Interaction Force Fields
for
Molecular Mechanics
and
Molecular Dynamics Simulations
Tahir Cagin
Molecular Modeling
•
•
•
•
•
Electrons
Atoms
Segments
Domains
Continuum
Multiscale hierarchy of modeling
time
ELECTRONS
ATOMS
DOMAINS
GRIDS
hours
Continuum
(FEM)
seconds
microsec
nanosec
MESO
First principles
Force Fields
MD
picosec
femtosec
QM
Å
nm
micron
mm
meters
distance
A molecular Model of DNA
Molecular Model of a Solid
Force Fields
• van der Waals interactions
• Metal force fields – many body effects/electronic effects
• Ligancy, ionicity/covalency
• Covalent interactions
• Chemical Reactions
Finally – compliance/consistency/transferability
Force Fields
• Simple Noble Gas Solid/Liquid/Gases
• Metals and Alloys
• Oxides, Carbides, Nitrides – Ceramics, Zeolites, Clays,
Minerals
• Organic Molecules, liquids, solids
• Polymers, Composites
• Proteins, Nucleic Acids
And more importantly:
Interfaces of any of the above
Valence Force Fields
AMBER, Assisted Model Building and Energy Refinement
AMBER/OPLS
CHARMM, Chemistry at HARvard Macromolecular Mechanics
DISCOVER, force fields of the Insight/Discover
DREIDING, force fields of POLYGRAF/BIOGRAF
GROMOS, GROningen MOlecular Simulation package
MM2/MM3/MM4 , Allinger molecular mechanics FF
MMFF94, the Merck Molecular Force Field
Tripos, the force field of the Sybyl molecular modeling program
UFF, Universal Force Fields of POLYGRAF/BIOGRAF/Cerius
DREIDING/UFF/References
•Mayo S L, Olafson B D, Goddard W A, “DREIDING - A Generic Force-field For
Molecular Simulations,” J Phys Chem 94 8897-8909 1990
•Rappe A K, Casewit C J, Colwell K S, et al. UFF, A Full Periodic-table Force-field
For Molecular Mechanics And Molecular-dynamics Simulations
J Am Chem Soc 114 10024-10035 1992
•Casewit C J, Colwell K S, Rappe A K, Application Of A Universal Force-field To
Main Group Compounds
J Am Chem Soc 114: 10046-10053 1992
Force Fields
hb
Interaction types
• Non Bonded Interactions
• Electrostatics
• van der Waals interactions
• Valence Interactions
• Bond Stretch
• Angle Bending
• Torsion/Dihedrals/Improper Torsions
• 4+ body interactions
• Special Interactions
• Hydrogen bonding
• Constraints
Non Bonded Interactions
ELECTROSTATICS
Electrostatics describes the force resulting from the interaction
between two charged particles. The electrostatic energy between
two atoms i and j is described by Coulomb's Law as
qi = charge of atom i; qj = charge of atom j
= Dielectric constant
, the vector from atom i to atom j
= length of
, distance between atoms i and j
Non Bonded Interactions
The van der Waals energy between two atoms i and j is described by
where
A = Repulsion constant
B = Attraction constant
= distance between atoms i and j
Valence Interactions
The energy of a bond between atoms i and j is given by
k = spring constant
= distance between atoms i and j
r0 = equilibrium distance of the bond
The energy of such angle bending between atoms i, j, and k is given by
= force constant this angle-bending type
= calculated angle between the vector i and j and vector k and j.
0 = equilibrium value of the bond angle
Dihedrals/Improper Torsions
Dihedral and improper bonds model the interaction between 4 bonded
atoms. They are modeled by an angular spring between the planes
formed by the first 3 atoms and the second 3 atoms. The energy for a
dihedral or improper between atoms i, j, k, and l is given by
k = force constant
= angle between the plane formed by atoms (i, j, k) and the
plane formed by atoms (j, k, l)
n = periodicity of the bond specified in the parameter file for this
bond type
= phase shift
Harmonic Constraints
Harmonic constraints provide a mechanism for holding certain parts of a
molecule relatively immobile during a simulation. For instance: it allows specific
atoms to be held at a reference position. The energy for an atom i that is
constrained is
ki = force constant defined for atom i
= current position of atom I
= reference position atom i is constrained to
Evaluating Non Bond Interactions
• Exclusion issues
•Finite Systems
• All non bond interactions are evaluated
• Cut off distances are used
• Direct Cut off
• Shifted Energy
• Shifted Force
• Splines (continuity issues)
• Screening issues
• Infinite Systems
• Cut offs
• Minimum Image
• Ewald/PME/FMM methods
Splines/Switching Functions
For van der Waals interactions, the energy is modified as
switching function defined by
inner cutoff value for switching
outer cutoff value for switching
Exclusions of Valence Interactions
4
2
1
3
5
The bonded terms are excluded in evaluating non bond interactions
Bond stretch terms: 1-2; 2-3; 3-4; 4-5
Angle bend terms : 1-3; 2-4; 3-5
Optional exclusion for torsion terms: 1-4; 2-5
scale : 0 exclude
scale : 0.5 as in AMBER
scale : 1 include
AMBER/References
•Cornell, W. D., Cieplak, P., Bayly, C. I., Gould, I. R., Merz, K. M. Jr., Ferguson, D. M.
Spellmeyer, D. C., Fox, T., Caldwell, J. W., and Kollman, P. A. (1995) A second
generation force field for the simulation of proteins, nucleic acids and organic
molecules, J. Am. Chem. Soc. 117, 5179-5197.
•Pearlman, D. A., Case, D. A., Caldwell, J. C., Seibel, G. L., Singh, U. C., Weiner, P., &
Kollman, P. A., (1991) AMBER 4.0, University of California, San Francisco.
•Weiner, P. K., & Kollman, P. A., (1981) AMBER: Assisted Model Building with
Energy Refinement. A General Program for Modeling Molecules and Their Interactions,
J. Comp. Chem. 2, 287-303.
•Weiner, S.J., Kollman, P.A., Case, D.A., Singh, U.C., Ghio, C., Alagona, G., Profeta,
S., Jr., Weiner, P.K. (1984) A new force field for molecular mechanical simulation of
nucleic acids and proteins. J. Am. Chem. Soc. 106, 765-784.
•Weiner, S. J., Kollman, P. A., Nguyen, D. T., and Case, D. A., (1986) "An All Atom
Force Field for Simulations of Proteins and Nucleic Acids," J. Comp. Chem. 7, 230252.
OPLS/References
•Damm, W., A. Frontera, J. Tirado-Rives and W. L. Jorgensen (1997) "OPLS All-Atom
Force Field for Carbohydrates," J. Comp. Chem. 18, 1955-1970. Jorgensen, W. L.;
Maxwell, D. S. and Tirado-Rives, J. (1996) "Development and Testing of the OPLS AllAtom Force Field on Conformational Energetics and Properties of Organic Liquids" J.
Am. Chem. Soc., 118, 11225-11236.
•Jorgensen, W. L., & Tirado-Rives, J.,(1988) The OPLS Potential Functions for
Proteins. Energy Minimization for Crystals of Cyclic Peptides and Crambin, J. Am.
Chem. Soc. 110, 1657-1666.
•Kaminski, G., Duffy, E. M. Matsui, T., and Jorgensen, W. L. (1994) J. Phys. Chem. 98,
13077-13082.
CHARMM/References
•Brooks, B.R., Bruccoleri, R.E., Olafson, B.D., States, D.J., Swaminathan, S., Karplus, M. (1983)
CHARMM: A program for macromolecular energy, minmimization, and dynamics calculations. J. Comp.
Chem. 4, 187-217. Feller et al.,(1997) Molecular Dynamics Simulation of Unsaturated Lipids at Low
Hydration: Parameterization and Comparison with Diffraction Studies. Biophys. J. 73, 2269-2279
•MacKerell, A D ; Bashford, D; Bellott, M; Dunbrack, R L; Eva seck, J D; Field, M J; Fischer, S; Gao, J;
Guo, H; Ha, S; JosephMcCarthy, D; Kuc nir, L; Kuczera, K; Lau, F T K; Mattos, C; Michnick, S; Ngo, T;
Nguyen, D T; Pro hom, B; Reiher, W E; Roux, B; Schlenkrich, M; Smith, J C; Stote, R; Straub, J; W tanabe,
M; WiorkiewiczKuczera, J; Yin, D; Karplus, M (1998) All-atom empirical potential for molecular modeling
and dynamics studies of proteins. J. Phys. Chem., B 102, 3586-3617
•Mackerell A D ; Wiorkiewiczkuczera J; Karplus, M (1995) An all-atom empirical energy function for the
simulation of nucleic acids. J. Amer. Chem. Soc.117, 11946-11975
•Momany, F. A., & Rone, R., (1992) Validation of the General Purpose QUANTA 3.2/CHARMm Force Field,
J. Comp. Chem. 13, 888-900.
•Pavelites, J. J., J. Gao, P.A. Bash and A. D. Mackerell, Jr. (1997) "A Molecular Mechanics Force Field for
NAD+, NADH, and the Pyrophosphate Groups of Nucleotides," J. Comp. Chem. 18, 221-239.
•Schlenkrich et al. (1996), Empirical Potential Energy Function for Phospholipids: Criteria for Parameter
Optimization and Applications in "Biological Membranes: A Molecular Perspective from Computation and
Experiment," K.M. Merz and B. Roux, Eds. Birkhauser, Boston, pp 31-81, 1996
DISCOVER/References
•Dauber-Osguthorpe, P.; Roberts, V. A.; Osguthorpe, D. J.; Wolff, J.; Genest, M.;
Hagler, A. T. (1988) "Structure and energetics of ligand binding to proteins: E. coli
dihydrofolate reductase- trimethoprim, a drug-receptor system", Proteins:
Structure, Function and Genetics, 4, 31-47.
•Hagler, A. T.; Ewig, C. S. "On the use of quantum energy surfaces in the
derivation of molecular force fields", Comp. Phys. Comm., 84, 131-155 (1994).
•Hwang, M.-J.; Stockfisch, T. P.; Hagler, A. T. "Derivation of Class II force fields.
2. Derivation and characterization of a Class II force field, CFF93, for the alkyl
functional group and alkane molecules", J. Amer. Chem. Soc., 116, 2515-2525
(1994).
•Maple, J. R.; Hwang, M.-J.; Stockfisch, T. P.; Dinur, U.; Waldman, M.; Ewig, C.
S; Hagler, A. T. "Derivation of Class II force fields. 1. Methodology and quantum
force field for the alkyl functional group and alkane molecules", J. Comput. Chem.,
15, 162-182 (1994a).
•Maple, J. R.; Hwang, M.-J.; Stockfisch, T. P.; Hagler, A. T. "Derivation of Class II
force fields. 3. Characterization of a quantum force field for the alkanes", Israel J.
Chem., 34, 195 -231 (1994b).
Gromos/References
•Hermans, J., Berendsen, H. J. C., van Gunsteren, W. F., & Postma, J. P. M., (1984) "A
Consistent Empirical Potential for Water-Protein Interactions," Biopolymers 23, 1 Ott,
K-H., B. Meyer (1996) "Parametrization of GROMOS force field for oligosaccharides
and assessment of efficiency of molecular dynamics simulations," J Comp Chem 17,
1068-1084>
•van Gunsteren, W. F., X. Daura and A.E. Mark (1997) "The GROMOS force field" in
Encyclopaedia of Computational Chemistry ()
MM2/MM3/MM4/References
•Lii, J-H., & Allinger, N. L. (1991) The MM3 Force Field for Amides, Polypeptides and
Proteins, J. Comp. Chem. 12, 186-199.
•Lii, J-H., & Allinger, N. L. (1998) Directional Hydrogen Bonding in the MM3 Force
Field. II. J. Comp. Chem. 19, 1001-1016.
•Allinger, N. L., K. Chen, and J-H Lii (1996) "An Improved Force Field (MM4) for
Saturated Hydrocarbons," J. Comp. Chem. 17, 642-668. Allinger, N. L., K. Chen, J. A.
Katzenellenbogen, S. R. Wilson and G. M. Anstead (1996) "Hyperconjugative Effects
on Carbon-Carbon Bond Lengths in Molecular Mechanics (MM4)" J. Comp. Chem. 17,
747-755.
•Allinger, N. L., and Y. Fan (1997) "Molecular Mechanics Studies (MM4) of Sulfides
and Mercaptans," J. Comp. Chem.18, 1827-1847.
•Nevens, N., K. Chen and N. L. Allinger (1996) "Molecular Mechanics (MM4)
Calculations on Alkenes," J. Comp. Chem. 17, 669-694.
•Nevins, N., J-H. Lii and N.L. Allinger (1996) "Molecular Mechanics (MM4)
Calculations on Conjugated Hydrocarbons," J. Comp. Chem. 17, 695-729.
•Nevins, N., and N. L. Allinger (1996) "Molecular Mechanics (MM4) Vibrational
Frequency Calculations for Alkenes an Conjugated Hydrocarbons," J. Comp. Chem. 17,
730-746.
MM2/MM3/MM4/References
•Lii, J-H., & Allinger, N. L. (1991) The MM3 Force Field for Amides, Polypeptides and
Proteins, J. Comp. Chem. 12, 186-199.
•Lii, J-H., & Allinger, N. L. (1998) Directional Hydrogen Bonding in the MM3 Force
Field. II. J. Comp. Chem. 19, 1001-1016.
•Allinger, N. L., K. Chen, and J-H Lii (1996) "An Improved Force Field (MM4) for
Saturated Hydrocarbons," J. Comp. Chem. 17, 642-668. Allinger, N. L., K. Chen, J. A.
Katzenellenbogen, S. R. Wilson and G. M. Anstead (1996) "Hyperconjugative Effects
on Carbon-Carbon Bond Lengths in Molecular Mechanics (MM4)" J. Comp. Chem. 17,
747-755.
•Allinger, N. L., and Y. Fan (1997) "Molecular Mechanics Studies (MM4) of Sulfides
and Mercaptans," J. Comp. Chem.18, 1827-1847.
•Nevens, N., K. Chen and N. L. Allinger (1996) "Molecular Mechanics (MM4)
Calculations on Alkenes," J. Comp. Chem. 17, 669-694.
•Nevins, N., J-H. Lii and N.L. Allinger (1996) "Molecular Mechanics (MM4)
Calculations on Conjugated Hydrocarbons," J. Comp. Chem. 17, 695-729.
•Nevins, N., and N. L. Allinger (1996) "Molecular Mechanics (MM4) Vibrational
Frequency Calculations for Alkenes an Conjugated Hydrocarbons," J. Comp. Chem. 17,
730-746.
MMFF94/References
•Halgren, T. A. (1992) J. Am. Chem. Soc. 114, 7827-7843. Halgren, T. A. (1996)
"Merck Molecular Force Field. I. Basis, Form, Scope, Parameterization and
Performance of MMFF94," J. Comp. Chem 17, 490-519.
•Halgren, T. A. (1996) "Merck Molecular Force Field. II. MMFF94 van der Waals and
Electrostatic Parameters for Intermolecular Interactions," J. Comp. Chem. 17, 520-552.
•Halgren, T. A. (1996) "Merck Molecular Force Field. III. Molecular Geometrics and
Vibrational Frequencies for MMFF94," J. Comp. Chem. 17, 553-586.
•Halgren, T. A., and Nachbar, R. B. (1996) "Merck Molecular Force Field. IV.
Conformational Energies and Geometries," J. Comp. Chem. 17, 587-615.
•Halgren, T. A. (1996) "Merck Molecular Force Field. V. Extension of MMFF94 using
Experimental Data, Additional Computational Data and Empirical Rules," J. Comp.
Chem. 17, 616-641.
Molecular Dynamics
Tahir Cagin
Summary
• Molecular Dynamics Methods
– Equilibrium Molecular Dynamics Methods
• Standard MD
• Extended Hamiltonian Methods
• Coarse Grain Methods
– Non Equilibrium Molecular Dynamics Methods
• Steady State MD
• Synthetic Hamiltonian NEMD Methods
•
•
•
•
Molecular Dynamics Introduction
Statistical Mechanics and Ensembles
Static Property Calculations
Dynamic Property Calculations from Equilibrium and NE MD
–
–
–
–
Viscosity
Thermal Conductivity
Friction
Plasticity
Thermodynamic Equilibrium 1
T
P
µ
Thermodynamic Equilibrium 2
EVN
HPN
TVN
LVµ
TPN
RPµ
TVµ
TPµ
Equilibrium vs. Steady State: Mass transport
∞
1
D = ∫ dt Vαi (t )Vαi (0)
20
Mass transport: diffusion
Equilibrium vs. Steady State: Energy transport
∞


1
Λ = 2 ∫ dt ⟨ j (t ); j (0)⟩
kT V 0


∂
j (t ) = ∑ r hi
∂t i
H = ∑ hi
i
J q = −λ DT
Energy transport:
Thermal conductivity
Equilibrium vs. Steady State: Momentum transport
P+
∞
V
η=
dt σ αβ (t )σ αβ (0)
∫
kbT 0
σ =n γ
PMomentum transport:
Flow viscosity
“Molecular Dynamics” Methods
Ab initio Molecular Dynamics
Tight binding Molecular
Dynamics
Classical Molecular Dynamics
Langevin Equation
Brownian Dynamics or DPD
Kinetic Monte Carlo
Molecular Dynamics (MD)
• Pick particles, masses and forces (or potential).
• Initialize positions and momentum (boundary
conditions in space and time).
• Solve F = m a to determine r(t), v(t).
• Compute the properties along the trajectory
• Estimate errors.
• Try to use the simulation to answer physical
questions.
Criteria for an Integrator
•
•
•
•
simplicity (How long does it take to write and debug?)
efficiency (How fast to advance a given system?)
stability (what is the long-term energy conservation?)
reliability (Can it handle a variety of temperatures, densities,
potentials?)
• compare statistical errors (going as h-1/2 ) with time step errors
which go as hp where p=2,3…. Match errors to pick the time
step.
The Verlet Algorithm
The nearly universal choice for an integrator is the Verlet (leapfrog) algorithm
r(t+h) = r(t) + v(t) h + 1/2 a(t) h2 + b(t) h3 + O(h4)
r(t-h) = r(t) - v(t) h + 1/2 a(t) h2 - b(t) h3 + O(h4)
r(t+h) = 2 r(t) - r(t-h) + a(t) h2 + O(h4)
v(t) = (r(t+h) - r(t-h))/(2h) + O(h2)
1
2
3
4
5
6
7
8
Taylor expansion
Reverse time
Add
estimate vels
9
10
11
12
13
Time reversal invariance is built in: the energy does not drift either up or down.
How to set the time step
• Adjust to get energy conservation to 1% of kinetic energy.
• Even if errors are large, you are close to the exact trajectory of a
nearby physical system with a different potential.
• Since we don’t really know the potential surface that accurately, this is
satisfactory.
• Leapfrog algorithm has a problem with round-off error.
• Use the equivalent velocity-Verlet instead:
r(t+h) = r(t) +h [ v(t) +(h/2) a(t)]
v(t+h/2) = v(t)+(h/2) a(t)
v(t+h)=v(t+h/2) + (h/2) a(t+h)
Higher Order Methods?
• Higher order does not always
mean better
• Eg. Predictor Correctors have drift
in the Hamiltonian
• With proper time steps the higher
order predictor- corrector methods
may give better energy
conservation this may be
preferable for accurate evaluation
of fluctuations
Ergodicity
• Typically simulations are (assumed) ergodic:
– Roughly meaning that after a certain time the system loses memory of S0,
except possibly for certain conserved quantities like E, P etc.
–
Hence, there is a correlation time T.
– Ergodicity is often easy to prove for the random transition but usually
difficult for the deterministic simulation.
• For times much longer than T,
– all non-conserved properties are close to their average value.
• Warm up period at the beginning (equilibration)
• To get independent samples for computing errors.
– all properties of the system not explicitly conserved are unpredictable as if
randomly sampled from some distribution.
Averages and Errors
• If the system is ergodic, no matter what the initial state, one can
characterize the state of the system for t > T by a unique probability
distribution function F(S).
• At large time this will approach a unique distribution, the equilibrium state
F*(S).
– e.g., classically, the Boltzmann distribution:
P(E i ) = e−βE i / ∑i e−βE i
• One goal is to compute averages over S to get properties in equilibrium.
Another is to compute dynamics.
– e.g., Average (Mean) Energy: E = <e(S)> weighted by F*(S).
from the Trace of E: the plot of e(S) vs. n (computed).
See sampling in J.M. Haile “MD Simulation”,
Allen and Tildesley,
Averages
• We sample e(Sn) for n < Ttot, where Ttot is
total run time.
– Trace of E: the plot of e(S) vs. n.
– Must remove trace’s dependence on initial
transients (“warm-up”).
– Mean Energy: Formula vs. Estimator
• system’s data Ak (-∞ ≤ k ≤ ∞) vs computed data Ak (1
≤ k ≤ N)
• transient data: 1 ≤ k ≤ k1
Remove warm-up
• equilibrium
data: k1 ≤ k ≤ k2 with Neq= k2 - k1 +1
• <A> = limN→∞ N-1∑k Ak
vs.
a = N-1eq ∑k2k=k1 Ak
≈ <A>
Estimated Errors
• Next, how accurate is the estimate of the exact
value?

Simulation results without error bars are only suggestive.
• Error bar: the estimated error in the estimated
mean.




Error estimates based on Gauss’ Central Limit Theorem.
Average of statistical processes has normal (Gaussian)
distribution.
Error bars: square root of the variance of the distribution
divided by the number of uncorrelated steps.
Correlation time: number of steps for a fluctuation to
disappear.
Histogram of A
P(A)is Gaussian
Statistical vs. Systematic Errors
• What are systematic errors ?


Systematic error measures the deviation of the expected value
from the (trajectory) true average.
Systematic error is caused by sampling method, and round-off
error, etc.
• What are statistical errors?


•
Statistical error measures the distribution of the averages about
their avg.
Statistical error can be reduced by extending or repeating runs.
With <A> true avg, E[<a>] expected avg. (from many averages of many
runs), <a> estimated avg., and V is variance, can show that
mean-squared error is related as:
mse = (< A > −E[< a >])2 + V[E(< a >]− < x >]
= (systematic _ error)2 + (statistical _ error)2
E[(< A > − < a >)2 ] = (1−
M
)V 2 / M
M∞
problems with estimating errors
• Any good simulation quotes systematic and statistical errors for
anything important.
• The error and mean are simultaneously determined from the same
data. HOW?
• Central limit theorem: the distribution of an average approaches a
normal distribution (if the variance is finite).
– One standard deviation means ~2/3 of the time the correct answer is
within σ of the sample average.
• Problem in simulations is that data is correlated in time.
– It takes a “correlation” time T to be “ergodic”
• We must throw away the initial transient.
• Get rid of autocorrelation.
• We need ≥20 independent data points to estimate errors
Estimating Errors
• Trace of A(t):
• Equilibration time.
• Histogram of values of A ( P(A) ).
• Mean of A (a).
• Variance of A ( v ).
• estimate of the mean:
ΣA(t)/N
• estimate of the variance
• Autocorrelation of A (C(t)).
• Correlation time (κ ).
• The (estimated) error of the (estimated) mean (σ ).
• Efficiency [= 1/(CPU time * error 2)]
Ergodicity
•
•
•
In MD want to use the microcanonical (constant E) ensemble (just F=ma)!
Replace ensemble or heat bath with a SINGLE very long trajectory.
This is OK only if system is ergodic.
•
Ergodic Hypothesis: a phase point for any isolated system passes in succession through
every point compatible with the energy of the system before finally returning to its original
position in phase space. This journey takes a Poincare cycle.
•
In other words, Ergodic hypothesis: each state consistent with our knowledge is
equally “likely”.
– Implies the average value does not depend on initial conditions.
– Is <A>time= <A>ensemble so <Atime> = (1/NMD) = ∑t=1,N At is good estimator?
– True if: <A>= <<A>ens>time = <<A>time> ens = <A>time.
•
•
•
•
Equality one is true if the distribution is stationary.
For equality two, it is assumed that interchanging averages should not matter.
The third equality is only true if system is ERGODIC.
Are systems in nature really ergodic? Not always!
– Non-ergodic examples are glasses, folding proteins (in practice) and harmonic crystals
(in principle).
Different aspects of ergodicity
•
•
•
•
•
The system relaxes on a reasonable time scale towards a unique equilibrium
state.
This state is the microcanonical state. It differs from the canonical distribution
by order (1/N)
There are no hidden variable--conserved quantities--other than the energy,
linear and angular momentum, number of particles. (systems which do have
conserved quantities are integrable.)
Trajectories wander irregularly through the energy surface, eventually
sampling all of accessible phase space.
Trajectories initially close together separate rapidly. (sensitivity to initial
conditions) Coefficient is the Lyapnov exponent.
Ergodic behavior makes possible the use of statistical methods on MD
of small systems. Small round-off errors and other mathematical
approximations may not matter!
Particle in a smooth/rough circle
From J.M. Haile: MD Simulations
Static Correlations and Properties
What do we get out of a simulation?
• Static properties: energy, pressure, density, structure
• Response functions: specific heat, bulk modulus, thermal expansion, elastic
constants, etc.
• Pair correlation in real space and Fourier space.
• Order parameters and broken symmetry:
– How to tell a liquid from a solid
Thermodynamic properties
We can get Averages over Distributions
• Total (internal) Energy = kinetic energy + potential energy
• Kinetic Energy = kBT/2 per momentum (degree of freedom)
• Specific Heat = mean squared fluctuation in energy
• Pressure can be computed from the Virial Theorem.
• Compressibility, Bulk Modulus, Speed of Sound
We have Problems with
• Entropy and Free Energy because they are not ratios with respect to
Boltzmann distribution.
– We will discuss this later.
Molecular Level Modeling
BMGs
Model systems with the following compositions are
considered
Vitreloy 105: Al10-Ti5-Zr52.5-Cu17.9-Ni14.6 alloy
Vitreloy 101: Ti34Ni8Cu47Zr11 alloy
Vitreloy 4
: Be27.5 Ti8.25 Ni10 Cu7.5Zr46.75 alloy
V105: Structure from MD
4000 atom simulation cell
Elastic Properties of VIT4
ELASTIC CONSTANTS in GPa:
201.801
93.422
92.855
203.756
94.026
202.222
1.082
-0.373
-0.663
56.447
1.581
0.848
-0.247
-0.177
55.254
Isotropy Condition is satisfied : 202 - 2*56 = 90 ~ 93
Bs = 129.82 GPa
-0.999
0.360
-0.201
0.815
1.097
55.840
Elastic Properties of VIT101
ELASTIC CONSTANTS in GPa:
189.258
87.465
86.536
188.584
86.194
190.383
-0.170
1.198
0.669
50.382
0.708
0.272
-0.328
0.538
50.738
Isotropy Condition is satisfied : 190 - 2*51 = 88 ~ 86.5
Bs = 120.96 GPa
-0.742
-0.105
0.538
0.263
-0.187
51.640
Microscopic Density
Real-Space
ρ(r) = < Σi δ(r-r i) >
>
Fourier transform:
ρ k = < Σi exp(ikri)
This is a good way to smooth the density.
• In periodic boundary conditions, the k-vectors are on a grid:
k=2π/L (nx,ny,nz)
• In a solid Lindemann’s ratio gives a rough idea of melting
u2= <(ri-zi)2>/d2
When deviations about lattice are >~15%, solid has melted.
Pair Correlation Function, g(r)
Density-Density Correlations:
g(r) = ρ–2 < Σi Σj≠i δ(ri)δ (rj –r)>
= < Σi<j δ (r – (ri-rj))> (2Ω/N2)
• The last form is useful on computer – In practice, the delta-function is
replaced by binning and making a histogram.
• g(r) is related to thermodynamic properties.
Nρ 3
V = ∑i< j v(rij ) =
∫ d r rv(r)g(r)
2
g(r) for fcc and bcc lattices
1st n.n.
2nd n.n.
3rd n.n. …
distances are
arranged
in increasing
distances.
What happens at finite T?
What happens when potentials
are not hard spheres?
J. Haile, MD simulations
Variation of RDF upon cooling of
VIT105
(The Static) Structure Factor
S(K)
• The Fourier transform of g(r) is the structure factor:
S(k) = <|ρk|2>/N
S(k) = 1 + ρ dr exp(ikr) (g(r)-1)
(1)
(2)
• Problem with (2) is to extend g(r) to infinity
Why is S(K) important?
S(K) can:
• Be measured in neutron and X-ray scattering experiments.
• Provide a direct test of the assumed potential.
• Used to see the state of a system:
• Order parameter in solid is ρG where G is a wavevector.
Using S(K)
• In a perfect lattice S(k) will be non-zero only on the
reciprocal lattice vectors G: S(G) = N
• At non-zero temperature (or for a quantum system) this
structure factor is reduced by the Debye-Waller factor due
to thermal vibrations about the average lattice site:
S(G) = 1+ (N-1)exp(-G2u2/3)
• The compressibility is given by:
– We can use this is detect the liquid-gas transition since
the compressibility should diverge as k approaches 0.
– Order parameter is density.
– S(0) = N by definition, but limit of k-> 0 can be
different!
χ T = S(0)/(k BT ρ )
Basics of Statistical Mechanics
• Review of ensembles
– Microcanonical, canonical, Maxwell-Boltzmann
– Constant pressure, temperature, volume,…
• Thermodynamic limit
• Ergodicity
Thermodynamic Equilibrium 1
T
P
µ
Thermodynamic Equilibrium 2
EVN
HPN
TVN
LVµ
TPN
RPµ
TVµ
TPµ
Maxwell-Boltzmann Distribution
P(q, p)dqdp = exp(– βH(q, p)) /(N!Z)
•
•
•
Z=partition function. Defined so that probability is normalized.
Quantum expression : Z = Σ exp (-β Ei )
Also Z= exp(-β F) where F=free energy (more convenient)
•
•
In most cases H(q,p) = V(q)+ Σ p2i /2mi
Then the momentum integrals can be performed. One has simply an
uncorrelated Gaussian distribution of momentum.
On the average, there is no relation between position and velocity!
Microcanonical is different--think about harmonic oscillator.
Equipartition: each momentum d.o.f. carries 1/2 kBT of energy
•
•
•
<p2i /2mi>= (1/2)kB T
Thermodynamic limit
• To describe a macroscopic limit we need to study how systems
converge as N⇒∞ and large time.
• Sharp, mathematically well-defined phase transitions only occur in this
limit. Otherwise they are not sharp.
• It has been found that systems of as few as 20 particles with only
thousand of steps can be close to the limit if you are very careful with
boundary conditions (space/time).
• To get this behavior consider whether:
– Have your BCs introduced anything that shouldn’t be there? (walls,
defects, voids etc)
– Is your box bigger than the natural length scale of the considered phase.
(for a liquid/solid it is the interparticle spacing)
– The system starts in a reasonable state.
Temperature and Pressure
Controls
Ensembles
1.(E, V, N) microcanonical, constant energy
2.(T, V, N) canonical, constant volume
3.(T, P N) constant pressure
4.(T, V , µ) grand canonical
• Often #2, 3 or 4 are better to get macroscropic
properties
Constant Temperature MD
• Problem in MD is how to control the temperature.
• How to start the system?
– Starting from a perfect lattice upon thermalizing the
lattice absorbs the kinetic energy and cools down.
– Doubling T in assignment might help but not enough.
– Equilibrate the model first, by using a scaling method
• Strict temperature scaling methods.
• Differential feedback method of Berendsen
• Stochastic collisions with demons of Anderson
• Gaussian isokinetic method (lagrange multipliers)
• Nose-Hoover Thermostats
• Generalized Thermostats
Velocity Scaling Method
• Run for a while, compute kinetic energy, then rescale the
momentum to correct temperature T, repeat as needed.
1
∑i mivi2
2
TI =
3N − 3
Instantaneous TI
• Control is at best O(1/N), not real-time dynamics.
4 scaling: 400/100
vinew =
T old
vi
TI
Nose-Hoover thermostat
• MD in canonical ensemble (TVN)
• Introducing a friction force ζ(t)
dp
= F(q, t) − ζ (t ) p(t)
dt
SYSTEM
T Reservoir
Dynamics of friction coefficient through an equation of motion.
dζ
3N
1
2
Q
= ∑ mivi −
k BT
dt
2
2
dζ
=0
dt
Feedback makes
K.E.=3/2kT
Dynamics at steady-state
Q= fictitious “heat bath mass”. Large Q is weak coupling
Nose-Hoover thermodynamics
• Energy of physical system fluctuates. However energy of
system plus heat bath is conserved.
Q 2
d ln(s)
H'= H + ζ + gk BT ln(s) and
=ζ
2
dt
• Hamilton’s equation of motion for the system
– dr/dt=p, dp/dt= F - pζ /Q,
dζ/dt=pζ/Q etc.
• If the system is ergodic,
Then stationary state is canonical distribution:
1 2 Q 2
exp[–β (V +
p + ζ )]
2m
2
• Thermostats are needed in non-equilibrium situations where there
might be a flux of energy in/out of the system.
• It is time-reversible, deterministic and goes to the canonical
distribution but:
• How natural is the thermostat?
– Interactions are non-local. They propagate instantaneously
– Interaction with a single heat-bath variable-dynamics can be strange for
low dimensional systems. One must be careful to adjust the “mass”
REFERENCES
1. S. Nose, J. Chem. Phys. 81, 511 (1984); Mol. Phys. 52, 255 (1984).
2. W. Hoover, Phys. Rev. A31, 1695 (1985).
Constant Pressure Constant Stress
•
To generalize MD, follow similar procedure as for thermostats for constant
pressure. The size of the box is coupled to the internal pressure (Anderson).
• Volume is coupled to Pressure.


1 
dφ 
P=
2K − ∑ rij
3Ω 
•
i< j dr 

•Unit cell shape may also change (Rahman
Parinello method).
– System can switch between crystal structures.
– Method is very useful is studying the
transitions between crystal structures.
– Dynamics is unrealistic: Just because a system
can fluctuate from one structure to another does
not mean that probability is high for it to happen.
TPN
Parrinello-Rahman Dynamics
Consider:
• Internal coordinates: 0<ρ<1
• Physical coordinates: r
Hρ=r
• H is a 3 x 3 matrix which evolves in time.
– Do periodic boundary conditions with ρ.
– Calculate energy and forces with r.
Langevin Equation
• If we want to be more realistic we have to keep the momentum of the
heavy particle.
dr p
=
dt m
dp
= F + psolvent (t) − ςp
dt
• Where we have added a random force and a friction.
• To get detailed balance we must have:
psolvent (t) psolvent (t') = 2mkTδ(t − t')
• As friction goes to zero, MD is recovered
• As friction goes to infinity, Smoluchowski Eq. Is recovered.
• We can have more general memory also (Generalized LE) :
psolvent (t) psolvent (t') = C(t − t')
Brownian dynamics
• Put a system in contact with a heat bath
• Will discuss how to do this later.
• Leads to discontinuous velocities.
• Not necessarily a bad thing, but requires some physical
insight into how the bath interacts with the system in
question.
• For example, this is appropriate for a large molecule
(protein or colloid) in contact with a solvent. Other heat
baths in nature are given by phonons and photons,…
Hydrodynamic effects
• Brownian dynamics neglects long-time correlations that can arise from
hydrodynamic effects.
• One can treat this within Brownian dynamics by using the Oseen tensor.
D(R) is calculated in perturbation theory. Then the noise on 2 particles
becomes correlated.






rij rij
kT
Dij (R) =
(1+ 2 )
8πηrij
rij


• This is only a linear approximation to the hydrodynamics.
Implicit solvent mesoscale simulations
85,000 atoms
1)
atomistic macromolecule is represented
by a coarse grain model.
2)
atomistic solvent is represented as a
continuum by its average effect on the
solute solvated-effective potentials
and a friction term in Brownian
Dynamics simulations.
1fs step
365 beads
10 fs step
ISMM Simulation is 10,000 times
faster than atomistic
Frechet star in different solvents
Non polar solvent
Non polar dendrimer blocks
exposed to the solvent
Polar solvent
Intermediate polarity
Both exposed
Polar PEG exposed to the solvent εpp=εnn=εpn= 0.2 kT
(εpp = 2, εnn = 0.2 and εpn = 0.5 kT)
and
0.1 kT
Represent two solvent qualities
(εpp = 0.2, εnn = 2 and εpn = 0.5 kT)
The quality of the solvent (based on the relative solubility of each component) determines the
equilibrium shape, size and nature of the exposed area
Solvation Mechanism
1) polar wraps non polar
2) Collapse
Solvation Dynamics
Dissipative Particle Dynamics
• Another way to treat hydrodynamics more realistically
without including explicit solvent atoms.
Fi = ∑ [ f (rij ) − ω(rij ){vij • rˆij }rˆij + ςrˆij ]
j
< ς >= 0
< ς >= 2kTω(rij )
2
• Conserves momentum because equal and opposite random
forces are applied to each pair.
• Implies hydrodynamics.
• Steady state distribution is canonical (prove that is unique
stationary state)
• w(r) is the range of randomized force.
• Hybrid scheme to ensure detailed balance.
Dynamic Correlations -transport
coefficients
Dynamic Correlations is central to molecular dynamics
• Perturbation theory
• Linear response theory.
• Diffusion constants, velocity-velocity auto correlation function
• Transport coefficients
–
–
–
–
Diffusion:
Particle flux
Viscosity:
Stress tensor
Heat transport:
energy current
Electrical Conductivity:
electrical current
Transport coefficients
• Define as the response of the system to some dynamical or
long-term perturbation, e.g., velocity-velocity
• Kubo form: integral of time (auto-) correlation function.
Perturbation response
γ=
µ = ∫0∞ dt < A(t)A(t + s) >
Velocity + P.B.C. needs care
Consider: Diffusion Coefficient, D
where v = center of mass velocity
D=
1 t
∫0 dt < v i (t) • vi (0) >
3
Diffusion Constant
• Diffusion constant is defined by Fick’s law and controls how
systems mix
dρ
= D∇ 2ρ(r, t )
dt
D=
lim
t →∞
1
<| ri (t) – ri (0) |2 >
6t
1 t
D = ∫0 dt < v i (t) • vi (0) >
3
Linear response +
Mass conservation
Einstein relation (no PBC!)
Kubo formula
Dynamical Properties
• Fluctuation-Dissipation theorem:
∞
dA(0)
χ(ω) = β∫ dte < B(t)
>
dt
0
Here [A e-iwt ] is a perturbation and [χ (w) e-iwt] is the
response of B. We calculate the average on the lhs in
equilibrium (no external perturbation).
it ω
•
Fluctuations we see in equilibrium are equivalent to how a nonequilibrium system approaches equilibrium. (Onsager regression
hypothesis)
• Density-Density response function is S(k,w). It can be measured by
scattering and is sensitive to collective motions.
1 +∞
Sk (ω ) =
∫ –∞ dt Fk (t)
2π
1
Fk (t) = < ρ k (t) ρ –k (0) >
2
Example: a sound wave
ε iqx−iωt –iqx+iωt
iqx−i
ω
t
δρ(x, t) = ε Re{e
} = [e
+e
]
2
ρ (t) = ∫ d 3r ρ (r,t)eiqr = ρ δ (k) + ε[δ (k + q)eiωt + δ (k − q)e –iωt ]
k
0
• We will see peaks in S(k,w) at q and -q.
• Damping of sound wave broadens the peaks.
• Inelastic neutron scattering can measure the microscopic
collective modes.
Dynamical Structure
Factor
Hard Spheres
Transport Coefficients: examples
•
•
•
•
•
Diffusion:
Viscosity:
Heat transport:
Electrical Conductivity:
J(t)=total electric current
Particle flux
Stress tensor
Energy current
Electrical current
σ = ∫0∞ dt < J(t)J(0) >
These can also be evaluated with non-equilibrium simulations.
Need to use thermostats to control:
• Impose a shear, heat or current flow
• Initial difference in particle numbers
Equilibrium vs. Steady State: Mass transport
∞
1
D = ∫ dt Vαi (t )Vαi (0)
20
Mass transport: diffusion
Equilibrium vs. Steady State: Energy transport
∞


1
Λ = 2 ∫ dt ⟨ j (t ); j (0)⟩
kT V 0


∂
j (t ) = ∑ r hi
∂t i
H = ∑ hi
i
J q = −λ DT
Energy transport:
Thermal conductivity
Equilibrium vs. Steady State: Momentum transport
P+
∞
V
η=
dt σ αβ (t )σ αβ (0)
∫
kbT 0
σ =n γ
PMomentum transport:
Flow viscosity
Non Equilibrium MD
Couette Flow - shear viscosity
Periodic image
Periodic image
Simulation box
Sliding brick PBC
Simulation box
Sheared box PBC
Sutton-Chen Many body Force Field
U tot
 1
1/ 2 
= ∑ U i = ∑ D ∑ V (rij ) − cρ i 
i
i
 i≠ j 2

local electron density at atom i
pair-wise repulsion
potential
α
V (rij ) = 
r
 ij




n
α
ρ i = ∑ φ (rij ) =∑ 
j ≠i
j ≠ i  rij




m
Viscosities under Different Pressures (T=2000K)
a22
P=42 GPa
a23
P=17 GPa
a23.6 P=8.8 GPa
a24
P=5.0 GPa
a24.6 P=1.1 GPa
a25
P=-0.6 GPa
η (mPa.s)
10
1
0.01
0.1
1
-1
dγ /dt ( ps )
10
Zero Shear-rate Viscosity (T=2000K)
•Experimental Data:
Simulations
Experiments
•Density from [1],
•Viscosities from [1],[2]
10
η0 ( mPa.s )
[1] Smithells Metals
Reference Book, 6th ed.,
Butterworths Co. (1983)
[2] Handbook of PhysicoChemical Properties at High
Temperatures, The 140th
Committee of Japan Scciety
for Promotion of Science,
ISIJ, Tokyo, (1998)
1
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
3
d ( kg/m )
95
Calculation of Transport Coefficient
Thermal Conductivity
J

J = λ∇ T
Kinetic Theory of
Continuum Model:
( Boltzmann eq.)
Speed of sound
1
λ = cvL
3
Heat capacity
Phonon mean
free path
Calculation of Thermal Conductivity
From microscopic quantities
Equilibrium Linear response Non-equilibrium property
ensemble
(transport phenomena)
Fluctuation - Dissipation theorem: (Green -Kubo)


1
λ=
dt ⟨ J (t ); J ⟩
2
∫
k BT V 0
∞
heat current
<a;b>
quantum canonical correlation function
Nanotubes: (Heterogeneity)
Single walled nanotube
nanotube bundles
High current flow along tube;
z
y
x
Low current flow perpendicular
Single walled (10,10) nanotube
34
λ (W/cm/K)
32
30
28
26
24
22
20
0
50
100
150
Tube Periodicity (A)
200
250
(10,10) nanotube with vacancies
14
λ (W/cm/K)
12
y = 0.11402 * x^(-0.78849) R= 0.99794
10
8
6
4
2
4
6
8
nv ( per 1000 atoms)
10
12
(10,10) nanotube with (5,7,7,5) defects
35
y = 22.733 * x^(-0.64179) R= 0.99082
λ (W/cm/K)
30
25
20
15
10
0.5
1
1.5
2
nd (per 1000 atoms)
2.5
3
Nanotube Bundles: (Heterogeneity)
Z: 9.5 W/cm/K
Graphite // plane: 10. W/cm/K
X,Y: 0.055 W/cm/K
z
y
x
Graphite plane: 0.056 W/cm/K
Friction Models: Static Friction
From theory and simulation:
Static Friction Fs:
•Commensurate Interface: strongly interaction
v=0 when Fx< Fc;
i.e. must overcome a threshold force to make motion occur
•Incommensurate Interface: weakly interaction
flat, atomically smooth, clean: Fs= 0;
with defects, roughness or third body: Fs ≠ 0
Friction Models: Kinetic Friction
Kinetic Friction Fk vs. velocity v
Fk(v)=Fk(0)+cvβ
•Dry Friction: β=0, Coulomb friction
kinetic friction has no dependence in the slow sliding limit
Observed in experiments for macroscopic system; MD simulation and other
atomic-level treatments fail to give this behavior
Viscous Friction: β=1, Stokes friction
Kinetic friction is proportional to the velocity
It is normally believed that dry friction and static friction are related
and if kinetic friction behaves viscous law, there is no static friction.
Constant Force Simulation
Constant force simulation to
determine the static friction
FN
z direction movements of
top layers are fixed to be zero
(one Al layer and one O layer for Al2O3,
one Al layer for Al metal)
Fex
f
The length of z-direction has been fixed,
so the samples are under compression
condition.
z-direction movements are
fixed to be zero
External force Fex is
applied to the outmost layers
Fex
f
Constant Velocity Simulation
Constant velocity simulation to
study the kinetic friction
z direction movements of
top layers are fixed to be zero
Vex
For atom n in top layers of top
slab:
Vn(t1)= Vn(t0)+(Vex-Vcm(t1)) × t
Xn(t1)=Xn(t0)+Vn(t1) × t
Vcm is the calculated velocity for
center of mass, so top slab moves
with Vex
x and z directions movements
are fixed to be zero
f
f
Tahir Cagin
Why to talk about mechanics of Nanomaterials?
•
•
•
Mechanics of materials is really about the microstructure
– Ultimate mechanical properties are on perfect systems
– Actual properties observed are something else, (are they?)
Properties and especially mechanical properties of small world
– Surface to volume ratio (1/L) increases
– Properties are determined more and more by this increasing ratio
Continuum elasticity is a tool for relating to macro-world, but not the
method
– Atomistic details are the main player in properties
– Applied loads have statistically distributed results
– Measured properties depend on small variations in the structure
Nano-mechanics of
•
•
•
•
Nano-dots, clusters
– metallic clusters
– dendrimers
Nano-wires
– nm-size wires of metals, ceramics, and semiconductors
Nano-shells
– Thin films of metals,
– Thin films of Ferro-, piezo- electric materials, semiconductors,
– self assembled monolayers of organic films, dendrons
Nano-tubes
– isolated, bundled single and multiple walled carbon nanotubes,
– inorganic and organic (cyclic peptides) nanotubes
Carbon Nanotubes
Carbon Nanotubes with Valence FF
(n1,n2)
R2
R1
Stable Forms: Circular / Collapsed?
(n,n) -armchair
(n,0) - zigzag
(2n,n) - chiral
Circular - Collapsed
Forms upto R=170A
Stable Forms: Circular to Collapsed transition
The cross sectional shape shows a broad transition range at R1
through R2, both circular and collapsed form exist. Below R1,
Circular form above R2 collapsed form are stable.
The radii of the first transition are:
• (n,n) armchair, R1 is between 10.77 A of (16,16) and 11.44 of (17,17).
• (2n,n) chiral, R1 is between 10.28 A of (20,10) and 11.31 of (22,11).
• (n,0) zigzag, R1 is between 10.49 A of (27,0) and 10.88 of (28,0).
The radii of the second transition are:
• (n,n) armchair, R2 is between 29.62 A of (45,45) and 30.30 of (46,46).
• (2n,n) chiral, R2 is between 29.82 A of (58,29) and 30.85 of (60,30).
• (n,0) zigzag, R2 is between 29.93 A of (77,0) and 30.32 of (78,0).
Zero Strain (axial and radial) Stable Cross
Sectional Shapes for (n,m)
Bending modulus of Graphene
Basic energetics by approximating the tube as a membrane with a curvature 1/R and bending
modulus of κ. Assuming a as the thickness of tube wall, the elastic energy stored in a slab of
width L, is given by
The per atom energy
N is the number of carbon atoms per slab
Eο is energy per carbon atom for tubes with 1/R ∼ 0, i.e. flat sheets.
Considering the number of carbon atoms per unit area of tube wall
a the spacing between two graphite sheets, 3.335(Å),
R0 = 1.410(Å) as the C-C bond distance,
κ(n,n) = 963.44 (GPa),
κ(n,0) = 911.64 (GPa), and
κ(2n,n) = 935.48 (GPa).
Stretching and compressing tubes at cross-over
radius
Stretching and compressing tubes at cross-over
radius
Structure and mechanical moduli of
SWNT Bundles
Single Walled CNT Bundles
(10,10) armchair lattice parameter a = 16.78 Å, density d = 1.33 (g/cm3).
(17,0) zigzag, they are a = 16.52 Å, and d = 1.34 (g/cm3).
(12,6) chiral form, they are a = 16.52 Å and d = 1.40 (g/cm3).
Young's modulus along the tube axis for triangular-packed SWNTs
(using the second derivatives of the potential energy.)
Y = 640.30 GPa, armchair
Y = 648.43 GPa, zigzag
Y = 673.49 GPa, chiral.
Normalized to carbon sheet these are within a few % of the graphite bulk value.
Bending Modulus of SWNT from Carbon Nanotori
and deformation kinks as deformation sinks
R=19.4 A (2000 atoms)
To
R=390 A (40,000 atoms)
Strain energy of Tori as a function of 1/R2
Rs > 183 A
Circular cross-section smooth torus
183 > Rs > 109
Circular cross section smooth torus,
But at RT MD yields dents/creases
If MD structure used as input small
dents diffuse and merge to make kinks
Emergence of Kinks
Structural Detail of Kinks upon bending
Strain energy of Tori
Kinks/Buckles for Same Rs around Transition
Lowest Energy
R = 109.6 A
(10,10,564)
with varying #
of buckles
Lower radii
down to 40 A
R< 40 A
unstable
Torsional Loads
Torsional Loads
Carbon Nanotubes with BO Potentials
(n1,n2)
R2
R1
Tensile failure of (10,10) nanotube
Tension
waves
(10, 10) Single Wall Nanotube Stretching
Tensile failure of (10,10)-(15,15) double walled nanotube
(10, 10)-(15,15) Double Wall Nanotube Stretching
Bending of (15,15)/(10,10) DWNT
0.8
DWNT (10,10)/(15,15)
0.7
Energy (kcal/mol)
0.6
0.5
BREN
0.4
0.3
0.2
l 3F 2
∆U =
6 EI
0.1
E=944 GPa
EBOD
0
0
0.2
0.4
Force (nN)
0.6
0.8
1
Plasticity of FCC Nanowires Under High Strain Rate
Apply uniform strain rate:
0.05%/ps, 0.5%/ps, 1%/ps,
2%/ps 5%/ps
for systems as Ni, NiCu
and glass former Uniaxial
CuAg Stra
and NiAu, staring with
perfect2nm
single crystal.
Infinity along c directio
the original unit length i
Stress of NiCu(50%) under different strain rate
10.0
0.05%/ps
0.5%/ps
1%/ps
2%/ps
Metallic glass
viscosity
5%/ps
σ = η dε/dt
η=0.8P
Strain Rate:
8.0
stress (GPa)
6.0
4.0
2.0
Fracture
0.0
0.0
0.1
0.2
0.3
0.4
0.5
-2.0
strain
0.6
0.7
0.8
0.9
1.0
Twinning processes at lower strain rate
5
NiCu @ 300K, strain rate = 0.05%/ps
4.5
Accommodate strain by twin formation
7% yield strain
Young Modulus is the same after twinning
Stress (GPa)
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
Strain
Perfect FCCFirst twinsTwins grow
Twin size decreases as strain rate increases
Perfect FCC searching method
•
Grain size
250
strain = 40%
200
150
100
50
0
0.5
1
2
strain rate (%/ps)
5
•
•
Start from an atoms with the
perfect FCC nearest neighbor
configuration(ABC packing).
Checking the further neighbors.
Stop at the boundary where the
atoms lost the FCC nearest
neighbor structure, and define the
number of atoms inside the
boundary as the grain size.
Strain rate induced Amorphization
Radial distribution function of Ni under slower and faster strain rates
•Strain rate = 5%/ps,
•Form amorphous,
•The second peak on GFR
disappeared at strain=15%
•Strain rate = 0.5%/ps,
•Twinning started from 7% strain
•Kept crystalline structure until 30% strain
7
7
strain = 0
strain = 0
6
strain = 10%
strain = 10%
strain = 20%
5
strain = 15%
5
strain = 30%
g(r)
g(r)
4
strain = 20%
1
4
0
3
3
g(r)
6
2
0.3
0.35
0.4
0.45
0.5
r(nm )
2
2
1
1
0
0
0.2
0.3
0.4
0.5
r (nm)
0.6
0.7
0.8
0.2
0.3
0.4
0.5
r (nm)
0.6
0.7
0.8
350
Elastic constants vs strain
3C66-C44
Strain rate = 5%/ps
300
Shear Moduli recovered
due to twinning
250
0.05%/ps
200
Cij (GPa)
Metal glass fully formed
C11,C22
150
C33
C12
C22-C23
C66
(C11-C12)/2 (GPa )
40
0.5%/ps
30
5%/ps
20
10
0
100
0
(C11-C12)/2
0
0.2
0.15
Tetragonal shear moduli
would vanish at strain = 0.11,
which initiated the transition.
C44,C55
0
0.1
strain
C23,C13
50
0.05
0.4
0.6
strain
0.8
1
0.2
• Deformation mechanism beyond defects movements is twinning and
amorphization.
• Strain rate plays a role similar to temperature.
• Disorder causes similar effect in strain rate induced amorphization, that
lager size differences lower the strain rate required for amorphization.
Cooling rate
strain rate
size ration
0.005
0.5
1
2
5
Ni
NiCu
1
1.025
Crystal
Crystal
Crystal
Crystal
Crystal
Crystal
Crystal
Amorphous
Amorphous Amorphous
NiAu
Amorphous
1.156
Amorphous
Amorphous
Amorphous
Amorphous
Amorphous
crystal
Strain rate
Size ratio