Open Journal of Com p osi t e M at e ri al s, 2011, 1, 19-37
doi:10.4236/ojcm.2011.11003 Published Online October 2011 (
Copyright © 2011 SciRes. OJCM
Multi-Scale Modeling of Metal-Composite Interfaces in
Titanium-Graphite Fiber Metal Laminates Part I:
Molecular Scale
Jacob M. Hundley1*, H. Thomas Hahn2, Jenn-Ming Yang3, Andrew B. Facciano4
1Mechanical and Aerospace Engineering Department, Multifunctional Composites Lab (MCL),
University of California Los Angeles, Los Angeles, USA
2Raytheon Distinguished Professor Emeritus, Mechani cal and Aerospace Engineering Department,
University of California Los Angeles, Los Angeles, USA
3Materials Sci ence and Engi n ee ri ng De p art me nt,
University of California Los Angeles, Los Angeles, USA
4Raytheon Missile Systems, SM3 Syste m s E ngineering 1151, Tucson, USA
E-mail: *
Received August 12, 2011; revised September 16, 2011; accept ed September 25, 2011
This study presents the first stage of a multi-scale numerical framework designed to predict the non-linear
constitutive behavior of metal-composite interfaces in titanium-graphite fiber metal laminates. Scanning
electron microscopy and x-ray diffraction techniques are used to characterize the baseline physical and
chemical state of the interface. The physics of adhesion between the metal and polymer matrix composite
components are then evaluated on the atomistic scale using molecular dynamics simulations. Interfacial me-
chanical properties are subsequently derived from these simulations using classical mechanics and thermo-
dynamics. These molecular-level property predictions are used in a companion study to parameterize a con-
tinuum-level finite element model of the interface by means of a traction-separation constitutive law. Exten-
sion of the proposed approach to other dissimilar metal- or metal oxide-polymer interfaces is also discussed.
Keywords: Fiber Metal Laminates (FML), Titanium-Graphite (TiGr), Molecular Dynamics (MD), Adhesion
1. Introduction
Fiber metal laminates (FMLs) are a unique class of
structural materials that combine traditional fiber-reinfor-
ced polymer matrix composite (PMC) laminae with thin,
adhesively bonded metal layers. In these FML systems,
the metal reinforcement layer(s) can be added to the
PMC structure either internally or externally, provided
that the fiber-reinforced polymer and metal layers are
bonded together during the composite consolidation
process. Consequently, the resultant structure is globally
homogeneous but locally heterogeneous, with distinct
material phases evident by visual inspection, Figure 1.
First developed at Delft University of Technology in the
late 1970s, commercially available examples of fiber
metal laminates include GLARE (glass fiber-reinforced
aluminum laminates) and ARALL (aramid fiber-
reinforced aluminum laminates) [1]. Recently, a novel
FML variant has been developed combining carbon fi-
ber-reinforced polymer matrix composites with adhe-
sively bonded titanium layers [2]. These FMLs are the
focus of the present study and are commonly referred to
as titanium-graphite (TiGr) or hybrid titanium co mposite
laminates (HTCL). TiGr FMLs represent an intriguing
material system as they possess increased stiffness and
strength in comparison to ARALL and GLARE. They
can also be utilized in elevated temperature applications,
provided that the polymer matrix material has sufficient
thermal degradation resistance.
The primary advantage of all fiber metal laminate
variants is their ‘best of both worlds’ design, in which
the beneficial properties of the individual metal and
PMC constituents are combined into a single structure
with customizable material properties [3]. In essence,
fiber metal laminates utilize the syn ergistic enhancement
of structural properties achieved by combining two ma-
Figure 1. Schematic illustration of the components in a fiber metal laminate.
terials (metals and PMCs) with counterbalanced
strengths and weaknesses. Because the limitations of
each constituent are effectively offset, FMLs can be used
in applications which are typically considered ill-suited
for either metals or polymer matrix composites, such as
impact-critical or mechanically fastened structures [4-9].
For each of these scenarios, it is important to note that
the amount of improvement offered by an FML over a
traditional metal or composite structure is heavily de-
pendent upon the state of the interface between the metal
and PMC constituents [10]. For instance, in impact-
critical structures the formation and propagation of a
delamination zone between the metal and PMC layers
reduces the overall damage tolerance and impact energy
dissipation potential of the FML. In addition to this loss
of damage tolerance, interfacial delamination resulting
from impact has also been shown to significantly reduce
the residual static and fatigue strength of the FML [11].
For mechanically fastened fiber metal laminate structures,
interfacial delamination prevents th e metal reinforcement
layers from alleviating the high stress concentration in
the region surrounding the fastener. As such, the result-
ing delaminated FML is highly notch sensitive and will
fail at lower than expected joint loads [6]. It is, therefore,
highly desirable to form and maintain a strong inter-
laminar bond between the polymer matrix adhesive and
the metal adhered in any FML system so that these
components do not delaminate.
The importance of the metal-composite interface with
respect to the structural performance of any FML v ariant
underscores the evident need for accurate predictive
models of interfacial mechanical behavior. However, it
should be noted that prediction of adhesion between any
two dissimilar components at a common interface is one
of the most challenging and complex problems in the
realm of materials modeling. The inherent difficulty is
primarily due to the multiple length scales and damage
modes associated with interfacial debonding and failure.
For FMLs, this complexity means that any metal-
composite interfacial model must be multi-scale. It must
incorporate molecular-level considerations arising from
separation of the interface, as well as continuum-level
damage and failure phenomena due to microcracking and
fracture of the polymer matrix. The present analysis is
concerned with the development of one such multi-scale
interface model, specifically applied and validated for
use with TiGr FMLs. Establishment of the molecu-
lar-level simulation methodology and physical/chemical
characterization of the metal-composite interface are
described herein. The coupled continuum-level modeling
approach and experimental validation of the multi-scale
framework will be covered in detail in a companion
2. Tigr Processing and Interfacial
2.1. Titanium Surface Treatment
In the fabrication of any fiber metal laminate system,
formation of a strong interlaminar bond between the
metal and polymer matrix composite layers of the FML
is of the utmost importance. In the specific case of tita-
nium-graphite FMLs, the metal-composite interface must
also be capable of withstanding the large residual ther-
mal stress state that is introduced during the FML con-
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solidation process. These residual stresses are caused by
a sizable mismatch in the thermal expansion coefficients
of adjacent titanium and PMC layers. In the as-received
state, the interlaminar bond strength between the un-
modified titanium and the polymer matrix is typically not
sufficient to endure these residual thermal stresses, re-
sulting in delaminatio n after completion of the cure cycle
[12,13]. Thus, in order to increase the strength and reli-
ability of the TiGr metal-composite interface, a recently
proposed chromium-free sequential acid-base titanium
surface treatment procedure was employed [14]. It
should be noted that the surface treatment process vari-
ables were slightly modified to adapt the process to the
un-alloyed commercially pure (CP) titanium used in this
The ability of this chromium-free chemical treatment
process to physically alter the CP titanium surface is
evident from comparison of Figure 2 and Figure 3,
which show scanning electron microscope (SEM) images
obtained using a JEOL 6700F field emission SEM for
pre- and post-treated CP titanium foil. In Figure 2, the
non-uniform preexisting surface oxide layer is preferen-
tially aligned with the rolling direction of the foil, and
only provides partial surface coverage. After completion
of the surface treatment process, this preferential orienta-
tion disappears and full surface coverage is achieved, as
shown in Figure 3. Coverage of the CP titanium surface
with a protective titanium dioxide (TiO2) layer is the
primary basis for enhanced interlaminar bond strength
obtained using this sequential acid-base surface treat-
ment method [14]. Additionally, comparison of the
as-received and surface treated titanium foil surfaces
shown respectively in Figure 2 and Figure 3 indicates
that the surface treated titaniu m possesses a much higher
degree of surface roughness in comparison the as-re-
ceived titanium. This increased surface roughness also
contributes to the interlaminar bond strength improve-
ment, since it represents a larger active surface area be-
tween the polymer matrix adhesive and the titanium ad-
2.2. Characterization of the Titanium Dioxide
Surface Layer
In the preceding discussion, it was assumed a priori that
changes in the CP titanium surface morphology, exhib-
ited in Figure 3, were attributable to the formation of a
titanium dioxide layer on the surface of the foil. How-
ever, it should be noted that such chemical changes can-
not be determined through visual inspection alone (i.e.
scanning electron or optical microscopy). Therefore,
elemental characterization of the surface treated CP tita-
nium foil was required to confirm an increase in oxygen
content at the foil surface, consistent with the presence of
a titanium dioxide layer. To quantify the chemical com-
position of the material, a total of four 6.35 mm by 6.35
mm samples were cut from random locations in two sur-
face treated and as-received titanium foil sheets. These
samples were then analyzed using an energy dispersive
X-ray (EDX) spectroscopy system attached to the JEOL
scanning electron microscope. Figure 4 provides the
representative EDX spectra for both the as-received and
surface treated titanium foil, with each spectrum normal-
ized and offset for clarity. From this figure and the ele-
mental reference peak energy values listed in Table 1, it
is evident that the surface treated titanium foil exhibits a
clear peak corresponding to the oxygen Kα x-ray radia-
tion energy (0.525 keV). Comparatively, the as-received
titanium foil contains only Ti peaks (Ti Kα, Ti Kβ, and
Ti LI), which is expected for an unalloyed and theoreti-
cally uncontaminated material. From the EDX measure-
ments, the oxygen Kα peak obtained for the surface
treated samples indicates the presence of the previously
assumed oxide layer on the titanium foil surface.
While the comparative EDX spectra of Figure 4 are
ideal for determining the elemental composition of a
given material, they provide no information as to its
crystalline structure. This distinction is particularly im-
portant for titanium dioxide, since it naturally occurs in
three distinct polymorphs; rutile, anatase, and brookite.
Of these three, rutile, represents the most common phase
[15]. Given that the positions of the titanium and oxygen
atoms within the TiO2 lattice and the atomic packing
density are both functions of the crystallin e phase, struc-
tural characterization is essential for accurate molecu-
lar-level modeling. In order to determine the crystalline
structure of the titanium dioxide surface layer, a total of
four 5.08 cm by 2.54 cm samples were cut from random
locations in the same as-received and surface treated
titanium foil sheets. These samples were subsequently
analyzed using a Panalytical X’Pert Pro X-ray powder
diffraction (XRD) system. X-ray intensity was measured
for each sample at room temperature over a detector an-
gle ( 2
) range of 25˚ to 45˚ using a Cu-target X-ray
tube with wavelength of 1.542 Å, operated at 1800 watts.
Normalized X-ray diffraction patterns obtained using
these settings are shown in Figure 5 and Figure 6 for the
as-received and surface treated CP titanium samples,
respectively. As with the EDX spectra discussed previ-
ously, there is an eviden t distinction between the diffrac-
tion patterns of the as-received and surface treated tita-
nium foil. As expected, detector angles corresponding to
the X-ray intensity peaks of Fig ure 5 show go od co rr ela-
tion with the powder diffraction file (PDF) reference
peak locations for unalloyed α-phase titanium, with the
highest intensity peaks occurring at detector angles of
Copyright © 2011 SciRes. OJCM
Figure 2. Scanning electron micrograph (2000× magnification) of the as-received CP titanium foil surface showing preexist-
ing oxide layer pref erentially align ed w it h foil rolling direction (length of the page).
Figure 3. Scanning electron micrograph (2000× magnification) of the treated CP titanium foil surface.
38.3˚ and 40.2˚ [16]. The detector angles which corre-
spond to observed peaks in the as-received samples and
correspond to PDF reference peak locations for α-phase
titanium are provided in Table 2. In comparison, inves-
tigation of the surface treated titanium XRD pattern of
Figure 6 shows the same α-phase Ti peaks, although at
reduced relative intensities and slightly shifted to lower
detector angles. Additionally, a series of peaks attribut-
able to the titanium dioxide surface layer are also evident
in the treated CP titanium samples. In Figure 6, the
highest intensity peak now resides at a detector angle of
27.4˚. Comparison of this XRD pattern with the PDF
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Figure 4. EDX spectra for as-received (bottom) and surface treated (top) CP titanium foil with elemental reference peak in-
Table 1. Measured EDX energy values for as-received and surface treated CP Ti.
X-Ray Energy (keV)
EDX Spectrum Type Ti L O Kα Ti Kα Ti Kβ
As-Received CP Titanium 0.39 - 4.52 4.94
Surface Treated CP Titanium 0.43 0.52 4.52 4.93
Reference Peak Energies 0.40 0.53 4.51 4.93
Table 2. Measured XRD peak detector angles for as-received surface treated CP Ti.
Peak Locations - Detector Ang le (2θ)
Spectrum Type Peak 1 Peak 2 Peak 3 Peak 4 Peak 5 Peak 6 Peak 7 Peak 8
As-Received CP
Titanium - 35.08 - 38.34 - 40.15 - -
Surface Treated CP
Titanium 27.44 34.90 36.07 37.72 39.18 39.79 41.25 44.02
PDF 44-1294
(α-Ti) - 35.09 - 38.42 - 40.17 - -
PDF 21-1276
(Rutile TiO2) 27.41 - 36.03 - 39.21 - 41.23 44.07
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Copyright © 2011 SciRes. OJCM
Figure 5. XRD pattern for as-received CP titanium with reference peak locations.
Figure 6. XRD pattern for surface treated CP titanium with reference peak locations.
references for titanium dioxide polymorphs indicates that
the titanium dioxide surface layer is in the rutile crystal-
line phase, as indicated by the peak location detector
angles provided in Table 2 [17].
2.3. Titanium-Graphite Fiber Metal Laminate
Following characterization of the titanium dioxide sur-
face layer, a set of two 10.16 cm by 15.24 cm tita-
nium-graphite fiber metal laminate panels were fabri-
cated, each with an identical [0/45/Ti/-45/90]s stacking
sequence. A [0/45/Ti/-45/90]s layup was chosen since it
had been shown previously to provide the optimum spe-
cific properties (property ratio with respect to density)
for mechanically fastened titanium-graphite fiber metal
laminate joints [18]. The polymer matrix composite lay-
ers of these TiGr FML panels consisted of unidirectional
carbon fiber-reinforced toughened epoxy prepreg tape
(CYCOM 977-3/IM7), obtained from Cytec Industries.
Panel consolidation was performed in a 2 ft by 5 ft
Thermal Equipment autoclave, according to the prepreg
manufacturer’s suggested cure cycle, with a maximum
temperature of 177˚C and a sustained application pres-
sure of 0.586 MPa [19]. After completion of the panel
cure cycle, an optical microscopy investigation of each
TiGr panel cross-section indicated that the panels were
well consolidated, with the metal-composite interfaces
free of discontinuities, as shown in Figure 7. In addition,
enhanced magnification SEM images obtained at the
metal-polymer interfaces revealed excellent continuity
between the dissimilar materials down to the mi-
cron-scale, as evidenced by Figure 8. In this figure, the
bright line at the boundary between the titanium rein-
forcement and the epoxy matrix corresponds to the tita-
nium dioxide surface layer. Since titanium dioxide is
highly resistive, particularly with respect to the titanium
substrate and the IM7 carbon fibers, charging effects
from the SEM electron beam make the TiO2 layer highly
3. Simulation of the Interfacial Molecular
3.1. Computational Methods for Molecular
In order to numerically reproduce the bonded metal-
composite interface shown in Figure 8, the molecular
basis for adhesion between the titanium dioxide and ep-
oxy constituents was evaluated using molecular dynam-
ics (MD) simulations. Molecular dynamics is essentially
a numerical solution to a general n-body problem, in
Figure 7. Optical micrograph (100× magnification) of a
[0/45/Ti/-45/90]s TiGr FML cross-section.
which the interaction physics between neighboring bod-
ies (atoms) are approximated using a pre-defined force-
field or potential energy expression. Given that these
simulations are classical in nature, quantum mechanics
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Figure 8. Scanning electron micrograph (3000 × magnification) of the titanium/45° composite interface in a [0/45/Ti/-45/90]s
effects or other related phenomena are not considered. In
this analysis, commercially available MD simulation
software (Materials Studio) from Accelrys Inc. was used
to generate molecular-level structure and property pre-
dictions for metal-composite interfaces in TiGr FMLs.
The Materials Studio software suite contains a version of
the condensed-phase optimized molecular potentials for
atomistic simulation studies (COMPASS) forcefield, an
ab initio forcefield which has been parameterized and
validated using condensed-phase properties [20]. In the
COMPASS forcefield, the general relationship between
any two atoms in the simulation can be defined by the
following functional fo rm of the potential energy expr es-
sion, which is a linear combination of all possible
bonded (B), cross-term (CT), and non-bonded (NB) in-
teractions [20]:
totalB CTNB
UUUU (1)
Specific forms of the bonded and cross-term interac-
tions in the above potential energy expression can be
obtained from literature. However, since this analysis is
primarily concerned with the non-bonded interactions
between atoms that describe the behavior of physisorbed
interfaces, the functional form of UNB is provided in
Equation (2):
i jijij
NB ij
ij ij
qqA B
Ur rr
In the above equation, the non-bonded interaction en-
ergy component consists of electrostatic and Leo-
nard-Jones potential terms, where rij is the spatial posi-
tion vector between the two atoms and qi and qj are the
atomic charges. Additionally, Aij, Bij, and ε are mate-
rial-dependant constants, which must be determined for
any unique interaction between two atoms in the simula-
tion. A major advantage of a commercially available
software package, such as Materials Studio, is that the
material constants necessary in the evaluation of Equa-
tion (2) have been previously determined for a variety of
atomic configurations. Therefore, no additional parame-
terization of the COMPASS forcefield is required.
3.2. Molecular Modeling of the Metal-Composite
To simulate the mechanics of adhesion at the titanium
dioxide-epoxy interfaces in a TiGr FML, the COMPASS
forcefield and the functional forms of the system’s total
potential energy and the non-bond interaction energy
provided in Equations (1)-(2) were used in a series of
MD simulations to construct represen tative models of the
molecular-level interfacial structure. Formation of a tita-
nium dioxide simulation cell was the first step in the
construction of this model, requiring knowledge of the
TiO2 lattice parameters and atomic positions, as well as
the exposed crystalline surface. From the X-ray diffrac-
tion pattern of Figure 6, it was previously determined
that the CP titanium reinforcement layers in the TiGr
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FMLs possessed a rutile-phase titanium dioxide surface
structure. The unit cell of this particular TiO2 polymorph
is tetragonal, with a TiO6 octahedron forming the basic
structural unit of the crystal. The lattice parameters and
atomic coordinates necessary for molecular modeling of
rutile TiO2 are well established in the literature, and are
listed in Table 3 [22]. However, it should be noted that
the exterior faces of the unit cell formed by these lattice
parameters do not necessarily correspond to the exposed
TiO2 crystallographic plane at the surface of the CP tita-
nium reinforcement. In rutile TiO2, the predominately
exposed crystalline cleave plane is the (110) face, and
thus all titanium dioxide surface models presented in this
study utilize TiO2 crystals terminated with the (110)
plane [23]. Because the dimensions of a single TiO2 unit
cell are much too small to produce physically relevant
results, the simulation cell surface area (As) was in-
creased by extending the crystal in each of the in-plane
directions, as illustrated in Figure 9. This TiO 2 supercell
was approximately tetragonal with dimensions of
A=20.71 Å, B=19.47 Å, and C=14.94 Å. Prior to layer-
ing the polymer molecules on top of this TiO2 supercell,
a minimization of the sup ercell total p oten tial en ergy was
performed using the COMPASS forcefield to obtain the
equilibrium configuration of the exposed surface. As
expected, the positions of all atoms in the TiO2 supercell
did not significantly change following energy minimiza-
tion, confirming the accuracy of the lattice parameters
and atomic coordinates provided in Table 3.
After building the titanium dioxide surface model, the
interface between the metal and composite constituents
in a TiGr FML was replicated at the molecular-level by
layering a cross-linked epoxy structure on top of the en-
ergy minimized TiO2 surface. Exclusion of the reinforc-
ing fibers from the molecular-level interface model is a
reasonable assumption, since SEM investigation showed
that the carbon fiber-titanium separation distance was
several orders of magnitude larger than the MD unit cell.
It should also be noted that the stipulation of a
cross-linked epoxy layer complicated the analysis
somewhat, since the number and location of these
cross-links was not known a priori. Therefore, it was
first necessary to model the transformation of the linear
epoxy molecules into a highly networked structure be-
fore analyzing the resultant interface. Given that the be-
havior of the epoxy resin near the interface will, in gen-
eral, deviate from the behavior of the bulk epoxy, the
MD cross-linking simulations were performed in the
presence of the TiO2 surface. Although the actual for-
mulation of the epoxy resin (CYCOM 977-3) is proprie-
tary, it is widely assumed that this material is a tetragly-
cidyl methylene dianiline (TGMDA) and dia-
mino-diphenyl sulfone (DDS) blend, where the chemical
Figure 9. Titanium dioxide unit cell and supercell configurations used for molecular simulation.
Table 3. Lattice parameters and symmetry atomic coordinates for rutile TiO2 [22].
TiO2 Lattice Quantity a b c
Lattice Parameters (Å) 0.4595 0.4595 0.2959
Ti Atomic Position (Fractional Coordinates) 0.0000 0.0000 0.0000
O Atomic Position (Fractional Coordinates) 0.3048 0.3048 0.0000
Copyright © 2011 SciRes. OJCM
structures and molecular models for each linear epoxy
constituent are shown in Figure 10 [24]. The individual
epoxy constituents presented in Figure 10 were then
energy-minimized and a total of eight distinct amorphous
polymer cells were created, containing 74.5% TGMDA
to 25.5% DDS by weight. Due to the highly unstructured
nature of the liquid resin, these eight independent epoxy
configurations were required to effectively ‘average out’
configuration-space effects. The planar dimensions of the
amorphous epoxy cells were set equal to the lattice pa-
rameters of the TiO2 surface (A=20.71 Å and B=19.47
Å), with the cell height constrained by the liquid-state
density of the epoxy resin (1.1 6 g/cm3), determined from
the manufacturer’s datasheet [19]. Each of the eight in-
dependent configurations were then layered on top of the
titanium dioxide surface and a 20 Å vacuum spacer layer
was added to the top of the simulation cells. Figure 11
provides two-dimensional views for two of the eight unique
Figure 10. Epoxy resin monomer (TGMDA) and cross-linking agent (DDS) molecular structure.
Figure 11. MD interface model for the TiO2-line a r ep o x y system.
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TiO2-linear epoxy interface configurations, each con-
taining 1630 atoms. In the figure, it is important to note
that periodic boundary conditions were applied for the
in-plane directions (1- and 2-directions in Figure 11),
but the presence of the vacuum spacer eliminated perio-
dicity in the thickness (3) direction.
In order to imitate formation of the networked epoxy
structure, molecular dynamics cross-linking simulations
were performed using the COMPASS forcefield with an
isochoric-isobaric (NVT) canonical ensemble for each of
the eight TiO2-linear epoxy simulation cells. In each case,
the equilibrium configuration of the interface was deter-
mined from 100 ps of dynamics time at 450K, which
corresponded to the maximum applied temperature in the
TiGr consolidation process [19]. The time step between
successive dynamics increments was set at 1 fs, and a
cutoff radius of 9.5 Å was used to determine the maxi-
mum separation distance (rij) at which the non-bond en-
ergy of Equation (2) was evaluated for each atom pair.
Given that the potential energy of the TiO2 surface had
previously been minimized, the positions of all atoms in
the titanium dioxide supercell were fixed in place during
the molecular dynamics run. A fixed surface assumption
is valid since the TiO2 lattice parameters and atomic co-
ordinates do not significantly vary (0.16%) over the
temperature range of interest (298K to 450K) [22].
During the MD analysis, trajectories for all epoxy atoms
in the model were stored at every 5 ps of simulation time,
and were used to predict the resultin g cross-linked struc-
ture according to the method outlined by Yarovsky and
Evans [25]. In this method, cross-link formation in a
thermosetting polymer is assumed to be directly related
to the distance between the active sites of the linear
polymer and cross-linking agent molecules. When these
sites are within a specified interaction radius (6.0 Å in
this study), cross-linking will occur. For a TGMDA/DDS
system, the active sites for each molecule correspond to
the epoxy (-C-O-C-) and amine (NH2) functional groups,
as illustrated in Figure 12. Replication of this cross-
linking process was performed by calculating the posi-
tion of each active epoxy site from the MD trajectories
and applying a nearest neighbor search algorithm to
identify the corresponding amine site in closest prox-
imity. If the distance between these two groups was less
than or equal to the interaction radius, then the corre-
sponding TGMDA and DDS molecules were connected
in the MD simulation cell according to Figure 12. Ap-
plication of this method to all eight TiO2-linear epoxy
interface models resulted in an average cross-link effi-
ciency of 83.9% after 100 ps of dynamics time.
It is important to point out that the geometric ap-
proximation of an essentially chemical process was ne-
cessitated by the classical nature of molecular dynamics
simulations. Because MD analyses are based upon nu-
merical solution of the Newtonian equations of motion,
charge transfer phenomena associated with the formation
and breakage of chemical bonds cannot be explicitly
modeled using MD techniques without the inclusion of
quantum mechanics effects. Given that quantum calcula-
tions for a system containing thousands of atoms become
prohibitively exp ensive from a computational standpo int,
the above geometric method represents the best predic-
tive strategy for obtaining the fin al cross-linked configu-
ration of the epoxy resin. Once the networked epoxy
configurations had been formed, the final structure of
each TiO2-epoxy interface was determined from a 200 ps
molecular dynamics run at room temperature (298K).
MD analyses of the cooled interface were, once again,
performed using the COMPASS forcefield and an NVT
ensemble. In this manner, a total of 300 ps of MD simu-
lation time were used to determine the equilibrium state
of each interface configuration; 100 ps for the linear ep-
oxy at 450K and 200 ps for the cross-linked system at
Figure 12. TGMDA/DDS epoxy cross-linking reaction.
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4. Molecular-Level Interfacial Property
4.1. Interfacial Predictions across Multiple
Length Scales
The room temperature molecular configurations obtained
for each of the eight simulation cells were then used to
generate mechanical property predictions for the metal-
composite interfaces in TiGr FMLs. In this analysis, the
properties of interest are the transverse elastic constants
(E33, G13, G23), ultimate normal and shear interfacial
strength values (33
), corresponding strains at
failure (33
), and fracture energies (33
U). However, it should be noted that only three of
these component groups are independent, due to the in-
tegral relationship between fracture energy and the
stress-strain curves in each of the normal and shear
modes. In this analysis, strain at failure was chosen as
the interfacial dependent variable set, since its resolution
from MD simulations contains the greatest ambiguity. To
determine the remaining independent interfacial vari-
ables, a series of calculations were performed using the
predicted TiO 2-epoxy molecular structures obtained after
300 ps of MD simulation time. As will be discussed in
the companion study, these MD-derived mechanical
properties were integrated into a continuum-level finite
element analysis via a traction-separation constitutive
law. This propagation of molecular-level considerations
up to the continuum-level is what gives the current
model its multi-scale nature.
4.2. Calculation of Interfacial Fracture Energy
Of these interfacial mechanical properties, fracture en-
ergy is by far the most straightforward to determine from
a molecular dynamics simulation. This is due to the
analogous relationship between fracture energy and work
of adhesion (Wad) for a bonded interface, in which ‘fail-
ure’ is defined by separation of the heterogeneous inter-
face into two distinct homogeneous components. For a
chemical or mechanical bond between two materials,
work of adhesion is the fundamental thermodynamic
quantity which defines their dissociation at the molecu-
lar-level. Experimental measurements of the work of
adhesion are difficult to obtain and often overestimate
the stability of the interface This is due to secondary en-
ergy dissipation mechanisms in the sample which do not
contribute to debonding (e.g. specimen relaxation and
plasticity) [26]. Calculation of the work of adhesion in a
molecular dynamics simulation is comparatively simple,
since Wad is linearly dependent on the interaction energy
(Uint) between the interface components:
Where (As)I represents the surface area of the interface
with normal direction (I) and the interaction energy is
defined by the total system energy and the surface free
energies of each component of the interface (TiO2 and
epoxy) as per the Dupré relationship [27]:
int TiO
total epoxy
UUUU (4)
Computation of each term in Equation (4) in an MD
analysis is straightforward because potential energy must
be determined at each time step in order to solve the
governing equations of motion. Therefore, for each of
the eight TiO2-epoxy interface configurations, the work
of adhesion at 298K was determined by calculating the
potential energy o f the total system as well as that of the
epoxy and titanium dioxide surfaces using the COM-
PASS forcefield at the final MD time step (t = 200 ps).
Using this method, the predicted work of adhesion for a
TiO2-epoxy interface was found to be 1.12 ± 0.03 J/m2,
where the uncertainty represents one standard deviation
over all eight configurations. Because the surface area
component of Wad in Equation (3) is always normal to
the interface, it was assumed that the work of adhesion
and fracture energy were mode-independent (i.e. Wad=
U, 13
U, 23
U). It should also be noted that while use
of the final molecular trajectory (t = 200 ps) in the work
of adhesion calculations was somewhat arbitrary, it did
not greatly impact the predictio ns. This is due to the fact
that the potential en ergy of each configur ation converg ed
to an equilibrium state after approximately 7 ps of dy-
namics time, as shown in Figure 13.
4.3. Calculation of Interfacial Normal and Shear
In addition to fracture energy, the interfacial strength in
the shear and normal deformation modes was also calcu-
lated from the equilibrium configurations of the
TiO2-epoxy interface using energy considerations and
the COMPASS forcefield [28]. From the potential en-
ergy form in Equation (1), the inter- and intra-molecular
forces (FI) acting on any atom in the MD simulation are
related to the atomic total potential energy through the
gradient operator:
II ij
where summation is not applied for all lower case sub-
scripts. While this equation is essential for the solution of
the classical equations of motion in any MD analysis, it
can also be used to calculate the tensile or shear stresses
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Figure 13. TiO2-epoxy total potential energy as a function of MD simulation time at 298K for a representative interface con-
acting on an interface. Focusing solely on the interaction
between the TiO2 surface and the cross-linked epoxy
layer, the intra-atomic forces required for bonding (FIint)
can be obtained from Equation (5) if the total potential
energy is replaced by the interaction energy, previously
defined in Equation (4 ). Utilizing th e classical mechanics
definition of stress (σIJ) as the action of a force (FJ) on a
differential area with normal vector (nI), the state of
stress at the interface can be completely described by the
interaction energy between the constituents:
Since the interfacial surface area remains approxi-
mately constant in an MD simulation performed using
the NVT canonical ensemble, its value can be incorpo-
rated into the gradient operator. The interfacial stress
tensor can be further simplified using the definition of
work of adhesion provided in Equation (3):
 (7)
This represents the general form of the stress tensor
associated with any arbitrary deformation of the interface.
However, this analysis is only concerned with three dis-
tinct stress states (33
), and thus the gradient in
Equation (7) can be replaced with an ordinary differen-
tial operator, provided that deformation of the interface
occurs in a purely normal or shear mode. For example,
considering a state of pure tensile stress applied to the
33 3
where δ3 in the above equation represents the interfacial
spacing between the titanium dioxide surface and the
cross-linked epoxy layer. In order to evaluate the
right-hand side of Equation (7), the work of adhesion
was calculated at a finite number of interfacial separation
distances, shown schematically for two extreme values
of δ3 in Figure 14. Interfacial stresses were then com-
puted at these discrete sampling points using a sec-
ond-order centered finite difference approximation of
Equation (7). Considering a purely tensile deformation
mode, Equation (7) reduces to Equation (8) and this ap-
proximation becomes:
333333 33
28 82
adadad ad
 
pyright © 2011 SciRes. OJCM
Figure 14. Interfacial spacing convention for calculation of tensile strength.
where () represents the incremental spacing between
successive measurements of Wad. Application of Equa-
tion (9) to the TiO2-epoxy MD equilibrium structures
generated in the preceding MD analysis produced a se-
ries of tensile stress separation curves, shown for two
representative interface configurations in Figure 15. As
expected, the interfacial stress is highly compressive at
small separation distances (δ3 1.5Å), with its limit be-
coming unbound ed as δ3 decays to zero. This behavior is
a consequence of the repulsive component of the Leo-
nard-Jones potential energy, which dominates non-
bonded interaction s at small atomic separation distances.
Focusing on the bounded region of Figure 15, the inter-
facial tensile strength for each configuration was deter-
mined from the maximum point on the respective
stress-separation curve. The average predicted tensile
strength (33
) over all eight config urations was found to
be 23.77 ± 2.34 MPa, where the uncertainty once again
represents the measured standard deviation. This same
analysis was also applied to determine the shear strength
of the interface (13
), with periodicity in one of the
in-plane directions eliminated to simulate the presence of
a free surface. The numerical methods outlined in Equa-
tions (8)-(9) were analogous for the determination of
shear strength, with the only difference being that the
interfacial separation distance (δ3) was replaced in these
equations by the interfacial shear displacement between
the TiO2 surface and the cross-linked epoxy layer. The
calculated strength values in each shear mode were ap-
proximately equal, and thus it was assumed that the
transverse shear strength showed no directional depend-
ence (13
). Averaged over both shear deformation
modes for all eight configurations, the predicted shear
strength of the interface was equal to 18.61 ± 1.54 MPa.
4.4. Calculation of the Interfacial Elastic
After determination of the fracture energy and strength in
both the normal and shear modes, the interfacial elastic
constants (E33, G13, G23) represented the only remaining
parameters required for characterization of the trac-
tion-separation constitutive model. In this analysis, the
elastic constants were determined from the MD-pre-
dicted structure of the TiO2-epoxy interface using the
small-strain procedure of Theodorou and Suter [29]. In
this method, the stiffness tensor of a material can be at-
omistically determined through slight perturbations of
the system away from equilibrium, with subsequent
measurements of the resulting change in total potential
energy. The ‘small-strain’ provision of this method im-
plies equivalence of the material and spatial configura-
tions, such that all second-order deformation gradient
terms are much smaller than their first-order counterparts.
Assuming that this assumption is valid, the isothermal
stiffness tensor (CIJKL) of the interface can be defined in
terms of the second derivative of the Helmholtz free en-
ergy (A) with respect to the strain tensor (εIJ), normalized
by the volume of the simulation cell (V):
In the above equation, the isothermal requirement is
relatively inconsequential in this analysis since all MD
simulations were performed using the NVT ensemble.
Thus, the temperature of the system rapidly converged to
its specified value after only a few picoseconds of dy-
namics time, illustrated for a representative interface
configuration in Figure 16. Equation (10) can be further
generalized using the definition of Helmholtz free energy
as a function of the system total potential energy, tem-
perature (T), and entropy (S):
UTS (11)
Substitution of Equation (11) into Equation (10) and
expansion of the second-order strain tensor derivative for
an isothermal system results in:
Copyright © 2011 SciRes. OJCM
Figure 15.TiO2-epoxy interfacial tensile stress vs. separation distance for selected configurations.
Figure 16. TiO2-epoxy simulation cell temperature convergence to the NVT control point (298K) for a representative con-
pyright © 2011 SciRes. OJCM
22 2total
 
 
 
(12) (12)
In their detailed analysis of amorphous glassy poly-
mers, Theodorou and Suter showed that entropic contri-
butions to the above equation are relatively minor, and
thus their effects can be neglected [29]. Using the same
assumption, the material stiffness tensor can be ap-
proximated by the second derivative of the system’s total
potential energy with respect to the strain tensor, pro-
vided that the vibratory contributions to the internal en-
ergy of the system are negligible:
In their detailed analysis of amorphous glassy poly-
mers, Theodorou and Suter showed that entropic contri-
butions to the above equation are relatively minor, and
thus their effects can be neglected [29]. Using the same
assumption, the material stiffness tensor can be ap-
proximated by the second derivative of the system’s total
potential energy with respect to the strain tensor, pro-
vided that the vibratory contributions to the internal en-
ergy of the system are negligible:
Expression of any mechanical quantity in terms of the
system’s total potential energy is highly advantageous
since determination of the potential energy is a necessary
condition for solution of the classical equations of mo-
tion in an MD analysis. Using the approximation of
Equation (13), the individual stiffness components were
determined by subjecting the equilibrium interface con-
figurations to a series of small strain deformations and
measuring the resultant change in potential energy.
Switching from tensorial to reduced index notation and
considering a state of pure tensile strain normal to the
interface, the strain second-derivative term greatly sim-
plifies, resulting in the following diagonal component of
the stiffness matrix:
33 2
The other 35 diagonal and off-di agonal componen ts of
CIJ were determined analogously using small perturba-
tions of the interface away from its equilibrium configu-
ration in various deformation modes. It is important to
note that calculation of all 36 components of the stiffness
matrix was necessary, since the method outlined in
Equation (10)-(13) will generally produce a fully popu-
lated, asymmetric stiffness matrix due to spatial
non-uniformity on the molecular-level. For each of the
eight TiO2-epoxy interface configurations, the system
potential energy was measured using the COMPASS
forcefield, with the strain second derivative evaluated
using a finite difference approximation, akin to that of
Equation (9). This produced the following average and
standard deviation values (in GPa) of the TiO2-epoxy
interfacial stiffness matrix components: (Equation (15))
In order to simplify the above result and to remove
spurious measurements related to the effects of spatial
non-uniformity, it was assumed that any component of
the stiffness matrix which was within one standard de-
viation of zero could be effectively neglected. This as-
sumption reduced Equation (15) from a generally ani-
sotropic form to an orthotropic form. Furthermore, be-
cause asymmetry of the stiffness matrix violates the re-
ciprocity theorems of classical mechanics, the remaining
non-zero off-diagonal components were averaged with
respect to their symmetry counterparts. The resulting
symmetric, orthotropic stiffness matrix averaged over all
eight TiO2-epoxy interface configurations is provided in
Equation (16) :
11.06 6.126.79000
6.12 12.606.57000
6.796.57 13.04000
0 0 02.380 0
Comparison of Equations (15) and (16) shows that av-
eraging the non-zero off-diagonal components (C12|C21,
C13|C31, C23|C32) does not significantly change any of
their values; therefore averaging was performed solely to
conform to the tenets of classical mechanics. Inversion of
the stiffness matrix in Equation (16) yields the compli-
ance matrix, from which the requisite elastic constants of
the interface were found to be; E33 = 8.01 GPa, G13 =
2.38 GPa, and G23 = 2.37 GPa. As was the case for the
interfacial shear strength and fracture energy, the shear
moduli in each mode are effectively equivalent, and thus
their average was used as the ‘true’ value of the
mode-independent interfacial shear modulus. As a vali-
dation of the methodology of Equations (10)-(16), the
same molecular-level analysis was repeated for the bulk
epoxy material (i.e. no TiO2 surface structure included).
 1.890.521.580.181.89
6.45 2.6612.60 3.236.764.430.682.200.61 1.770.151.52
6.11 3.716.392.6213.048.931.21 1.60 1.810.252.48
  
 
 2.38 0.670.040.860.230.36
0.741.040.42 2.101.86 5.950.432.102.370.580.05 0.89
0.45 0.831.071.533.037.621.213.770.02 0.632.44 0.59
 
 
 
 
 
 
 
 
 
 
pyright © 2011 SciRes. OJCM
The bulk system, shown in Figure 17, underwent an
identical simulation procedure; amorphous cell construc-
tion, elevated temperature cross-linking, room tempera-
ture cooldown, and small-strain perturbations from equi-
librium, for a total of eight different spatial configura-
tions. The resultant stiffness matrix for the bulk epoxy
was passed through the same anisotropy and asymmetry
filters. Neglecting small molecular-level spatial non-
uniformity, the isotropic elastic constants of the material
(E = 3.66 GPa an d G = 1.41 GPa) were very close to the
values provided by the manufacturer (E = 3.52 GPa and
G = 1.30 GPa), thus va lidating th e procedu res outlined in
Equations (10)-(16) [19]. With the calculation and vali-
dation of these interfacial elastic constants, all requisite
molecular mechanical properties of the TiO2-epoxy in-
terface were fully determined, as outlined in the method-
ology of Equations (3)-(16). These MD obtained material
properties were subsequently used in a continuum-level
finite element model to define the mechanical behavior
of the metal-composite interfaces in a TiGr FML, de-
scribed in the second portion of th is study.
5. Summary and Conclusions
Titanium-graphite fiber metal laminates are promising
candidate materials for structural applications in which
either polymer matrix composite or monolithic metal
components are considered unfit. However, TiGr imple-
mentation has been slowed by the relatively poor metal-
composite interlaminar bond strength obtained during the
FML consolidation process. This interface often repre-
sents the ‘weakest link’ in the titanium-graphite layup,
Figure 17. MD bulk epoxy model for elastic constant vali-
and its comparatively poor mechanical performance di-
minishes the strength and weight advantages associated
with TiGr FMLs. Therefore, a comprehensive under-
standing of the mechanics of adhesion at these metal-
composite interfaces represents the first step towards
enhancement of TiGr FML int erlaminar properties.
In this analysis, construction of a molecular-level in-
terface model was discussed, with interfacial mechanical
properties obtained from thermodynamics and structural
mechanics considerations. The properties derived herein
will be propagated up to the continuum-level in the sec-
ond portion of this study, allowing for experimental
validation of the multi-scale method. It should be noted
that, as a standalone system, the molecular-level model
showed good correlation when used to predict the be-
havior of the bulk TiGr polymer matrix material. As such,
it can be concluded that the presented molecular-level
model can be utilized independent of the multi-scale
framework. Furthermore, there is no need for this model
to be restricted only to bonded metal-composite inter-
faces in titanium-graphite fiber metal laminates, or any
other FML variant. In fact, the methodology presented
herein could easily be applied to any physisorbed metal-
polymer interface; provided that the length scales associ-
ated with individual constituent materials are such that
their structure can be completely defined within the con-
fines of a molecular dynamics simulation cell. The phy-
sisorption requirement associated with this model sym-
bolizes one potential area of improvement, as only sec-
ondary bonding effects (van der Waals and electrostatic)
are presently considered between the two components of
the interface. Inclusion of covalent bonding between the
constituents would allow for extension of this method to
functionalized metal-polymer interfaces, which possess a
mixture of chemical, mechanical, and electrostatic ef-
fects. Recent advances in computational methods and
chemical simulation techniques have made this prospect
feasible, although a significant amount of research is
required for its actual implementation.
6. Acknowledgements
This study was principally supported by Raytheon Mis-
sile Systems with Andrew Facciano as the program
monitor. Additional funding was provided by the United
States Department of Education through the Graduate
Assistance in Areas of National Need (GAANN) fel-
lowship program. SEM and EDX results presented in this
study utilized additional support from NSF IGERT: Ma-
terials Creation Training Program (MCTP)—DGE-
0654431 and the California NanoSystems Institute (CNSI).
The authors would like to thank Julio de Unamuno IV of
Accelrys Inc. for his helpful discussion of the molecular
pyright © 2011 SciRes. OJCM
dynamics simulations employed in this study. Addition-
ally, special thanks go to Rachel Hundley for her in-
valuable editorial comments.
7. References
[1] J. W. Gunnink, A. Vlot, T. J. de Vries and W. van der
Hoeven, “Glare Technology Development 1997-2000,”
Applied Composite Materials, Vol. 9, No. 4, 2002, pp.
201-219. doi:10.1023/A:1016006314630
[2] J. L. Miller, D. J. Progar, W. S. Johnson and T. L. St.
Clair, “Preliminary Evaluation of Hybrid Titanium Com-
posite Laminates,” NASA TM-109095, 1994.
[3] C. A. J. R. Vermeeren, T. Beumler, J. L. C. G. de Kanter,
O. C. van der Jagt and B. C. Out, “Glare Design Aspects
and Philosophies,” Applied Composite Materials, Vol. 10,
No. 4, 2003, pp. 257-276. doi:10.1023/A:1025581600897
[4] S. Bernhardt, M. Ramulu and A. S. Kobayashi, “Low-
Velocity Impact Response Characterization of a Hybrid
Titanium Composite Laminate,” Journal of Engineering
Materials and Technology, Vol. 129, No. 4, 2007, pp.
220-226. doi:10.1115/1.2400272
[5] H. S. Seo, J. M. Hundley, H. T. Hahn and J. M. Yang,
“Numerical Simulation of Glass-Fiber-Reinforced Alu-
minum Laminates with Diverse Impact Damage,” Ameri-
can Institute of Aeronautic s and A stronautics Jo urnal, Vol .
48, No. 3, 2010, pp. 676-687. doi:10.2514/1.45551
[6] L. B. Greszczuk, “Stress Concentrations and Failure Cri-
teria for Orthotropic and Anisotropic Plates with Circular
Openings,” Proceedings of the Composite Materials Test-
ing and Design 2nd Conference, Anaheim, 1971, pp. 363-
[7] J. M. Hundley, H. T. Hahn, J. M. Yang and A. B. Fac-
ciano, “Three-Dimensional Progressive Failure Analysis
of Bolted Titani um-Graphite Fiber Metal Laminate Joints,”
Journal of Composite Materials, Vol. 45, No. 7, 2011, pp.
751-769. doi:10.1177/0021998310391047
[8] B. Kolesnikov, L. Herbeck, and A. Fink, “CFRP/Tita-
nium Hybrid Material for Improving Composite Bolted
Joints,” Composite Structures, Vol. 83, 2007, pp. 368-380.
[9] R. G. J. van Rooijen, J. Sinke, T. J. de Vries and S. van
der Zwaag, “The Bearing Strength of Fiber Metal Lami-
nates,” Journal of Composite Materials, Vol. 40, No. 1,
2008, pp. 5-19. doi:10.1177/0021998305053509
[10] G. Lawcock, L. Ye, Y. W. Mai and C. T. Sun, “The Ef-
fect of Adhesive Bonding Between Aluminum and Com-
posite Prepreg on the Mechanical Properties of Carbon-
Fiber-Reinforced Metal Laminates,” Composites Science
and Technology, Vol. 57, 1997, pp. 5-45.
[11] G. Wu, J. M. Yang and H. T. Hahn, “The Impact Proper-
ti es and Damage Tolerance of Bi-Directionally Reinforced
Fiber Metal Laminates,” Journal of Materials Science, Vol.
42, No. 3, 2007, pp. 948-957.
[12] T. Q. Cobb, W. S. Johnson, S. E. Lowther and T. L. St.
Clair, “Optimization of Surface Treatment and Adhesive
Selection for Bo nd Durability in Ti-15-3 La minates,” Jour-
nal of Adhesion, Vol. 71, 1999, pp. 115-141.
[13] P. Molitor, V. Barron and T. Young, “Surface Treatment
of Titanium for Adhesive Bonding to Polymer Compos-
ites: A Review,” Journal of Adhesion and Adhesives, Vol.
21, 2001, pp. 129-136.
[14] S. E. Lowther, C. Park and T. L. St. Clair, “A Novel Sur-
face Treatment for Titanium Alloys,” NASA TM-99-22,
[15] C. Ohara, T. Hongo, A. Yamazaki and T. Nagoya, “Syn-
thesis and Characterization of Brookite/Anatase Complex
Thin Film,” Applied Surface Science, Vol. 254, 2008, pp.
6619-6622. doi:10.1016/j.apsusc.2008.04.030
[16] Joint Committee on Powder Diffraction Standards—
International Center for Diffraction Data, “Powder Dif-
fraction File Card Number 44-1294,” Powder Diffraction
File, 1996.
[17] Joint Committee on Powder Diffraction Standards—
International Center for Diffraction Data, “Powder Dif-
fraction File Card Number 21-1276,” Powder Diffraction
File, 1996.
[18] J. M. Hundley, J. M. Yang, H. T. Hahn and A. B. Fac-
ciano, “Bearing Strength Analysis of Hybrid Titanium
Composite Laminates,” American Institute of Aeronautics
and Astronautics Journal, Vol. 46, No. 8, 2008, pp. 2074-
2085. doi:10.2514/1.36242
[19] Cytec Engineered Materials, “CYCOM 977-3 Toughened
Epoxy Resin Technical Datasheet,” 2005.
[20] Accelrys Inc., “Materials Studio User’s Manual,” Version
4.4, 2008.
[21] H. Sun, “COMPASS: An ab Initio Force-Field Optimized
for Condensed-Phase Applications, Overview with De-
tails on Alkane and Benzene Compounds,” Journal of
Physical Chemistry B, Vol. 102, No. 38, 1998, pp. 7338-
7364. doi:10.1021/jp980939v
[22] D. W. Kim, N. Enomoto, Z. E. Nakagawa and K. Kawa-
mure, “Molecular Dynamic Simulation in Titanium Di-
oxide Polymorphs: Rutile, Brookite and Anatase,” Jour-
nal of the American Ceramic Society, Vol. 79, No. 4,
1996, pp. 1095-1099.
[23] H. Knözinger, “Specific Poisoning and Characterization
of Catalytically Active Oxide Surfaces,” Advances in
Catalysis, Vol. 25, 1976, 184-271.
[24] W. Zhao, “Modeling of Ultrasonic Processing,” MS the-
sis, Massachusetts Institute of Technology, Cambridge,
[25] I. Yarovsky and E. Evans, “Computer Simulation of
Structure and Properties of Crosslinked Polymers: Ap-
plication to Epoxy Resins,” Polymer, Vol. 43, 2002, pp.
963-969. doi:10.1016/S0032-3861(01)00634-6
Copyright © 2011 SciRes. OJCM
Copyright © 2011 SciRes. OJCM
[26] S. Kisin, J. B. Vukić, P. G. T. van der Varst, G. de With
and G. C. E. Koning, “Estimating the Polymer-Metal Work
of Adhesion from Molecular Dynamics Simulations,”
Chemistry of Materials, Vol. 19, 2007, pp. 903- 907.
[27] I. Yarovsky, “Atomistic Simulation of Interfaces in Ma-
terials: Theory and Applications,” Australian Journal of
Physics, Vol. 50, 1997, pp. 407-424.
[28] D. J. Henry, C. A. Lukey, E. Evans and I. Yarovsky,
“Theoretical Study of Adhesion between Graphite, Poly-
ester and Silica Surfaces,” Molecular Simulation, Vol. 39,
No. 6-7, 2005, pp. 449-455.
[29] D. N. Theodorou and U. W. Suter, “Atomistic Modeling
of Mechanical Properties of Polymeric Glasses,” Mac-
romolecules, Vol. 19, 1986, pp. 139-154.