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* if distances between center of spheres, h < (D + d)/2 . Then the small spheres are excluded from the space between the large spheres [3] |
* if distances between center of spheres, h < (D + d)/2 . Then the small spheres are excluded from the space between the large spheres [3] |
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== Derjaguin Approximation == |
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== Density functional theory == |
== Density functional theory == |
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Causes
Gibbs Free Energy
The increase in entropy leads to a decrease in Gibbs free energy.
Entropy
An entropic force arises when there is overlap of restricted volumes of the (large) spheres or plates in solution. The overlap increases the volume accessible to solute which increases the systems entropy. This process happens spontaneously at constant temperature since the Hemholtz energy decreases.
Sterics
When two parallel plates are in a suspension of small particles at a distance less than two times the radius of the small particles, the small particles cannot fit in between the plates dude to spacial restrictions. There is a region surrounding large spherical particles that small spherical particles in the suspension are excluded from. When spheres get close enough (i.e. h< (Dlarge+dsmall)/2), the regions overlap which reduces the volume unavailable to small spheres.
Osmotic Pressure
When small spheres are excluded from between two parallel plates, osmotic pressure acts of the outer sides of the plates, resulting in aggregation.
Dispersion forces
Asakura-Oosawa model
Asakura-Oosawa Model Earliest model (1954) force is always attractive force is proportional to osmotic pressure p0=kTN/V Theory Distance Between Plates (general case)[1] *two plates in a solution of rigid spherical macromolecules
*If distance between two plates, a, is smaller than the diameter of solute molecules, d, no solute can enter between the plates.
*Pure solvent between plates *Force=osmotic pressure acts on plates * if medium is a very dilute and monodisperse solution, p=kTN ∂lnQ/ ∂a, force p, N is the total number of solute molecules *force causes entropy of macromolecules to increase [2] * force attractive for a < d (or z<2r) [2] Other Cases Plates in a solution of rod like macromolecules *Macromolecules of length (l), where (l^2<<A) A= area of plates Plates in a solution of polymers * The configurational entropy of chain molecules is decreased in the neighborhood of the plates * approximate as diffusion in a vessel with walls which absorb diffusing particles
* P = -Apo{(1-f)- a∂f/∂a}
1-f = attraction from osmotic effect a∂f/∂ repulsion due to chain molecules confined between plates P is on order of <r>, the mean end-to-end distance of chain molecules in free space Large hard spheres in a solution of small hard spheres * Large Spheres D = 2RB * Small spheres D= 2 Rs * if distances between center of spheres, h < (D + d)/2 . Then the small spheres are excluded from the space between the large spheres [3]