Module17: Intermolecular Force between Surfaces and Particles Lecture 23: Intermolecular Force between Surfaces and Particles 1
We now try to understand the nature of spontaneous instability in a confined ultra thin film. In order to understand it, it is essential to have an understanding on the Intermolecular forces are forces that act between neighboring particles, molecules, atoms or ions, which are not actually bonded. They are weak compared to intermolecular forces which hold all the atoms together making up a molecule and include all types of chemical bonds. The key intermolecular forces are 1. Van der Waal s forces 2. Electrostatic forces: If one of the surfaces carries a charge, it will induce a mirror charge on the other surface because of the pressure of the free or mobile ions in a liquid medium. These mobile charges screen the surface charge by accumulating near the surface having opposite charge. Therefore these forces are also called double layer forces. 3. Acid-base type interaction or the polar interactions. 4. There can be various other types of forces depending on the nature of the solid surface, liquid etc, which are related to adsorption, depletion, steric forces etc. Steric forces are always repulsive. Intermolecular forces can be: 1. Permanent dipole- permanent dipole forces: This is electrostatic interaction between permanent dipoles in molecules. These forces tend to align the molecules to increase the attraction. 2. Permanent dipole-induced dipole forces (Debye forces): These forces appear from induction or polarization. Example is HCl as its electrons are either attracted (by H + ) or repelled (by Cl - ). This type of interaction occurs in case of any polar molecule. 3. Hydrogen bonds: An attractive force between a lone pair of an electro-negative atom and a hydrogen atom that is bonded to nitrogen, oxygen or fluorine. While the hydrogen bond is often described as electro static dipole-dipole interaction, it also has certain features of a covalent bond, its directional, stronger than van der Waal interaction, produces inter 2
atomic distances shorter than van der Waal s radii and involves limited number of interaction partners depending on valency. 4. The induced dipole induced dipole type interactions, which are present in all types of materials, including the ones that are non polar and is thus the most fundamental form of interaction forces. Since polymers are predominantly a-polar, these forces dominate in a polymer thin film. The potential energy of interaction between two molecules or atoms due to van der Waal s force which is attractive, and mathematically is given as (23.1) where x is the separation distance between two particles, β is a material specific property which determines the precise magnitude of the long range inter-molecular attraction and φ a represents the potential energy of interaction between two molecules in case of attraction. When the two particles are in very close proximity, the electrons in the outer orbital overlap, resulting in stiff Born repulsion, which has a faster decay as compared to the attractive interaction and is given as (23.2) Figure 23.1: Typical Lennard Jones Potential Curve Where φ R represents the repulsive energy of interaction between two molecules and ξ represents a material specific parameter which determines the short range repulsion. The overall nature of inter particle interaction is given by the well known Lennard Jones (L-J) Potential, which is 3
also known as the 6 12 potential. The mathematical expression is obtained by combining equations 23.1 and 23.2, and has the following very well known form (23.3) A typical L-J potential curve is shown in figure 23.1 23.1 Interaction between two different surfaces Equation 23.3 gives the expression for the interaction between two atoms or fundamental particles. However, in reality, majority of the interactions is between two surfaces. We now develop a formulation that provides an expression for the interaction between two surfaces. For that, we consider two infinitely long blocks of materials 1 and 2, as shown in figure 23.2. First we consider a single molecule of material 2 located at a distance z from the surface of material 1. Figure 23.2: Schematic of interaction between two surafces (blocks of 1 and 2) Consider a volume of material 1, (23.4) Number of molecules present in unit volume of material 1, (23.5) Where N A is number of molecules per unit mole of A, M A is the molecular weight of A and ρ 1 is the density of material 1. Number of molecules present in the ring of volume dv is Based on the assumption of additivity, the net interaction energy of a molecule and a planar surface is 4
(23.6) Upon integrating the equation, it gives the total interaction of all molecules in the ring with a single molecule residing at a distance z from surface of material 1. Net interaction energy for a molecule at a distance d is given as (23.7) Now, Applying Pythagoras theorem in figure 23.2, we have (23.8) Therefore, Energy of interaction of 1 molecule of material 2 with all of material 1 (23.9) (23.10) Energy of interaction of shaded region of block 2 with whole of block 1 is where a is the cross sectional area. Energy of interaction per unit area can be given as (23.11) In equation 23.12, we substitute (23.12) (23.13) 5
A 12 is known as Hamakar constant which depends on bulk properties of material as well as nature of interaction at molecular level or the intermolecular potential. It can be observed that (23.14) Typical order of magnitude of Hamakar constant is 10-21 -10-19 J (23.15) Therefore, energy of interaction due to van der Waal s interaction between two surfaces can be given as (23.16) It can be observed from equation 23.16 that as d 0, G lw as no repulsive terms has been considered in the formulation. Change in force when surfaces are brought in contact from infinite distance can be given as (23.17) Where, d 0 is the minimum permissible separation distance between two surfaces (figure 23.3). Another limiting case can be d 1 and d 2, and then the energy of interaction can be given as (23.18) Which is the energy of interaction between two large blocks in close proximity with each other. Also, we know that (23.19) Therefore, (23.20) From expression of A 12 in equation 23.13 requires the magnitude of β 12 to evaluate A 12. However, the expression in equation 23.20 allows the calculation of A 12 entirely from physically measurable quantities.. Now, in case both materials 1 and 2 are same, Hamakar constant can be written as (23.21) 6
Further, we know that (23.22) Combining equation 23.19 and 23.22, we note that for pure van der Waal s interaction, A 12 will always be positive, with expression (23.23) 23.2 Self interaction energy of molecules So far, self interaction energy of the molecules has not been considered. We consider a semi infinite block extending to, as shown in figure 23.3. We create an interface at a distance h from the top surface. Figure 23.3: Schematic of interaction between two surafces (blocks of 1 and 2) Now, we consider the following in figure 23.3. Total Energy of molecules in the film (0, ) = Energy of molecules between (0, h) + Energy of interaction between (h, ) + Interfacial interaction between (0, h) and (h, ) at h. Since h is small, Energy of molecules in the film (0, ) Energy of interaction between (h, ) Thus, it can be argued that Energy of molecules between (0, h) = Interfacial interaction between (0, h) and (h, ) at h. (23.24) 7
Now, combining figure 23.3 and equation 23.16, we get (23.25) Combining equation 23.24 and 23.25 we get (23.26) Further, as, we have (23.27) Excess energy in the film can be given as (23.28) Physically this term is significant till h is around 50 ~ 100 nm, beyond that it is ~ 0. The effective interfacial potential which is the derivative of excess free energy with respect to the thickness can be given as (23.29) The negative derivative excess free energy is called Disjoining pressure. This is the measure of the excess pressure in a thin film due to interaction of two interacting interfaces. (23.30) 23.3 Thin Film on a sem-inifinite Substrate Now we consider a more A more realistic situation is where a film is present on a substrate. From figure 23.4 we can write Therefore, upon substitution we have (23.31) As d 1 ~ we can further simplify equation 23.32 as (23.32) 8
Figure 23.4: Schematic of a thin film of material 2 on a semi infinite substrate of material 1. (23.33) Where A e = A 22 A 12, which is the Effective Hamakar Constant. A 22 is related to energy of cohesion of and A 12 gives the energy of adhesion. For a liquid film on a solid surface, whether one will get complete wetting or not can be ascertained from the sign of A e. Now, from equation 23.21, we have Therefore, (23.34) (23.35) 23.4 Spreading Coefficient Before coating a film on a substrate, the surface energy of the system is γ 1. After coating a film of surface energy γ 2 on top, the surface energy of the system becomes (γ 2 + γ 12 ). Spreading coefficient can be defined as the difference between the initial and final surface energies. (23.36) Combining equations 23.35 and 23.36 we can write (23.37) Whether a film on a substrate is stable or not can be determined from the sign of the spreading coefficient or the Effective Hamakar Constant. It can be noted that a negative S 21 would imply positive A e, which essentially implies that the film is Unstable, as more energy gets trapped in the 9
film after coating than that in the substrate. This is the case when a high energy film is coated on a low energy substrate. A positive S 21 would imply negative A e, which implies a stable film, as is the case with a low energy film on a high energy substrate. 10