Chimica Fisica dei Materiali Avanzati Part 4 – Forces between particles and surfaces

Chimica Fisica dei Materiali AvanzatiPart 4 – Forces between particles and surfaces Laurea specialistica in Scienza e Ingegneria dei Materiali Curriculum Scienza dei Materiali Corso CFMA. LS-SIMat

Combining relations • The fundamental forces involved in the interaction among particles and between particles and surfaces are the same as those already described for atoms and molecules (electrostatic, VdW, solvation forces, etc.). • However, they can manifest themselves in quite different ways • There are also similarities expressed by certain semiquantitative relations known as combining relations. • We may express the binding energy of molecules A and B in contact as where A and B are the appropriate molecular properties of the corresponding molecules (e.g., charge, dipole, polarizability, etc.; cf. Part 1, Slide 6). Corso CFMA. LS-SIMat

Combining relations (cont’d) Difference in energy between dispersed and associated clusters In the general case where n is the number of like bonds that have been formed upon association. Since is always positive ( ) the associated state of identical molecules is energetically favored, i.e., there is always an effective attraction between like particles in a binary mixture. Corso CFMA. LS-SIMat

Surface and interfacial energy • The association energy can be recast in different forms • Consider two flat macroscopic surfaces of A, each of unit area, in a liquid of B. • The above eqn. can be read as (negative) free energy change for bringing the two surfaces into adhesive contact; equivalent to twice the interfacial energy, i.e. . • Let’s assume n bonds per unit area: is the adhesion energy in vacuum per unit area of the A-B interface • is the (negative) energy change for bringing unit area A into contact with unit area A in vacuum, known as cohesion energy. The cohesion energy is related to the (positive) surface energy or surface tension by . Similarly, . • Combining these definitions with the above eqn., one gets Corso CFMA. LS-SIMat

Surface and interfacial energy (cont’d) • Because of their definition, . Hence, i.e., the interfacial energy can be estimated solely from the surface energies or surface tensions of the pure liquids in the absence of any data on the energy of adhesion. Particles or surfaces in a third medium • By similar arguments, in a three-component mixture one gets for the association of two unlike molecules A and B in a solvent composed of molecules C. • This result shows that the association energy can be positive or negative. If positive, the particles effectively repel each other and remain dispersed. • This happens when C is intermediate between A and B. Corso CFMA. LS-SIMat

Summary • There is always an effective attraction between like molecules or particles in a multicomponent mixture. • Unlike particles may attract or repel each other depending on the properties of the medium. Proviso • There are two very important exceptions: • Coulomb interactions between atomic or molecular ions: the sign of is reversed; the dispersed state is favored • Hydrogen-bonding molecules: the strength of the H-bond between different molecules cannot be expressed simply in terms of . • Example: repulsive forces due to hydration of hydrophilic molecules. Corso CFMA. LS-SIMat

Long-range forces between macroscopic bodies • The properties of gases and the cohesive energies of condensed phases are determined mainly by the interaction energies of molecules at contact (Coulomb forces may be an exception). • For macroscopic bodies, when all the pair potentials between the molecules in each body is summed, we find • The net interaction energy is proportional to the size of the particles • It can be much larger than kBT even at separations of 100 nm or more • The energy and force decay much more slowly with the separation • A variety of different behaviors may arise depending on the specific form of the long-range distance dependence of the interaction Corso CFMA. LS-SIMat

Interaction potentials between macroscopic bodies Molecule-surface interaction Assume an additive molecular pair potential: By integration of the interactions between the isolated molecule and those contained in rings of volume , for the total interaction of a molecule at a distance D from the surface we get Assuming VdW forces with i.e., an interaction potential with much longer range than the original pair potential. Corso CFMA. LS-SIMat

Interaction potentials between macroscopic bodies (cont’d) Sphere-surface interaction The volume of a thin circular section of radius x within the sphere is (use is made of the chord theorem). Using the previous result for and integrating over a number of molecules at a distance from the planar surface, we get For , only small values of z contribute to the integral Corso CFMA. LS-SIMat

Interaction potentials … (cont’d) For VdW forces with The interaction energy is proportional to the radius of the sphere and decays as . For we may replace in the denominator by D, and obtain Since is the number of molecules in the sphere, the result is equivalent to that for the interaction of a molecule with a surface. Corso CFMA. LS-SIMat

Interaction potentials between macroscopic bodies (cont’d) Surface-surface interaction Consider the energy per unit surface area; starting from a sheet of thickness dz and unit area at a distance z from an extended surface of larger area. The interaction energy of this sheet with the surface is . Thus for the two surfaces For n = 6 When D is small compared to the lateral dimensions, this result holds to two unit areas of both surfaces. Corso CFMA. LS-SIMat

Non-retarded VdW interactions between macroscopic bodies • Assume a pair potential (in vacuum, additive and non-retarded) • The resulting interaction laws for some common geometries are given in terms of the conventional Hamaker constant • Typical values of Hamaker constants for condensed phases are in the range of 10-19 J. Corso CFMA. LS-SIMat

Van der Waals forces in condensed media • The definition just given for the Hamaker constant ignore the influence of neighboring atoms on the interaction between any pair of atoms; but, straightforward additivity breaks down in condensed media. • Lifshitz theory avoids the problem of additivity: large bodies are treated as continuous media characterized by bulk properties as the dielectric constant and refractive index. • As a result, although the expressions for the interaction energies remain valid, the Hamaker constant is calculated in a different way. • Example: for two identical phases 1 interacting across a medium 3 the Hamaker constant is • The first term accounts for the Keesom and Debye energies, the second for dispersion ( is an effective electronic excitation frequency) • The Hamaker constant of metals and metal oxides can be an order of magnitude higher Corso CFMA. LS-SIMat

Surface Tension • For any material, it costs energy to make a surface. To make a small element dA, the work done is i.e. if we make new surface, we get the first term, but if we stretch or change the surface as we create more, the second term will contribute. There is strain created. • For a simple one component system: • For liquids, we measure g using a variety of techniques such as: drop-weight, the Du Nouy ring method, static drop, capillary rise. • For solids: crystal cleavage or heat of solution Corso CFMA. LS-SIMat

Typical Values of Surface Tension* * Somorjai, Principles of Surface Chemistry, 1972. Corso CFMA. LS-SIMat

Computed Values of Specific Surface Energies* * Somorjai, Principles of Surface Chemistry, 1972. Es (erg/cm2) = Es(0) + DEs HereEs(0)is the specific surface energy of the rigid lattice and DEs is the relaxation energy of opposite sign. Corso CFMA. LS-SIMat

Pin Pout Curved Interfaces - Laplace Pressure If we imagine a bubble of radius r, the work done to expand it is but this is opposed by the creation of more surface, which would cost At equilibrium dWV = - dWS,hence With a little more work we can see that The equilibrium pressure inside a curved surface, whether solid or liquid, rises as the sphere decreases in size. Corso CFMA. LS-SIMat

solid glg gsg  gsl liquid solid gas liquid Young’s Equation - Contact Angle and Adhesion Consider the work done in separating a solid phase and a liquid phase. By balancing the surface forces, Young showed (1805) Measurement of the equilibrium contact angle allows one to measure (You can measure the difference but not the absolute values.) Consider the work done in separating a solid phase and a liquid phase Combining Young’s and Dupre’s equations we get In principle, one should work in vacuum for reference work. Corso CFMA. LS-SIMat

M(g) F M(liq) K = peq M M M M Nucleation Consider the simplest case where n moles of gas M are transferred to the liquid phase at constant temperature. The work of isothermal compression is If the pressure of gas is greater than the equilibrium value, then growth is favorable. If it is less than , the liquid will evaporate. However, for a curved droplet, the growing liquid must also create new surface as it forms Corso CFMA. LS-SIMat

Nucleation (cont’d) • If Vm = MW/r is the molar volume of M in the liquid phase, then n moles will cause a volume increase of, hence • There is a balance point when • Also Corso CFMA. LS-SIMat

Nucleation (cont’d) Corso CFMA. LS-SIMat

Ostwald ripening The growth of large particles at the expense of smaller ones, owing to a difference in solubility rates of different size particles • There is only one critical radius for any given set of S, T, p,g. Hence all crystallites in an ensemble are either above or below Rcrit. If they are above, they will grow, if they are below, they will evaporate. • Ostwald ripening starts once nucleation is complete and near the end of the growth phase as monomer decreases. Corso CFMA. LS-SIMat

Chimica Fisica dei Materiali Avanzati Part 4 – Forces between particles and surfaces