n
Figure 2.4: The skeletal formula of cellulose, composed of nunits of cellobiose.
solvent, cellulose has been shown to undergo a glass transition at temperatures around 500 K [39], when extrapolated to dry conditions.
In nature, cellulose exist as microfibrils structures of sizes down of 36-18 chains [40, 41]. For many applications, particularly for packaging and fabric ap-plications, it would be beneficial to have the cellulose as single chains. There is a great interest in understanding how cellulose might be dissolved. A bundle of cellulose is held together by hydrogen bonds in-plane and van der Waals bonds that act perpendicularly. The perhaps most dominant challenge in obtaining dis-solution is that the cellulose bonds are both hydrophilic and hydrophobic [42, 43].
In the laboratory, the cellulose nanofibrils are typically isolated from wood-based fibres by mechanical treatment with high shear forces, with the optional chemical or enzymatic pretreatment step [44]. One common solvent for cellulose is composed of hydroxide and urea dissolved in water, which seems to be most effective at temperatures around 258 K [45]. The exact role of individual solvent constituents have been heavily investigated, but experimental and computational findings are not consistent between studies. One hypothesis, perhaps the most common, is that sodium and urea is thought to penetrate the bundle, where sodium then disrupts the interchain bonding and urea stabilize the dissolved chains to prevent re-agglomeration.
2.5 Applications for the articles
2.5.1 Article I
The first article is a computational study of tensile properties of PEO nanofibres, composed of linear chains of 33 monomers each. The geometry of the system is inspired by the fiber bundle model: the end-points of each chain are attached to stiff parallel planes, such that each chain is elongated by the same amount. The particles at the end-points of each chain are however allowed to move within these
planes. The tensile loading was studied both with constant force and constant strain rate. For the constant force simulations, the load on the chains are equally distributed through the clamps, similar to the equal load sharing fiber bundle model. The simulations with constant strain rate focus on the interchain interac-tions, and do not include any redistribution of forces through the clamps. Going far beyond the complexity of the fiber bundle model, we aim to study polymeric fibres from a material science point of view and in this way probe the limitations of these models.
While the fiber bundle model exhibits quenched disorder, the disorder in these simulations originates in the chaotic potential energy landscape and in fluctuations from a thermal bath, to be discussed more later. The effect of configuration defects is also explored. The covalent bonds are modeled with Morse potentials, allowing for the breaking of bonds. While these computational experiments targets creep, their high energy and limited time scale more closely approach the mechanical breaking process.
Crystallinity plays an important role, as we have a transition from a disordered state to a semi-crystalline state during stretching. In experimental studies of PEO, the samples are highly amorphous, and the tensile properties are dominated by the interchain interactions between the loosely coiled chains. The semi-crystalline fibres in this computational study resemble more closely the structure of polyethy-lene, to which it display similar mechanical properties.
2.5.2 Article II-III
Continuing with the stretching of PEO nanofibres, the second and third article focus on single-molecule stretching with various degrees of polymerization at theta conditions. Here entropic elasticity is explored in addition do the enthalpic elastic-ity, and the system is compared to an idealized FJC model. The system is verified by studying the behaviour in the entropic region according to Eq. (2.23), which can be written as
fS= 3kBT
Neffb2effl. (2.34)
Note that equation 7 in article II is off by a factor of 6: this expression was obtained by using the radius of gyration rather than the end-to-end distance as the length in the expression for the entropy, shown in 2.19. As the molecule undergoes a transition from helical to planar conformations, the unfolded length relevant for the entropic regime is the length of the molecule before this transition. Accounting for this, one obtains a Kuhn length of 2 Å. In this article N = 51 denotes the number of beads in the chain, and the resulting effective number of beads for the FJC model is Neff = 12. I consider this to be reasonable for a PEO-molecule composed ofn= 16monomers.
16
2.5.3 Article IV
The fourth article has a more applied objective, and concerns the dissolution of cellulose. The aim of the work is to understand the specific details concerning the dissolution process of cellulose in a solution of water, sodium hydroxide and urea.
As this is a highly complex system, it is not well described by idealized models.
3 Molecular Dynamics
“We adore chaos because we love to produce order”
M.C. Escher
Molecular dynamics (MD) is a incredibly versatile tool for computing the time evolution of systems. The complexity of the systems can be increased far beyond what can be described analytically. Similar to experiments in the laboratory, the outcome is prone to statistical errors. One also has to pay great attention into making sure that the simulation conditions are physical, and that the model is appropriate for what one is trying to achieve. Nonetheless, these brute force com-puter experiments have been shown to be highly useful tools for a wide range of applications. One attractive feature of molecular dynamics is that it permits di-rect visualization of the detailed motions of individual atoms in a system, thereby providing a window into the microscopic world. While the instantaneous move-ments of a singe trajectory might not be representative for the system, they can be useful as a guide toward understanding the mechanisms underlying a given process. Moreover, the average values have been shown to be reliable in most cases. We will here go through some basic principles of molecular dynamics, with an emphasis on properties that have been relevant for this work. For more details see [46].
3.1 Length and time scale
One of the first points to consider when preparing a computer simulation is the relevant length and time scale. With fully atomistic simulations one explicitly solves Newton’s equations of motion for every single atom, and one typically work on the nano-scale in length and time. If one is interested in significantly larger scales, it is often a good idea to coarse-grain: one particle in our simulation can represent multiple atoms or whole groups of atoms. This reduction of degrees of freedom allows for considerably larger systems or significantly longer simulation times. The interpretation of time scales in coarse grained simulations is however a complicated topic of its own, especially relevant for dynamic processes.
For systems with long-range interactions, the computation of the resulting long-range forces is typically dominating for the computational cost. For a sim-ulation where N particles interact with all other particles, a naive summation over pairs would give a running time of O(N2). By making use of Fast Fourier Transforms (FFT), the time complexity is often reduced toO(NlogN). Some al-gorithms even obtain linear time alal-gorithms, at the cost of limited accuracy. This will be discussed more in Section 3.5.
A common strategy to reduce the finite size effect of small systems is to make use of periodic boundaries in one or multiple dimensions, such that a particle interacts not only with the other particles in the system, but also replicas of these through the boundaries. As we will discuss more later, having periodic boundaries is even a requirement for using FFT. One should note that the computational cost of FFT-based methods scales with the volume of the simulation box.
When modeling systems such as polymers in a solvent, a common approach to speed up the simulations and increase the available time and length scales is to make use of implicit solvents. While there are different approaches to achieving it, they all aim to modify the non-bonded interaction potential to account for solvents without explicitly including the solvent-particles. This does not however account for the hydrophobic effect, viscosity or hydrogen bonding. If those effects are not relevant, implicit solvents can often be a good choice.