Dispersing Powders in Liquids, Part 2, by Ralph D. Nelson, Jr.

-- 8: Discussion of Hamaker Constants --

Multi-phase Hamaker Constants from Single-phase Values The Hamaker constant characterizes the resonance interactions between electronic orbitals in two particles and the intervening medium in much the same way as polarizability did in the case of two atom. AHl0l characterizes the interaction between two drops of the liquid (l) separated by a vacuum (0). AHp0p characterizes the interaction between two particles of the material p separated by a vacuum (0). AHq0q characterizes the interaction between two particles of the material q separated by a vacuum (0). The interaction between two different materials (x and y) when they are separated by a vacuum can be characterized by a combined constant, [8a]      AHx0y = [AHx0x AHy0y]0.5 The interaction of a particle of p with a particle of q when they are separated by a liquid (l) can be characterized by a combined constant, [8b]      AHplq = AHl0l + AHp0q - AHp0l - AHq0l. Using [8a] and substituting p, q, and l for x and y as needed, we get a relationship that allows determining AHplq from measurements or estimates based on the individual materials in a vacuum. [8c]      AHplq = AHl0l + AHp0p AHq0q]0.5             - [AHp0p AHl0l]0.5 - [AHl0l AHq0q]0.5 Polarizability interactions can cause apparent repulsion: If AHl0l is intermediate in value between AHp0p and AHq0q, then AHplq will have a negative value. You may check this by replacing AHl0l with (1 + ) AHp0p and AHq0q with (1 + + ) AHp0p, where and are small positive fractions which may approach zero. This yields a coeffient that is negative for any small positive values of delta and epsilon [8d]      AHplq = {1 + + (1 + + )0.5             - (1 + )0.5 - [(1 + )(1 + + )]0.5} AHp0p In this case the liquid is attracted into the space between the particles more strongly than the particles attract one another, with the result that the particles are pushed apart. This is functionally equivalent to particle-particle repulsion. We can take advantage of this phenomenon to disperse a powder in a liquid. This is discussed further in the section on Surface Thermodynamics. Normally interactions are attractive: Usually the two particles are made of the same material, in which case [8d] reduces to [8e]      AHplp = (AHp0p0.5 - AHl0l0.5)2 Since the square of either a positive or a negative quantity is positive AHplp always positive, and the rsonance interaction between two particles of the same material in a liquid always causes attraction. This simplified treatment of the Hamaker constant indicates that if the liquid's Hamaker constant equals that of the particles there will be NO attraction between the particles. A more detailed approach (see Lifschitz discussion below) indicates that there will be some attraction between two particles of the same material unless the liquid's adsorption spectrum is nearly the same as that of the particles. Determining Hamaker Constants A simple fundamental approach: For two spheres composed of identical harmonic oscillators and separated by a vacuum Hiemenz (his page 647) derived [8f]      AHp0p = (3 he / 4) [N0 / (4 0 M)]2 A practical experimental approach: Fowkes (1964) found that the Hamaker constant for two particles of the same material separated by a vacuum, AHx0x, could be estimated from the dispersion (resonance) portion of the surface tension disp,x [N/m] as [8g]      AHx0x (10 / 3) [Mx / (N0 x)] 2/3 disp,x where x [kg/m3] is the material's density and Mx [kg/mol] is the molar mass. Sections 5 and 6 have tables of these quantities for several common solids and liquids For organic materials with no strongly hydrogen-bonding groups Israelachvili (his page 158) developed the approximation [8h]      AHx0x KHest p where KHest = 2.1 * 10-18 m2. A more sophisticated treatment of polarizability: Lifschitz provided a more complete model for the polarizability interaction (see the discussion in Visser). He defined the Hamaker constant in terms of the complex dielectric permittivity (relative to a vacuum) * [dimensionless] integrated over the entire spectrum of the complex frequency i [radians]. [8i]      The above relationship is not easy to evaluate even in those few instances for which the complex dielectric permittivity is known throughout the entire electromagnetic spectrum. Mahanty and Ninham's text provides further discussion of this approach and how the equations may be reduced to some of the approximations used in this section. Israelachvili (his page 144) shows how to use dielectric and refractive index data without doing the full Lifschitz integration. This method treats AH as a free energy with enthalpic and entropic contributions, AH = AHH - T AHS. The enthalpic term is based on mutual polarizability, [8j]      AHHplq = [3 he/ (8 * 20.5] Kpl Kql / Kden where       Kplp = (nRI0,p2 - nRI0,l2) / [nRI0,p2 + nRI0,l2]0.5       Kqlq = {nRI0,q2 - nRI0,l2) / [nRI0,q2 + nRI0,l2]0.5       Kden = [nRI0,p2 + nRI0,l2]0.5 + [ nRI0,q2 + nRI0,l2]0.5 The entropic term is based on the ability of the liquid's molecular dipoles to rotate so as to offset the effect of electric fields in the system. It is determined from (measured at radio frequency, below the dipolar rotational relaxation frequency) as [8k]      AHSplq = (3kT/4) [(p - l) / (p + l)] [(q - l) / (q + l)] This entropic term is always less than 3kT/4, so it is never large compared to the thermal energy kT. For two particles made of the same material, nRI0,p = nRI0,q and p = q, the equations simplify to [8l]      [8m]      AHH will be negative if nRI,l is either larger or smaller than both nRI,p and nRI,l. AHS will be negative if l is either larger or smaller than both p and q. If the overall AH is negative, the mutual polarizability interaction will be an attraction. For nonpolar organic particles dispersed in a nonpolar organic liquid, nRI,p nRI,l, so AHH may be nearly zero. Both AHH and AHS MUST be positive, so since AH = AHH -T AHS, the polymer may be dispersed at low temperatures (where AH is positive), but agglomerate at higher temperatures as AH becomes negative. Distance Dependence of Polarizability Attraction The polarizability attraction between two spheres of different diameters is (Hiemenz, page 648) [8n]      ureso = (- AHplq / 12) [1 / ZN + 1 / ZD + 2 ln (ZN / ZD)] where ZN = zp zq + zp + zq, ZD = ZN + 1, zp = s / dp and zq = s / dq When two identical spheres are close together so that s dp, then       up,reso,close = - AHplp dp / (12 s). Note that for s = dp / 12, u = -AHplq For two identical spheres coated by an adsorbed layer (subscript f) of thickness [m] made up of either adsorbed surfactant or bound liquid, then the polarizability attraction is (Parfitt, page 22) [8o]      ucoated = (1/12) [- AHflf Sf - AHfsf Ss - 2 Sfs (AHflf AHfsf)0.5] where the subscripted distance scaling factors Sw are determined from       Sw = y / YN + y / YD + 2 ln (YN / YD)       and YN = z2 + zy + z, YD = YN + y. For w = f, y = 1, and z = s/(dp + 2 ) For w = s, y = 1, and z = (s + 2 )/dp For w = fs, y = 1, + 2 / dp, and z = (s + 2 ) / dp For this model the distance between outer surfaces along the line-of-centers, s [m], is defined as including the adsorbed or bound layer. Here are some notes on several cases of interest that have been approached by more advanced theory but are not easily applied to practical sitations: For particles at intermediate distances where x > 10 nm the expression defining ureso and AH should include a dependence on x or s to account for retardation -- the time it takes for an electromagnetic wave to propagate from one particle to another, roughly 1/e. The energy equations must be multiplied by a correction factor (see Hiemenz, page 635) which decreases from a value of unity (when the particles are close together) to 23 c/(6 2 e s) when s c e. Thus, at s > 100 nm, the attraction drops at a somewhat faster rate than if not retarded. For atoms the dependence becomes s-7 instead of s-6. For particles the dependence becomes s-2 instead of s-1. For charged particles in the presence of ions in solution AHS must include a multiplier which decreases exponentially with separation to account for effect of the counterion atmosphere around the particles and their effect on charge-charge repulsion.