3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
I
LECTURE 10: MOLECULE-SURFACE INTERACTIONS
Outline :
LAST LECTURE : BRIDGING THE GAP BETWEEN LENGTH SCALES ................................................. 2
MOLECULE-SURFACE INTERACTIONS: MOLECULAR ORIGINS OF BIOCOMPATIBILITY ................ 3
CALULATION OF THE NET POTENTIAL FOR INTERACTING BODIES .............................................4-8
Volume Integration Method : Procedures and Assumptions ................................................ 4
Molecule-Surface Interactions...........................................................................................5-8
Geometry ................................................................................................................ 5
Derivation 1 ............................................................................................................ 6
Derivation 2 ............................................................................................................. 7
Dispersion, London ................................................................................................. 8
Objectives: To mathematically scale up intermolecular potentials to intersurface and interparticle
potentials
Readings: Course Reader documents 22 & 23 and Israelachvili, Chapter 10.
Multimedia : Bonding and Protein Structure Demo (California Lutheran University).
1
3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
BRIDGING THE GAP BETWEEN LENGTH SCALES
-A typical inter- atomic, ionic, or intamolecular
potential (e.g. LJ potential)
A typical intersurface or interparticle
force vs. separation distance curve
0.01
repulsive
regime
0
-0.01 0
Force, F (nN)
force, f(r) (nN)
0.02
0.2 0.4 0.6 0.8
1
-0.02
→
-0.03
0
-0.04
Interatomic or
Intermolecular Separation
Distance, r(nm)
w(r) or U(r) → f(r)
(one atom, ion, or
molecule)
w(r)= -Ar -6
kc
attractive
regime
Tip-Sample Separation
Distance, D (nm)
W(D) → F(D)
r
(net interaction
between larger bodies,
i.e. assemblies of
atoms, ions, or
molecules)
D
2
3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
MOLECULE-SURFACE INTERACTIONS : MOLECULAR ORIGINS OF
BIOCOMPATIBILITY
BLOOD FLOW
blood plasma
proteins
.
PLATELETS!
BLOOD PRESSURE+
ATTRACTIVE
FORCES
Photo of platelets
courtesy of Tokyo
Metropolitan Institute
of Medical Science
(RINSHOKEN).
Used with permission.
denatures
Photo courtesy of David Gregory & Debbie Marshall, Wellcome Images.
BLOOD CLOT!
-acute occlusive thrombosis
- infection / inflammation
- neointimal hyperplasia
Solid-Liquid
Solid-Liquid
Interface
adsorbs
BIOMATERIAL SURFACE
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3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
CALCULATION OF THE NET POTENTIAL FOR INTERACTING BODIES :
VOLUME INTEGRATION METHOD : PROCEDURES AND ASSUMPTIONS
1) Choose the mathematical form of the interatomic/ionic/molecular potential, w(r) (e.g. in
A
this case we will use an arbitrary power law : w(r)= - n )
r
2) Set up the geometry of the particular interaction being derived (e.g. molecule-surface,
particle-surface, particle-particle, etc.)
3) Assume "pairwise additivity"; i.e. the net interacion energy of a body is the sum of the
individual interatomic/intermolecular interactions of the constituent atoms or molecules
which make up that body
4) A solid continuum exists : the summation is replaced by an integration over the volumes
of the interacting bodies assuming a number density of atoms/molecules/m3, ρ
5) Constant material properties : ρ and A are constant over the volume of the body
↓ volume integration
W(D)= ∫ ∫ ∫ w(r) • ρ dV
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3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
\INTERACTION POTENTIAL BETWEEN AN ATOM / MOLECULAR AND SURFACE:
GEOMETRY
z
Geometry :
atom or
molecule
z=0
z = direction perpendicular to the sample surface
v
D (nm)= normal molecule-surface separation
distance
x (nm) = direction parallel to sample surface
r=(z2+x2)1/2
planar surface
= circular ring radius (m)
A = infinitesimal cross-sectional area (m2) = dx dz
z=D
x=0
V = ring volume (m3)= 2πx (dxdz)
N = # of atoms within the ring = ρ (2πx) dx dz
dz
x
dx
ρ = number density of atoms in the material
constituting the surface (atoms/m3)
r = distance from molecule to differential area
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3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
INTERACTION POTENTIAL BETWEEN AN ATOM / MOLECULAR AND SURFACE:
DERIVATION
A
z
w(r)= - n (1)
r
atom or
molecule
z=0
Substitute (2) into (1) :
v
w(r)= -
(z
A
2
+x
)
2 n/ 2
r = z 2 + x 2 (2)
(3)
Net Interaction Energy :
r=(z2+x2)1/2
W(D)= ∫ ∫ ∫ w(r) • ρ dV
Z=∞ x= ∞
planar surface
z=D
x=0
W(D)=
w(r)
ρ ( 2π x ) dz dx (4)
{ 14
4244
3
∫ ∫
Z=D x=0 potential of
number of molecules
each molecule
substitute (1) → (2)
Z=∞ x=∞
dz
x
W(D)=
dx
∫ ∫
Z=D x=0
-
(z
A
2
+x
)
2 n/ 2
ρ ( 2π x ) dzdx (5)
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3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
INTERACTION POTENTIAL BETWEEN AN ATOM / MOLECULAR AND SURFACE:
DERIVATION
Pull out constant terms :
z
atom or
molecule
z=0
W(D)= Aρ 2π
Z=∞
dz
Z=D
v
∫
-
x
dx (6)
x=0 ( z + x )
14442444
3
2
2 n/ 2
Integral 1
W(D)= -Aρ 2π
W(D)MOL-SFC =
z=D
x=0
Z=∞
1
dz (7)
n-2
∫
2
n
z
(
)
Z=D
1442443
Integral 2
r=(z2+x2)1/2
planar surface
∫
x=∞
-2π Aρ
(8)
( n - 2 )( n - 3 ) D n-3
n = determined by the type of interaction (see slide 2);
dz
x
dx
related to the range of the interaction
A = molecular level parameter;
related to strength of the interaction
ρ = atomic density
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3.052 Nanomechanics of Materials and Biomaterials Tuesday 03/13/07
Prof. C. Ortiz, MIT-DMSE
INTERACTION POTENTIAL BETWEEN AN ATOM / MOLECULAR AND SURFACE:
DERIVATION
z
W(D)MOL-SFC =
atom or
molecule
z=0
-2π Aρ
( n - 2 )( n - 3 ) D n-3
London Dispersion Interactions n = 6 ;
v
-π Aρ
6D 3
∂W(D) -π Aρ
F(D)MOL-SFC =
=
2D 4
∂D
W(D)MOL-SFC =
r=(z2+x2)1/2
planar surface
z=D
x=0
dz
x
dx
8