Creep characterization of vapor-grown carbon nanofiber/vinyl ester nanocomposites using a response surface methodology

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AbstractMethodologyMaterial ModelInput DataResultsAcknowledgmentsReferences


File:CCD Design.gif
Central Composite Design.

TThe effects of selected factors such as vapor-grown carbon nanofiber (VGCNF) weight fraction, applied stress, and temperature on the viscoelastic responses (creep strain and creep compliance) of VGCNF/vinyl ester (VE) nanocomposites were studied using a central composite design (CCD). Nanocomposite test articles were fabricated by high-shear mixing, casting, curing, and post curing in an open-face mold under a nitrogen environment. Short-term creep/creep recovery experiments were conducted at prescribed combinations of temperature (23.8–69.2C), applied stress (30.2–49.8 MPa), and VGCNF weight fraction (0.00–1.00 parts of VGCNF per hundred parts of resin) determined from the CCD. Response surface models (RSMs) for predicting these viscoelastic responses were developed using the least squares method and an analysis of variance procedure. The response surface estimates indicate that increasing the VGCNF weight fraction marginally increases the creep resistance of the VGCNF/VE nanocomposite at low temperatures (i.e., 23.8–46.5C). However, increasing the VGCNF weight fraction decreased the creep resistance of these nanocomposites for temperatures greater than 50C. The latter response may be due to a decrease in the nanofiber-to-matrix adhesion as the temperature is increased. The RSMs for creep strain and creep compliance revealed the interactions between the VGCNF weight fraction, stress, and temperature on the creep behavior of thermoset polymer nanocomposites. The design of experiments approach is useful in revealing interactions between selected factors, and thus can facilitate the development of more physics-based models.

Author(s): Daniel A. Drake, Rani W. Sullivan, Thomas E. Lacy, Charles U. Pittman, Jr., Hossein Toghiani, Janice L. DuBien, Sasan Nouranian, Jutima Simsiriwong

Corresponding Author: [ Rani W. Sulivan, Ph.D.]

Figure 1. An example of the bicrystal simulation cell used for the grain boundary structure calculations. In this particular example, the grain boundary is a <110> tilt grain boundary. (click on the image to enlarge).
Figure 2. This is a plot of the accessibility of each grain boundary structure for an asymmetric tilt grain boundary. For complex boundaries, a large number of boundaries may need to be sampled to find the "global" minimum energy boundary. (click on the image to enlarge).


For each grain boundary structure, the crystal orientations and their relationship to the grain boundary plane -- five degrees of freedom -- are often used to describe the grain boundary character. A three-dimensional periodic simulation cell is used for these calculations (see Fig. 1), whereby there are two crystal orientations and two grain boundaries (one in the middle and one at the top/bottom). This simulation cell allows the user to manipulate the five grain boundary degrees of freedom to investigate their relationship on properties. The distance between the two boundaries must be large enough for the grain boundary energy to converge, i.e., in the input script below, a distance of 12 nm was used. Figure 2. This is a plot of the accessibility of each grain boundary structure for an asymmetric tilt grain boundary. For complex boundaries, a large number of boundaries may need to be sampled to find the "global" minimum energy boundary.

In many cases, at the atomistic level, the minimum energy structure cannot be found by merely adjoining the two crystal lattices in the simulation cell. The relative translation of the two crystal lattices with respect to each other is important as well as the number of atoms within the grain boundary region. Therefore, to find the minimum energy structure, a large number of configurations with different relative translations is vital to finding the stable minimum energy structure and not just a metastable grain boundary structure. Figure 2 shows an example for an asymmetric tilt grain boundary, where the minimum energy structure was only found (accessed) 8.76% of the time with various translations. In more complex boundaries, the accessibility may be less than 0.1%.

The following input script shows how multiple translations and an atom deletion criteria are used to calculate the minimum energy structure. This input script for LAMMPS[1] can be called with a command of the form, "lmp_exe < input.script." This script contains loops over x-translations, z-translations, and atom overlap distances (an atom is deleted when an atom pair with a nearest neighbor distance is less than this distance). The unique minimum energy structures are saved as a dump file with the energy appended to the filename in a new folder specified by the 'gbname' variable. The dump files can then be easily scanned through for the global minimum energy structure.

Material Model

Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)

Input Data

See LAMMPS Input Deck for Grain boundary generation


This methodology has resulted in structures that agree with HRTEM images and grain boundary energies that agree with experimentally-measured grain boundary energies. Figure 3 shows an example plot of the grain boundary energy versus inclination angle for a complex grain boundaries in Cu (Sigma 3 asymmetric grain boundaries). This system of boundaries displays a phase transformation at the boundary to the orthorhombic 9R phase for inclination angles of 70-90 degrees. The methodology used above agrees nicely with experimental results and previous atomistic calculations of Wolf and coworkers.
Al <100> symmetric tilt grain boundaries.
Al <110> symmetric tilt grain boundaries.
Cu <100> symmetric tilt grain boundaries.
Cu <110> symmetric tilt grain boundaries.
Figure 3. Plot of the grain boundary energy as a function of inclination angle for asymmetric tilt grain boundaries in Cu.


M.A. Tschopp would like to acknowledge funding provided under an NSF graduate fellowship for the initial work. Continued funding for investigating structure-property relationships in grain boundaries under the NEAMS (Nuclear Energy Advanced Modeling and Simulation) program is also acknowledged.


The initial methodology was used in the following papers:

  1. S. Plimpton, "Fast Parallel Algorithms for Short-Range Molecular Dynamics," J. Comp. Phys., 117, 1-19 (1995).
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