In this article, the effects of size and confinement at the nanometre size scale on both the melting temperature, Tm, and the glass transition temperature, Tg, are reviewed. Although there is an accepted thermodynamic model (the Gibbs–Thomson equation) for explaining the shift in the first-order transition, Tm, for confined materials, the depression of the melting point is still not fully understood and clearly requires further investigation. However, the main thrust of the work is a review of the field of confinement and size effects on the glass transition temperature. We present in detail the dynamic, thermodynamic and pseudo-thermodynamic measurements reported for the glass transition in confined geometries for both small molecules confined in nanopores and for ultrathin polymer films. We survey the observations that show that the glass transition temperature decreases, increases, remains the same or even disappears depending upon details of the experimental (or molecular simulation) conditions. Indeed, different behaviours have been observed for the same material depending on the experimental methods used. It seems that the existing theories of Tg are unable to explain the range of behaviours seen at the nanometre size scale, in part because the glass transition phenomenon itself is not fully understood. Importantly, here we conclude that the vast majority of the experiments have been carried out carefully and the results are reproducible. What is currently lacking appears to be an overall view, which accounts for the range of observations. The field seems to be experimentally and empirically driven rather than responding to major theoretical developments.

https://iopscience.iop.org/article/10.1088/0953-8984/17/15/R01/pdf

Effects of confinement on material behaviour at the nanometre size scale

The relation between atomistic chemical structure, molecular archi- tecture, molecular weight, and material properties is of basic concern in modern soft material science and includes standard properties of bulk materials and surface and interface aspects, as well as the relation between structure and function in nanoscopic objects and molecular assemblies of both synthetic and biological origin. This all implies a thorough understanding on many length and correspondingly time scales, ranging from (sub)atomistic to macroscopic. Presently, com- puter simulations play an increasingly important, if not central, role. Some problems do not require specific atomistic details, whereas others require them only locally. However, in many cases this strict separation is not sufficient for a comprehensive understanding of systems, and flexible simulation schemes are required that link the different levels of resolution. We here give a general view of the problem regarding soft matter and discuss some specific examples of linked simulation techniques at different resolution levels. We then discuss a recently developed flexible simulation scheme, the AdResS method, which allows one to adaptively change the resolution in certain regions of space on demand.

https://www.researchgate.net/profile/Luigi_Delle_Site/publication/5785059_Multiscale_Simulation_of_Soft_Matter_From_Scale_Bridging_to_Adaptive_Resolution/links/0fcfd5134e36be4806000000.pdf

Multiscale simulation of soft matter: from scale bridging to adaptive resolution.

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the behavior of such systems can be obtained by energy arguments. The claim would be that there exists a critical temperature Tc, such that for T > Tc the free energy F (c, T ) of the system is a single welled function of the concentration c of one of the species, whereas for T < Tc the free energy is double welled. Referring to Figure 1, a system which was spatially uniform at temperature T1, when cooled to temperature T2, would find it energetically preferrable to separate itself into two systems, one at concentration cA and one at concentration cB. To be more specific consider now the system at temperature T < Tc. Assume that the free energy F (c) per unit volume (Gibbs free energy or Landau-Ginzburg free energy) of the spatially homogeneous system has the convex/concave shape indicated in Figure 2. More precisely F is concave in the spinodal interval cA < c < c s B and convex elsewhere. The points cA and cB where the supporting tangent touches the graph are sometimes referred to as the binodal points. The derivative f(c) = F (c) is depicted in Figure 3. The free energy of a spatially heterogeneous system would then be given by

/pdf/the-cahn-hilliard-equation-4ma76sukyt.pdf

The cahn-hilliard equation

A coarse grain model and a Monte Carlo sampling formalism are proposed for simulations of self-assembly in block copolymer melts and nanoparticle−copolymer composites. Our approach relies on a particle-based representation of the system, it does not invoke a saddle point approximation, and it permits treatment of large three-dimensional systems. We provide a detailed description of the model and methods and discuss their relationship to results from self-consistent-field theory and single chain in mean field simulations. The validity of the proposed approach is addressed by applying it to study systems whose description within existing approaches would be demanding. In particular, we use it to examine the directed assembly of copolymer blends and nanoparticles on nanopatterned substrates. We show that results from simulations are in good agreement with experiment, and we use our theoretical findings to help explain the experimental observations.

Monte Carlo Simulations of a Coarse Grain Model for Block Copolymers and Nanocomposites

Our aim in this article is to discuss recent issues related with the Cahn-Hilliard equation in phase separation with the thermodynamically relevant logarithmic potentials; in particular, we are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Neumann boundary conditions and then dynamic boundary conditions which account for the interactions with the walls in confined systems and have recently been proposed by physicists. We also present, in the case of dynamic boundary conditions, some numerical results.

The Cahn-Hilliard Equation with Logarithmic Potentials

Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions

A soft ellipsoid model for Gaussian polymer chains is studied, following an idea proposed by Murat and Kremer [J. Chem. Phys. 108, 4340 (1998)]. In this model chain molecules are mapped onto ellipsoids with certain shapes, and to each shape a monomer density is assigned. In the first part of the work, the probabilities for the shapes and the associated monomer densities are studied in detail for Gaussian chains. Both quantities are expressed in terms of simple approximate formulas. The free energy of a system composed of many ellipsoids is given by an intramolecular part accounting for the internal degrees of freedom and an intermolecular part following from pair interactions between the monomer densities. Structural and kinetic properties of both homogeneous systems and binary mixtures are subsequently studied by Monte Carlo simulations. It is shown that the model provides a powerful phenomenological approach for investigating polymeric systems on semimacroscopic time and length scales.

Soft ellipsoid model for Gaussian polymer chains

A soft particle model for diblock (AB) copolymer melts is proposed. Each molecule is mapped onto two soft spheres built by Gaussian A- and B-monomer distributions. An approximate analytical expression for the joint distribution function for the distance between both spheres and their radii of gyration is derived, which determines the entropic contribution to the intramolecular free energy. Adding a mean-field expression for the intermolecular interactions, we obtain the total free energy of the system. Based on this free energy, Monte Carlo simulations are carried out to study the kinetics of microphase ordering in the bulk and its effect on molecular diffusion. This is followed by an analysis of thin films, with emphasis on pattern transfer from walls with a periodic structure. It is shown that the level of coarse graining in the soft particle model is suitable to describe structural and kinetic properties of copolymers on mesoscopic scales.

/pdf/soft-particle-model-for-block-copolymers-5ec2245vgh.pdf

Soft particle model for block copolymers.

Polymer systems in slab geometries are studied on the basis of the recently presented Gaussian ellipsoid model [F. Eurich and P. Maass, J. Chem. Phys. 114, 7655 (2001)]. The potential of the confining walls has an exponential shape. For homogeneous systems in thermodynamic equilibrium we discuss density, orientation, and deformation profiles of the polymers close to the walls. For strongly segregated mixtures of polymer components A and B equilibrium profiles are studied near a planar interface separating A and B rich regions. Spinodal decomposition processes of the mixtures in the presence of neutral walls show upon strong confinement an increase of the lateral size of A and B rich domains and a slowing down of the demixing kinetics. These findings are in agreement with predictions from time dependent Ginzburg–Landau theory. In the case, where one wall periodically favors one of the two mixture components over the other, different equilibrium structures emerge and lead to different kinetic pathways of spinodal decomposition processes in such systems.

Gaussian ellipsoid model for confined polymer systems

A soft ellipsoid model for Gaussian polymer chains is studied, following an idea proposed by Murat and Kremer [J. Chem. Phys. 108, 4340 (1998)]. In this model chain molecules are mapped onto ellipsoids with certain shapes, and to each shape a monomer density is assigned. In the first part of the work, the probabilities for the shapes and the associated monomer densities are studied in detail for Gaussian chains. Both quantities are expressed in terms of simple approximate formulae. The free energy of a system composed of many ellipsoids is given by an intramolecular part accounting for the internal degrees of freedom and an intermolecular part following from pair interactions between the monomer densities. Structural and kinetic properties of both homogeneous systems and binary mixtures are subsequently studied by Monte-Carlo simulations. It is shown that the model provides a powerful phenomenological approach for investigating polymeric systems on semi-macroscopic time and length scales.

https://arxiv.org/pdf/cond-mat/0008425.pdf

Frank Eurich

Papers

Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions

Soft ellipsoid model for Gaussian polymer chains

Soft particle model for block copolymers.

Gaussian ellipsoid model for confined polymer systems

Soft Ellipsoid Model for Gaussian Polymer Chains