion results are analyzed, and the technical and conceptual challenges of reproducing them are explained. Force matching exhibits promising results for capturing the swing effect. This is the first time these CG methods, widely applied in polymer and biomolecule communities, are deployed to model porous solids. We highlight the challenges of fitting CG force fields for these materials.

Machine Learning Potentials with the Iterative Boltzmann Inversion: Training to Experiment.

Methodologies for training machine learning potentials (MLPs) with quantum-mechanical simulation data have recently seen tremendous progress. Experimental data have a very different character than simulated data, and most MLP training procedures cannot be easily adapted to incorporate both types of data into the training process. We investigate a training procedure based on iterative Boltzmann inversion that produces a pair potential correction to an existing MLP using equilibrium radial distribution function data. By applying these corrections to an MLP for pure aluminum based on density functional theory, we observe that the resulting model largely addresses previous overstructuring in the melt phase. Interestingly, the corrected MLP also exhibits improved performance in predicting experimental diffusion constants, which are not included in the training procedure. The presented method does not require autodifferentiating through a molecular dynamics solver and does not make assumptions about the MLP architecture. Our results suggest a practical framework for incorporating experimental data into machine learning models to improve the accuracy of molecular dynamics simulations.

Analysis of static and dynamic mechanical properties of carbon black/carbon nanotubes hybrid filled natural rubber nanocomposites using coarse‐grained molecular dynamics

AbstractNanoparticle (NP)‐filled natural rubber (NR) has attracted much attention owing to their prominent mechanical strength, stiffness, and wear resistance. In this study, the static and dynamic mechanical properties of carbon black (CB)/carbon nanotubes (CNTs) hybrid filled NR nanocomposites are investigated using coarse‐grained molecular dynamics (CGMD) simulations. The non‐bonded coarse‐grained force fields for the CB/CNTs hybrid filled NR nanocomposites are constructed by combining several methods (i.e., effective force coarse‐graining, iterative Boltzmann inversion, and energy matching methods). The uniaxial tensile results indicate that the gradual replacement of CB by the same weight percentage of CNTs significantly improves the mechanical performances of the nanocomposites and increases the heterogeneity of the stress and strain distributions in the nanocomposites, which can be ascribed to the high mechanical strength of CNTs and the formation of the rigid interface networks between CNTs and the NR matrix due to the wrapping behavior of NR molecular chains on the surface of CNTs. Furthermore, the dynamic shear results demonstrate that the introduction of NPs significantly enhances the Payne effect of the nanocomposites, and the higher the CNT loading in hybrid NPs, the stronger the Payne effect. It is also found that the breakup of NP networks plays a dominant role for the Payne effect of filled rubber as compared to the interfacial separation between NPs and the rubber matrix. The present multiscale computational framework can be further extended to other polymer nanocomposites, and sheds new light on the design and exploration of polymer nanocomposites through bottom‐up prediction.

Development of a coarse-grained molecular dynamics model for poly(dimethyl-co-diphenyl)siloxane.

Polydimethylsiloxane is an important polymeric material with a wide range of applications. However, environmental effects like low temperature can induce crystallization in this material with resulting changes in its structural and dynamic properties. The incorporation of phenyl-siloxane components, e.g., as in a poly(dimethyl-co-diphenyl)siloxane random copolymer, is known to suppress such crystallization. Molecular dynamics (MD) simulations can be a powerful tool to understand such effects in atomistic detail. Unfortunately, all-atomistic molecular dynamics (AAMD) is limited in both spatial dimensions and simulation times it can probe. To overcome such constraints and to extend to more useful length- and time-scales, we systematically develop a coarse-grained molecular dynamics (CGMD) model for the poly(dimethyl-co-diphenyl)siloxane system with bonded and non-bonded interactions determined from all-atomistic simulations by the iterative Boltzmann inversion (IBI) method. Additionally, we propose a lever rule that can be useful to generate non-bonded potentials for such systems without reference to the all-atomistic ground truth. Our model captures the structural and dynamic properties of the copolymer material with quantitative accuracy and is useful to study long-time dynamics of highly-entangled systems, sequence-dependent properties, phase behaviour, etc.

Re-entrant percolation in active Brownian hard disks.

Non-equilibrium clustering and percolation are investigated in an archetypal model of two-dimensional active matter using dynamic simulations of self-propelled Brownian repulsive particles. We concentrate on the single-phase region up to moderate levels of activity, before motility-induced phase separation (MIPS) sets in. Weak activity promotes cluster formation and lowers the percolation threshold. However, driving the system further out of equilibrium partly reverses this effect, resulting in a minimum in the critical density for the formation of system-spanning clusters and introducing re-entrant percolation as a function of activity in the pre-MIPS regime. This non-monotonic behaviour arises from competition between activity-induced effective attraction (which eventually leads to MIPS) and activity-driven cluster breakup. Using an adapted iterative Boltzmann inversion method, we derive effective potentials to map weakly active cases onto a passive (equilibrium) model with conservative attraction, which can be characterised by Monte Carlo simulations. While the active and passive systems have practically identical radial distribution functions, we find decisive differences in higher-order structural correlations, to which the percolation threshold is highly sensitive. For sufficiently strong activity, no passive pairwise potential can reproduce the radial distribution function of the active system.

Resolving the dynamic properties of entangled linear polymers in non-equilibrium coarse grain simulation with a priori scaling factors.

The molecular weight of polymers can influence the material properties, but the molecular weight at the experiment level sometimes can be a huge burden for property prediction with full-atomic simulations. The traditional bottom-up coarse grain (CG) simulation can reduce the computation cost. However, the dynamic properties predicted by the CG simulation can deviate from the full-atomic simulation result. Usually, in CG simulations, the diffusion is faster and the viscosity and modulus are much lower. The fast dynamics in CG are usually solved by a posteriori scaling on time, temperature, or potential modifications, which usually have poor transferability to other non-fitted physical properties because of a lack of fundamental physics. In this work, a priori scaling factors were calculated by the loss of degrees of freedom and implemented in the iterative Boltzmann inversion. According to the simulation results on 3 different CG levels at different temperatures and loading rates, such a priori scaling factors can help in reproducing some dynamic properties of polycaprolactone in CG simulation more accurately, such as heat capacity, Young's modulus, and viscosity, while maintaining the accuracy in the structural distribution prediction. The transferability of entropy-enthalpy compensation and a dissipative particle dynamics thermostat is also presented for comparison. The proposed method reveals the huge potential for developing customized CG thermostats and offers a simple way to rebuild multiphysics CG models for polymers with good transferability.

Derived Coarse-Grained Potentials for Semicrystalline Polymers with a Blended Multistate Iterative Boltzmann Inversion Method.

In this article, we employ the multistate iterative Boltzmann inversion (MS-IBI) method to develop coarse-grained potentials capable of representing molecular structure in both the amorphous and crystalline phases of semicrystalline polymers with improved accuracy while allowing for tunable control over the dynamics governing the α-relaxation process. A unique feature of this method is that the potentials are blended using the product of the target structural distributions, for example, the radial density function, for each phase and a weighting factor. To demonstrate this approach, a family of potentials for polyethylene is developed where the weighting factor of the crystalline phase ranges is varied from zero, incorporating information only from the amorphous phase, to unity, where the model is trained from only the crystalline phase. The most accurate representation of structural distributions was obtained when the crystalline phases is weighted at 50%. However, we show that when the crystalline phase is weighted at 90%, the model more accurately represents dynamics of the α-relaxation process, with realistic predicted values of activation energy and diffusion rates, with relatively minor impact on accuracy in structure.

OpenMSCG: A Software Tool for Bottom-Up Coarse-Graining.

The "bottom-up" approach to coarse-graining, for building accurate and efficient computational models to simulate large-scale and complex phenomena and processes, is an important approach in computational chemistry, biophysics, and materials science. As one example, the Multiscale Coarse-Graining (MS-CG) approach to developing CG models can be rigorously derived using statistical mechanics applied to fine-grained, i.e., all-atom simulation data for a given system. Under a number of circumstances, a systematic procedure, such as MS-CG modeling, is particularly valuable. Here, we present the development of the OpenMSCG software, a modularized open-source software that provides a collection of successful and widely applied bottom-up CG methods, including Boltzmann Inversion (BI), Force-Matching (FM), Ultra-Coarse-Graining (UCG), Relative Entropy Minimization (REM), Essential Dynamics Coarse-Graining (EDCG), and Heterogeneous Elastic Network Modeling (HeteroENM). OpenMSCG is a high-performance and comprehensive toolset that can be used to derive CG models from large-scale fine-grained simulation data in file formats from common molecular dynamics (MD) software packages, such as GROMACS, LAMMPS, and NAMD. OpenMSCG is modularized in the Python programming framework, which allows users to create and customize modeling "recipes" for reproducible results, thus greatly improving the reliability, reproducibility, and sharing of bottom-up CG models and their applications.