Modeling of engineering objects with complex heterogeneous material structure at nanoscale level has emerged as an important research problem. In this research, we are interested in multiscale modeling and analysis of mechanical properties of the polymer structures created in the Step and Flash Imprint Lithography (SFIL) process. SFIL is a novel imprint lithography process designed to transfer circuit patterns for fabricating microchips in low-pressure and room-temperature environments. Since the smallest features in SFIL are only a few molecules across, approximating them as a continuum is not completely accurate. Previous research in this subject has dealt with coupling discrete models with continuum hyperelasticity models. The modeling of the post-polymerization step in SFIL involves computing solutions of large nonlinear energy minimization problems with fast spatial variation in material properties. An equilibrium configuration is found by minimizing the energy of this heterogeneous polymeric lattice. Numerical solution of such a molecular statics base model, which is assumed to describe the microstructure completely, is computationally very expensive. This is due to the problem size – on the order of millions of degrees of freedom (DOFs). Rapid variation in material properties, ill-conditioning, nonlinearity, and non-convexity make this problem even more challenging to solve. We devise a method for efficient approximation of the solution. Combining numerical homogenization, adaptive finite element meshes, and goaloriented error estimation, we develop a black-box method for efficient solution of problems with multiple spatial scales. The purpose of this homogenization method is to reduce the number of DOFs, find locally optimal effective material properties, and do goal-oriented mesh refinement. In addition, it smoothes the energy landscape. Traditionally, a finite element mesh is designed after obtaining material properties in different regions. The mesh has to resolve material discontinuities and rapid variations. In our approach, however, we generate a sequence of coarse meshes (possibly 1-irregular), and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem. This upscaling is done using Moore-Penrose pseudoinverse of the linearized fine-scale element stiffness matrices, and a material independent interpolation operator. This requires solution of a continuous-time Lyapunov equation on each element. Using the adjoint solution, we compute local error estimates in the quantity of interest. The error estimates also drive the automatic mesh adaptivity algorithm. The results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine-scale problem. Critical to the computational speed of local homogenization is computing Moore-Penrose pseudoinverse of rank-deficient matrices without using Singular Value Decomposition. To this end, we use four algorithms, each having different desirable features. The algorithms are based on Tikhonov regularization, sparse QR factorization, a priori knowledge of the null-space of the matrix, and iterative methods based on proper splittings of matrices. These algorithms can exploit sparsity and thus are fast. Although the homogenization method is designed with a specific molecular statics problem in mind, it is a general method applicable for problems with a given fine mesh that sufficiently resolves the fine-scale material properties. We verify the method using a conductivity problem in 2-D, with chessboard like thermal conductivity pattern, which has a known homogenized conductivity. We analyze other aspects of the homogenization method, for example the choice of norm in which we measure local error, optimum coarse mesh element size for homogenizing SFIL lattices, and the effect of the method chosen for computing the pseudoinverse. / text

We grow AB₂ random hyperbranched polymer structures in different ways and using different simulation methods. In particular we use a method of ad hoc construction of the connectivity matrix and the bond fluctuation model on a 3D lattice. We show that hyperbranched polymers split into two universality classes depending on the growth process. For a “slow growth” (SG) process where monomers are added sequentially to an existing molecule which strictly avoids cluster–cluster aggregation the resulting structures share all characteristic features with regular dendrimers. For a “quick growth” (QG) process which allows for cluster–cluster aggregation we obtain structures which can be identified as random fractals. Without excluded volume interactions the SG model displays a logarithmic growth of the radius of gyration with respect to the degree of polymerization while the QG model displays a power law behavior with an exponent of 1/4. By analyzing the spectral properties of the connectivity matrix we confirm the behavior of dendritic structures for the SG model and the corresponding fractal properties in the QG case. A mean field model is developed which explains the extension of the hyperbranched polymers in an athermal solvent for both cases. While the radius of gyration of the QG model shows a power-law behavior with the exponent value close to 4/5, the corresponding result for the SG model is a mixed logarithmic–power-law behavior. These different behaviors are confirmed by simulations using the bond fluctuation model. Our studies indicate that random sequential growth according to our SG model can be an alternative to the synthesis of perfect dendrimers.

The homo and hetero dimerisation of two spectroscopically different chromophores were studied, namely: 4,4-difluoro-4-bora-3a,4a-diazas-indacene (g-BODIPY) and its 5-styryl-derivative (r-BODIPY). Various spectroscopic properties of the r-BODIPY in different common solvents were determined. It was shown that g- and r-BODIPY in the ground state can form homo- as well as hetero dimers. We demonstrate that the ganglioside GM1 in lipid bilayers of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) exhibits a non-uniform lateral distribution, which is an argument in favour of self-aggregation of GM1 being an intrinsic property of the GM1. This was concluded from energy transfer/migration studies of BODIPY-labelled gangliosides. An algorithm is presented that quantitatively accounts for donor–donor energy migration (DDEM) among fluorophore-labelled proteins forming regular non-covalent polymers. The DDEM algorithm is based on Monte Carlo (MC) and Brownian dynamics (BD) simulations and applies to the calculation of fluorescence depolarisation data, such as the fluorescence anisotropy. Thereby local orientations, as well as reorienting motions of the fluorescent groups are considered in the absence and presence of DDEM among them. A new method, in which a genetic algorithm (GA) was combined with BD and MC simulations, was developed to analyse fluorescence depolarisation data collected by the time-correlated single photon counting technique. It was applied to study g-BODIPY-labelled filamentous actin (F-actin). The technique registered the local order and reorienting motions of the fluorophores, which were covalently coupled to cysteine 374 (C374) in actin and interacted by means of electronic energy migration within the polymer. Analyses of F-actin samples composed of different fractions of labelled actin molecules revealed the known helical organiszation of F-actin, and demonstrated the usefulness of this technique for structure determination of complex protein polymers. The distance from the filament axis to the fluorophore was found to be considerably less than expected from the proposed position of C374 at a high filament radius. In addition, polymerisation experiments with BODIPY-actin suggest a 25-fold more efficient signal for filament formation than pyrene-actin.