## 2.1 Crystalline cellulose

Celluloses can be presented in different crystalline phases as \({I}_{\alpha }\), \({I}_{\beta }, II\) and \(II{I}_{I}\). Native plant celluloses are mixture of\({I}_{\alpha }\).and \({I}_{\beta }\). Their ratio of \({I}_{\alpha }\) to \({I}_{\beta }\) depends on original fiber sources. In wood fibers, \({I}_{ \beta }\) is a major component and adapted in this simulation work.

First, \({I}_{\beta }\) crystalline celluloses are built by using Cellulose Builder developed by Gomes and Skaf [8]. One \(\beta\)1–4 linked D-glucose unit is shown in Fig. 1.

The simulated cellulose crystal has thirty-two chains, each chain with five repeated \(\beta\) 1–4 linked D-glucose units as shown in Fig. 2. The total atom number in this crystal is 7620, which are located in a parallelepiped box with a triclinic symmetry. The basis three vectors of the simulation box are **a**=(x, y, z)= (31.136, 0, 0), **b**=( -3.7135, 32.593, 0), and **c**=(0, 0, 51.90), respectively, where length unit is in Angstrom. The cellulose chains are along axial c- or z-direction. During simulations, periodical boundary conditions are imposed in all the three directions.

## 2.2 Composite of CNT re-enforced cellulose

In order to build a CNT-cellulose composite, one of cellulose chains is removed from the crystalline cellulose solid as shown in Fig. 3 (a) and replaced by a single walled carbon nanotube with (\({n}_{1}, {n}_{2})=\)(3,3) structure as shown in Fig. 3 (b). The single walled carbon nanotube, as shown in Fig. 3. (c), has a diameter of 4.07 Angstrom, which is smaller than the size of space of the removed cellulose chain. The distance between its neighboring cellulose chains is 7.78 Angstrom. The single walled carbon nanotube is built by mapping the atoms of a 2D graphite to the surface of a cylinder. (\({n}_{1}, {n}_{2})\) denotes a particular scheme to build the nanotube from the 2D Bravais lattice of the graphite. The diameter of the carbon nanotube depends on the integer numbers of \({n}_{1}\) and \({n}_{2}\). More details can be found in White et. al. work [9]. Following their work, a carbon nanotube builder software was developed at Western Michigan University [7].

## 2.3 MD simulations

A open software of molecular dynamics simulation package called Lammps [10] is employed to simulate various physical properties, such as potential energy, total energy, density, hydrogen bond energy etc..

In simulations, the popular Charmm force fields [11] are adopted for cellulose. In Charmm force fields, total energy \({E}_{t}\)in the cellulose system can be presented by

The first term in the right side of the equation represents the bonding energy, the second term is the energy contribution due to vibration of bonded two atoms (only Charmm force field has this term), the third term is the angular energy between two bonds, the fourth term stands for the dihedral energy, the fifth term is improper energy, the sixth term is Van der Waal nonbond potential energy, the last term is electric interaction energy. \({k}_{b}\), \({k}_{UB}\), \({k}_{a}\), \({k}_{d}\), \({k}_{i}\) are the spring coefficients of corresponding terms; \({ϵ}_{ij}\)is the depth of energy well of Lennard-Jones (LJ) potential; \({R}_{min,ij}\)is its minimum energy position; \({q}_{i}\) is the charge of atom i; \({ϵ}_{0}\) is di-electrical constant; \({r}_{0}\), \({s}_{0}\), \({\theta }_{0}\), \({{\psi }}_{0}\), and \({{\psi }}_{0}\) are corresponding equilibrium position or angles. Both the LJ and electric Coulombic forces are cut off at 12 Angstrom. From the gradient of total energy, the forces acting on each atom can be calculated. Then, based on the forces, a leap-frog algorithm is utilized to update velocities and positions of all atoms.

Carbon nanotubes have stable structure due to \(s{p}^{2}\)hydride bonds, resulting in large mechanical strength. It is evident that the Tripos 5.2 force fields [12] are suitable for description of \(s{p}^{2}\)bonds of carbon nanotube. This was verified by our previous work [7]. The Tripos 5.2 force fields are employed again for presenting carbon nanotube motion in this work.

Hydrogen bonds (H-bond) are critically important for stability of cellulose structure. The Dreiding model [13–14] is selected to track the hydrogen bonds. The distance between donor and acceptor is cut off at 4.5 Angstroms and the angle is cut off at 120 degrees in this three-body H-bond model.

In MD simulation, first, an energy minimization is imposed on all atoms, then they run in a NVT ensemble for 40000 time steps at a temperature of 298 \({K}^{0}\). Next, the system runs at a constant pressure of 1atm and at the same temperature for additional 260000 time steps, allowing the system to reach an equilibrium state. During the constant pressure run, 1atm pressure is kept in the x-, y-, and z- directions and a zero pressure is kept in the xy-, yz-, and xz-directions. The volume or density and different type of energies can be measured in the NPT time integration via Nose/Hoover algorithm [15–16]. One step time interval is 0.5 femtoseconds.