Colloid science

The colloid science behind TDA Logic

Most of TDA Logic’s loading budgets rest on classical colloid science: DLVO theory, double-layer screening, specific ion effects, depletion forces and Ostwald ripening. This page sets out each mechanism, the budget it informs, and what light scattering can report.

Published 6 November 2026 · Primary reference: Pensive Beauty® Nanoscience Labs (2026), Zenodo

What does DLVO theory say about particle stability?

DLVO theory (Derjaguin & Landau 1941; Verwey & Overbeek 1948) writes the interaction energy of two charged particles in an electrolyte as the sum of van der Waals attraction and the repulsion that arises when their electrical double layers overlap. For two equal spheres of radius a at surface separation h much smaller than a, standard approximate forms are (see Israelachvili 2011):

Here A is the Hamaker constant across water, ψ₀ the surface potential and κ⁻¹ the Debye length. The repulsive form is the constant-potential expression from linearised Poisson–Boltzmann theory; it assumes a low (roughly ≤ 25 mV) surface potential and κa large. When repulsion is strong and long-ranged relative to attraction, the total curve has a maximum, the energy barrier; if it is many times k_BT, particles seldom reach contact and the dispersion is kinetically stable. Anything that lowers ψ₀ or shortens κ⁻¹ lowers that barrier; for an architecture specified as anionic with electrostatic (DLVO) stabilisation, these are the quantities the budgets protect.

How does ionic strength set the Debye length?

The Debye length is the decay distance of the double-layer potential. In the linearised (Debye–Hückel) treatment:

For a 1:1 electrolyte such as sodium chloride, I equals the molar concentration.

Each tenfold rise in ionic strength shortens the Debye length by √10, roughly a factor of 3.2. As a sense of scale, 1 % w/w of a monovalent salt with a molar mass near 151 g/mol (sodium PCA, for example) is about 66 mM, which gives κ⁻¹ ≈ 1.2 nm.

Why do multivalent ions count disproportionately?

Because each ion enters I weighted by the square of its charge. At the same 10 mM concentration, sodium chloride gives I = 10 mM; calcium chloride gives ½(10 × 2² + 20 × 1²) = 30 mM; magnesium sulfate gives ½(10 × 2² + 10 × 2²) = 40 mM. The corresponding Debye lengths are about 3.0, 1.8 and 1.5 nm. Written per molar concentration c, these are the familiar forms 0.304/√c, 0.176/√c and 0.152/√c nm for 1:1, 2:1 and 2:2 salts. Hence the ionic-strength budget counts every ionic ingredient, chelators, buffers and counterions included, and cannot be read from weight percentages.

Why do multivalent ions do more than screen?

The Schulze–Hardy rule describes how steeply the critical coagulation concentration (CCC) falls with counterion valence z. For highly charged surfaces DLVO theory gives CCC ∝ z⁻⁶, so mono-, di- and trivalent counterions coagulate at concentrations in the ratio of roughly 1 : 1/64 : 1/729. At low surface potential the dependence weakens towards z⁻². For an anionic surface the counterions in question are cations.

Multivalent cations also act specifically. Calcium and magnesium coordinate carboxylate, phosphate and sulfate head groups; one ion can hold head groups on two surfaces and so bridge particles; some pairings precipitate as sparingly soluble salts at the interface. Stability is lost at lower concentrations than ionic strength alone predicts. Placement criterion 3 therefore minimises or excludes multivalent metal ions; chelating trace metals is a routine countermeasure.

What happens when cationic species meet an anionic surface?

Cationic surfactants, quaternary ammonium compounds and polycations adsorb onto an anionic surface. As the net charge falls towards zero the repulsive barrier disappears and particles aggregate; polycations can also bridge. Excess cation can reverse the charge, giving a different dispersion altogether. Either way the anionic design assumption fails, so cationic species are excluded rather than assigned a domain.

Why does free polymer cause depletion flocculation, and why does molecular weight matter?

Non-adsorbing polymer is excluded from a surface layer roughly as thick as the coil’s radius of gyration, R_g. When two particles come closer than twice that thickness, polymer leaves the gap and the osmotic pressure of the surrounding solution pushes them together (Asakura & Oosawa 1954). In the dilute limit the osmotic pressure is about n_pk_BT, and the attraction at contact is roughly that pressure multiplied by the overlap volume of the two depletion zones.

Molecular weight raises the risk in two ways. R_g grows as M^ν (ν about 0.5–0.6), so the depletion layer thickens and the attraction reaches further. And at a fixed weight percentage, larger coils fill more of the solution volume (the overlap concentration c* falls as M rises), so flocculating conditions are reached at lower weight percentages. Hence the free-polymer budget shrinks as molecular weight increases. A strong polymer network can restrain particles mechanically, but that rheological effect does not remove the attraction.

What drives Ostwald ripening, and how can it be slowed?

The Kelvin effect makes the solubility of a dispersed oil in water rise as particle radius falls: c(r) = c(∞) exp(2γV_m / rRT). Molecules therefore diffuse from smaller particles to larger ones. Lifshitz–Slyozov–Wagner (LSW) theory predicts that, at steady state, the cube of the mean radius grows linearly with time (Lifshitz & Slyozov 1961; Wagner 1961), at a rate:

where c(∞) is the bulk solubility of the oil in water, γ the interfacial tension, V_m the oil’s molar volume, D its diffusion coefficient in water and ρ its density (form as given by Tadros et al. 2004). The rate is proportional to solubility, so an oil with even slight water solubility ripens far faster than an insoluble one. This is the physics behind placement criterion 2.

Ripening is inhibited by a much less water-soluble component in the same particles. As the soluble component leaves small particles, the insoluble one concentrates there, lowering the chemical potential of what remains; this opposes the Kelvin excess and can halt net transfer (Kabalnov & Shchukin 1992). In practice, a slightly soluble ester sharing an oil core with a triglyceride ripens more slowly than the ester alone.

Kinetic or thermodynamic stability: which does TDA Logic assume?

Nanometre-scale oil-in-water dispersions made with energy input have a positive free energy of formation, because the interfacial term γΔA outweighs the configurational entropy gain. They are thermodynamically unstable but can stay kinetically stable for long periods when aggregation barriers are high and ripening slow (Tadros et al. 2004; McClements 2012). Microemulsions differ: they form spontaneously and are thermodynamically stable.

TDA Logic assumes kinetic stabilisation of coexisting populations by an electrostatic barrier. It does not assume thermodynamic stability or spontaneous formation. Molecular packing theory describes the equilibrium self-assembly of single amphiphiles into micelles and bilayers (Israelachvili, Mitchell & Ninham 1976); packing theory is silent on a multi-component structure assembled with energy input and held in a kinetically stabilised state, and it neither predicts nor excludes one. The budgets exist to keep barrier height and ripening rate where the design places them.

What do DLS Z-average and PDI actually measure?

Dynamic light scattering converts the autocorrelation of scattered-intensity fluctuations into a diffusion coefficient D, and then a hydrodynamic diameter via Stokes–Einstein, d_H = k_BT / (3πηD). Cumulants analysis (ISO 22412) reports the Z-average, an intensity-weighted harmonic mean hydrodynamic diameter, and the polydispersity index (PDI), a dimensionless width from the second cumulant, μ₂/Γ̄².

For particles much smaller than the wavelength, scattered intensity scales with roughly the sixth power of diameter, so a few larger particles dominate the signal and micelles contribute little even when numerous.

A tri-domain system holds three populations of different size by design; the cumulants fit treats them as one broadened peak, so PDI reads higher than for a single-population carrier of similar Z-average. A broader reading is a design expectation, not a fault in itself, and PDI benchmarks from single-population systems do not transfer. What carries information is batch-to-batch consistency against the architecture’s own specification, and change over time: a rising Z-average signals ripening or aggregation. Distribution algorithms struggle to separate populations closer than about a factor of two to three in size, so multimodal results need cautious reading. DLS reports size, not location: it cannot show which ingredient sits in which domain.

What is specified, what is measured and what is modelled?

References

1. Pensive Beauty® Nanoscience Labs (2026). TDA Logic: The Tri-Domain Placement Logic Succeeding 1940s HLB Logic in Nanoscale Cosmetic Formulation (Version 1.0) [Preprint]. Zenodo. doi:10.5281/zenodo.23384911
2. Derjaguin B, Landau L (1941). Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes. Acta Physicochimica URSS 14:633–662.
3. Verwey EJW, Overbeek JThG (1948). Theory of the Stability of Lyophobic Colloids. Elsevier.
4. Israelachvili JN (2011). Intermolecular and Surface Forces, 3rd ed. Academic Press.
5. Asakura S, Oosawa F (1954). On interaction between two bodies immersed in a solution of macromolecules. J Chem Phys 22:1255–1256.
6. Lifshitz IM, Slyozov VV (1961). J Phys Chem Solids 19:35–50.
7. Wagner C (1961). Z Elektrochem 65:581–591.
8. Kabalnov AS, Shchukin ED (1992). Ostwald ripening theory: applications to fluorocarbon emulsion stability. Adv Colloid Interface Sci 38:69–97.
9. Tadros T, Izquierdo P, Esquena J, Solans C (2004). Formation and stability of nano-emulsions. Adv Colloid Interface Sci 108–109:303–318. doi:10.1016/j.cis.2003.10.023
10. McClements DJ (2012). Nanoemulsions versus microemulsions: terminology, differences, and similarities. Soft Matter 8:1719–1729.
11. Israelachvili JN, Mitchell DJ, Ninham BW (1976). Theory of self-assembly of hydrocarbon amphiphiles into micelles and bilayers. J Chem Soc Faraday Trans 2 72:1525–1568. doi:10.1039/F29767201525