SOLUBILITY AND CRYSTAL RADIUS R nc Liquid Solid a a + b Liquid a + b a r

C sinf = 10  g/cm 3 (37°C, water) Carbon Nitrogen Oxigen Sulphur M w = NIMESULIDE (non steroidal antiinflammatory drug) (crystal cell side = 0.87 nm)

C sinf polymer nanocrystrals amorphous NIMESULIDE RELEASE FROM CROSSLINKED PVP (water 37°C)

Liquid a + b a r Kelvin equation 9  sl = solid-liquid surface tension v s = solid solute molar volume R = universal gas constant T = temperature C snc = nanocrystal solubility C sinf = macrocrystal solubility It holds for an ideal solution

 lv  Vapour  sv  sl Solid substrate Liquid drop EQUAZIONE DI YOUNG Per sostanza pura  = 0 ===>

Melting temperature and enthalpy dependence on crystal radius solid liquid vapor

 sl = solid-liquid interfacial tension  sv = solid-vapour interfacial tension  lv = liquid-vapour interfacial tension A sv = solid-vapour interfacial area A sl = solid-liquid interfacial area A lv = liquid-vapour interfacial area liquid-vapour surface first curvature liquid-vapour surface second curvaturesolid-liquid surface second curvature Solid-liquid surface first curvature solid-vapour surface second curvaturesolid-vapour surface first curvature constants For a sphere: r sl, r sv, r lv curvature radii

Closed system thermal equilibrium chemical equilibrium Remembering that: PsPs PlPl PvPv 1) 2) Young eq. for a pure substance

mechanical equilibrium S LV R sl R sv

Considering the Gibbs-Duhem equation k = 1 ===> only one component (pure substance) From the mechanical equilibrium conditions, it follows:

then: Assuming v l and v s << v v

R nc R lv ≈  TWO LIMITING CONDITIONS R sv does not exist R lv ≈ R sl =R nc R nc R lv

R nc R lv ≈  R lv ≈ R sl =R nc R nc R lv X ncr ≈ 1 Many nanocrystals X ncr ≈ 0 Very few nanocrystals

General equation  h mr and T mr dependence on R nc and X cnr requires an iterative solution of these equations assuming a starting value of X cnr [M. Zhang, et al., Physical Review B 62 (2000) 10548]

X ncr = X ncr 1A Yes Solution: X ncr,  h mr (R nc ), T mr (R nc ) No Numerical solution of: ?  d (drug mass fraction)  h mix (mixture melt. enthalpy) 01  h md (drug melt. enthalpy)  d  h r +  h T )

Nanocrystals size distribution volume occupied by crystals ranging in [R nc – (R nc +dR nc )]

Solubility dependence on crystal radius R nc thermodynamic equilibrium Liquid(a+b) a drug solubility fugacity of pure drug in the state of under-cooled liquid at the system temperature (T) and pressure (P)

1 Solid drug nanocrystals T, P 2 Solid drug nanocrystals T mr, P 3 Liquid drug T mr, P 4 Under-cooled liquid drug T, P isobaric heating Isobaric-isotermic melting isobaric cooling

 d is calculated knowing macro-crystal solubility in the desired solvent

Case study: nimesulide + crosslinked polyvinylpyrrolidone co-ground Ratio 1:3 Co-grinding time: 1, 2 and 4 hours DSC analysis

Nanocrystals differential size distribution

 h mr and T mr dependence on R nc and X ncr (crystal cell side = 0.87 nm)

Nanocrystals solubility dependence on R nc and X ncr (crystal cell side = 0.87 nm)