Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 6 Interaction forces II – Tip-sample interaction forces Ron Reifenberger Birck Nanotechnology Center Purdue University

Summary of last lecture Type of interaction Ion-ion electrostatic Dipole-charge electrostatic Dipole-dipole electrostatic Angle-averaged electrostatic (Keesom force) Angle-averaged induced polarization force (Debye force) Dispersion forces act between any two molecules or atoms (London force) Q1 Q2 r p Q r q p1 r q1 f p2 q2 r p1 r p2 p1 r p2 Adapted from J. Israelachvilli, “Intermolecular and surface forces”.

From interatomic to tip-sample interactions-simple theory First consider the net interaction between an isolated atom/molecule and a flat surface. Relevant separation distance will now be called “d”. Assume that the pair potential between the atom/molecule and an atom on the surface is given by U(r)=-C/rn. Assume additivity, that is the net interaction force will be the sum of its inter-actions with all molecules in the body – are surfaces atoms different than bulk atoms? No. of atoms/molecules in the infinitesimal ring is ρdV= ρ(2x dx d) where ρ is the number density of molecules/atoms in the surface. z=0 z √(z2+x2) dz x z=d dx

Number Densities of the Elements Introduction to Solid State Physics, 5th Ed. C. Kittel, pg. 32

From interatomic to tip-sample interactions-simple theory Next integrate atom-plane interaction over the volume of all atoms in the AFM tip. Number of atoms/molecules contained within the slice shown below is px2 dz = r p[R2-(R-z)2] dz = r p(2Rtip-z)z dz. Since all these are at the same equal distance d+z from the plane, the net interaction energy can be derived by using the result on the previous slide. d z=0 z z+d dz Rtip 2Rtip-z x

From interatomic to tip-sample interactions-some caveats If tip and sample are made of different materials replace r2 by r1r2 etc. Tip is assumed to be homogeneous sample also! Both made of “simple” atoms/molecules Assumes atom-atom interactions are independent of presence of other surrounding atoms Perfectly smooth interacting surfaces Tip-surface interactions obey very different power laws compared to atom-atom laws

Measuring Macroscopic Surface Forces An interface is the boundary region between two adjacent bulk phases L V G S L = Liquid G = Gas S = Solid V = Vapor We call (S/G), (S/L), and (L/V) surfaces We call (S/S) and (L/L) interfaces

Surface Energetics Atoms (or molecules) in the bulk of a material have a low relative energy due to nearest neighbor interactions (e.g. bonding). Performing work on the system to create an interface can disrupt this situation...

(Excess) Surface Free Energy Atoms (or molecules) at an interface are in a state of higher free energy than those in the bulk due to the lack of nearest neighbor interactions. interface

Surface Energy The work (dW) to create a new surface of area dA is proportional to the number of atoms/molecules at the surface and must therefore be proportional to the surface area (dA):  is the proportionality constant defined as the specific surface free energy. It is a scalar quantity and has units of energy/unit area, mJ/m2.  acts as a restoring force to resist any increase in area. For liquids  is numerically equal to the surface tension which is a vector and has units of force/unit length, mN/m. Surface tension acts to decrease the free energy of the system and leads to some well-known effects like liquid droplets forming spheres and meniscus effects in small capillaries.

high surface energy ↔ strong cohesion ↔ high melting temperatures Material Surface energy ( mJ/m2 ) mica 4500 gold 1000 mercury 487 water 73 benzene 29 methanol 23 Values depend on oxide contamination adsorbed contaminants surface roughness etc…. Materials with high surface energies tend to rapidly adsorb contaminants

Work of Cohesion and Adhesion For a single solid (work of cohesion): For two different solids (work of adhesion): 1 1 1 1 1 1 1 2 2 12 interfacial energy 2

Surface-surface interactions Following the steps in previous slides it is possible calculate the inter- action energy of two planar surfaces a distance of ‘d ’ apart, specifically ρdV = ρ dA d for the unit area of one surface (dA=1) interacting with an infinite area of the other. z z=d z=0 unit area dz

Typical Values for Hamaker Constant Typically, for solids interacting across a vacuum, C ≈ 10-77 Jm6 and ρ ≈ 3x1028 m-3 Typically, most solids have See Butt, Cappella, Kappl, Surf. Sci. Reps., 59, 50 (2005) for more complete list

Implications atom-atom interaction Equilibrium separation r=σ r*= Positive energy Repulsive force Negative energy Attractive force U or F U (r) Flocal max Umin r*= r r=σ rF=max= 15

Estimating the interfacial energy z z=d=r* z=0 unit area dz When d≈r*, then per unit area, we should have: This approach neglects the atomicity of both surfaces. Require an “effective” r*.

In general, vdW interactions between macroscopic objects depends on geometry Source J. Israelachvilli, “Intermolecular and surface forces”.

The Derjaguin approximation Plane-plane interaction energies are fundamental quantities and it is important to correlate tip-sample force to known values of surface interaction energies. For a sphere-plane interaction we saw that Comparing with previous slides we find that It can be shown that for two interacting sphere of different radii

Implications of Derjaguin’s approximation We showed this when U(r)=-C/rn - however it is valid for any force law - attractive or repulsive or oscillatory - for two rigid spheres. As mentioned before, if two spheres are in contact (assuming no contamination), then d=r*. The value of U(d=r*)plane-plane is basically dW11 the conventional surface energy per unit area to create a solid surface. Thus: This approximation is useful because it converts measured Fadhesion in AFM experiments to surface energy dW11 dW11

How to Model the Repulsive Interaction at Contact? Atom-Atom? Sphere-Plane? tip apex substrate Source: Capella & Dietler Maybe if the contact area involves tens or hundreds of atoms the description of net repulsive force is best captured by continuum elasticity models 20

Continuum description of contact - history Hertz (1881) takes into account neither surface forces nor adhesion, and assumes a linearly elastic sphere indenting an elastic surface Sneddon’s analysis (1965) considers a rigid sphere (or other rigid shapes) on a linearly elastic half-space. Neither Hertz or Sneddon considers surface forces. Bradley’s analysis (1932) considers two rigid spheres interacting via the Lennard-Jones 6-12 potential Derjaguin-Müller-Toporov (DMT, 1975) considers an elastic sphere with rigid surface but includes van der Waals forces outside the contact region. Applicable to stiff samples with low adhesion. Johnson-Kendall-Roberts (JKR, 1971) neglects long-range interactions o utside contact area but includes short-range forces in the contact area Applicable to soft samples with high adhesion. Maugis (1992) theory is even more accurate – shows that JKR and DMT are limits of same theory

Tip-sample Interaction Models Rigid tip-rigid sample Deformable tip and rigid sample* Rtip sample F aHertz aJKR as Equilibrium Pull-off 1 2 3 D (a) (b) (c) From the Derjaguin approximation for rigid tip interacting with rigid sample we have Real tips and samples are not rigid. Several theories are used to better account for this fact (Hertz, DMT, JKR) * These theories also apply to elastic samples, they are just shown on rigid sample to demonstrate key quantities clearly. For example D is the combined tip-sample deformation in (b)

I. Surface energies - notation Work of adhesion and cohesion: work done to separate unit areas of two media 1 and 2 from contact to infinity in vacuum. If 1 and 2 are different then W12 is the work of adhesion; if 1 and 2 are the same then W11 is the work of cohesion. Surface energy: This is the free energy change g when the surface area of a medium is increased by unit area. Thus While separating dissimilar materials the free energy change in expanding the “interfacial” area by unit area is known as their interfacial energy Work of adhesion in a third medium 1 2 3

II. What is the “Stiffness” of the Tip/Substrate? 1 Pa = 1 N/m2 five orders of magnitude 10 MPa http://www-materials.eng.cam.ac.uk/mpsite/interactive_charts/stiffness-density/NS6Chart.html

Standard results Contact radius a: F aHertz aJKR Equilibrium D (b) Deformation D: Source: Butt, Cappella, Kappl

Standard results (cont.) Pull-off Force F: F as Pull-off D (c) Source: Butt, Cappella, Kappl

Example Hertz contact: Rtip = 30 nm; Fapp= 1 nN Etip=Esub=200 Gpa; Poisson ratio = νtip=vsub=0.3=v Fapp aHertz D 60 nm Contact radius: Deformation: Pull-off Force=0 Contact Pressure:

2 click work of adhesion jump from contact

Contact forces: Maugis’ Theory a: normalized contact radius δ: normalized penetration P: normalized force Non-contact Contact Contact Radius Non-contact Contact Repulsive Penetration Loading Force 1 click λ 0: DMT (stiff materials) λ : JKR (soft materials) Attractive Penetration D. Maugis, J. Colloid Interface Sci.150, 243 (1992).

Validity of different models – converting measured adhesion force to work of adhesion

Comments on these theories JKR predicts infinite stress at edge of contact circle. In the limit of small adhesion JKR -> DMT Most equations of JKR and Hertz and DMT have been tested experimentally on molecularly smooth surfaces and found to apply extremely well Most practical limitation for AFM is that no tip is a perfect smooth sphere, small asperities make a big difference. Hertz, DMT describe conservative interaction forces, but in JKR, the interaction itself is non-conservative (why?) …for a force to be considered conservative it has to be describable as a gradient of potential energy.

Combining van der Waals force & DMT contact tip=Si A : Hamaker constant (Si-HOPG) R : Tip radius E* : Effective elastic modulus a0 : Intermolecular distance z 1. To model tip surface interaction we use van der Waals forces when tip is separated from the sample - A is Hamaker constant, R is tip radius, and z is tip-sample gap. 2. And when tip contacts the sample we use the DMT contact model shown here where E* is the effective elastic modulus. a0 =r* sample = HOPG Raman et al, Phys Rev B (2002), Ultramicroscopy (2003)

Effect of capillary condensation – modifications to DMT model capillary neck breaks Capillary force models range from simple to complex Strong dependence on humidity

Next lecture Couple cantilever mechanics to tip sample interaction forces F-Z vs. F-d curves