Chemical potential as a functionof surface curvature

The chemical potential is dependent on the radius of curvature ofa surface

To understand the relationship between chemical potential and surfacecurvature, let us consider transferring material from an infinite flatsurface to a spherical solid particle.

The volume change dV of the spherical particle (R radius) is equal to the atomic volume ( ) times dn (number of atoms transferred):

The work per atom transferred equals to the change of chemicalpotential is given by:

with surface energy (tension), cchemical potential on the particlesurface and chemical potentialon the flat surface.

Combining the two equations:

It describes the chemical potential of an atom in a spherical surface withrespect to a flat reference surface.

Generalization for any type of curved surfaces which can be described bytwo principal radii of curvature, R1 and R2:

Young-Laplace equation

For a convex surface, the curvature ispositive, and thus the chemicalpotential of an atom on such a surfaceis higher than that on a flat surface.Mass transfer from a flat surface to aconvex surface results in an increase insurface chemical potential.Thermodynamically, an atom on a convexsurface possesses the highest chemicalpotential, whereas an atom on a concavesurface has the lowest chemical potential.

Vapour pressure and solubilityAn atom on a convex surface possesses the highest chemical potential. Asimilar relationship is reflected by the difference in vapor pressure andsolubility of a solid.

For a flat surface (assuming the vapor of solid phase obeys the ideal gaslaw):

where µ is the chemical potential of a vapor atom, chemical potentialof an atom on the flat surface, k, the Boltzmann constant and P∞ theequilibrium vapor pressure of flat solid surface. For a curved surface:

where Pc is the equilibrium vapor pressure of the curved solid surface.Combining the two equations:

and

For a spherical particle: Kelvin equation

The vapor pressure of small particles is notably higher than that of the bulk material.

A relation similar to that of vapour pressure can be derived for thedependence of the solubility on surface curvature:

where Sc is the solubility of a curved solid surface, S∞ is the solubility of aflat surface.

Gibbs-Thomson relation

Solubility of thesmaller particles islarger than that ofthe larger particles.

Ostwald ripeningIf two particles with different radii (R1>>R2) are in a solvent, each particlewill develop an equilibrium with the surrounding solvent.

Solubility of the smaller particle is larger than that of the largerparticle. Consequently, there would be a net diffusion of solute fromproximity of the small particle to proximity of the large one.

To maintain the equilibrium, solute will deposit onto the surface of thelarge particle, whereas the small particle has to continue dissolving so asto compensate for the amountof solute diffused away. Thefinal result is that the smallparticle gets smaller andthe large particle getslarger. Ostwald ripening isoften not desirable. Insintering of polycrystallinematerials, it can result inabnormal grain growth,leading to inhomogeneousmicrostructures and lowermechanical properties.

Ostwald ripening has been explored also in the synthesis ofnanoparticles.

It can be used to narrow the size distribution of nanoparticles byeliminating small particles. Many relatively large particles grow atthe expense of a relatively small number of smaller particles.

Ostwald ripening can be promoted by varying processingtemperatures. In the synthesis of nanoparticles from solution, afterthe initial nucleation and subsequent growth, the temperature israised, and thus the solubility of solid in solvent increases topromote Ostwald ripening. As a result, the concentration of solid insolvent falls below the equilibrium solubility of small nanoparticles,and the small particles dissolve into the solvent. Such a growthprocess would stop when the concentration of solid in the solventequals the equilibrium solubility of these relatively largenanoparticles.

Stabilization against agglomeration

Without stabilization mechanisms, the nanostructured materials aremost likely and readily to form agglomerates.

There are two major stabilization mechanisms:

- Electrostatic stabilization- Steric stabilization

Main differences:

- a system using electrostatic stabilization is kinetically stable

- A system using steric stabilization is thermodynamically stable

Agglomeration is an alternativemechanism to reduce the overallsurface energy. In nanostructurefabrication is important toprevent their agglomeration.

Electrostatic stabilizationThe surface charge on nanoparticles produced in polar solvent or inan electrolyte solution will be developed through one or more of thefollowing mechanisms:

(1) Preferential adsorption of ions(2) Dissociation of surface charged species(3) Isomorphic substitution of ions(4) Accumulation or depletion of electrons at the surface(5) Physical adsorption of charged species onto the surface

The electrical charge density (electrode potential) E for a solidsurface in a liquid medium is given by the Nerst equation:

With E0 is the standard electrode potential when the concentration ofions is unity, ni is the valence state of ions, ai is the activity of ions, Rgis the gas constant, T is temperature and F is the Faraday’s constant.The surface potential of a solid varies with the concentration ofthe ions in the surrounding solution and can be either positive ornegative.

The surface charge in oxides is mainly derived from preferentialdissolution or deposition of ions. Ions adsorbed on the solid surfacedetermine the surface charge, they are called charge determining ionsor co-ions.In the oxide systems, typical charge determining ions are protons andhydroxyl groups and their concentrations aredescribed by pH (pH = -log [H+]).As the concentration of charge determining ionsvaries, the surface charge density changesfrom positive to negative or vice versa. Theconcentration of charge determining ionscorresponding to a neutral or zero-chargedsurface is defined as a point of zero charge(p.z.c.). At pH > p.z.c., the oxide surface isnegatively charged, since the surface is coveredwith hydroxyl groups, OH-. At pH < p.z.c., H+ isthe charge determining ions and the surface ispositively charged.The surface charge density E in volt isrelated to the pH and the Nernst equation:

At RT:

Electric potential at the solidsurface

When a surface charge density of a solid surface is established, therewill be an electrostatic force between the solid surface and thecharged species in the proximity to segregate positive and negativelycharged species.In the solution, there always exist both surface charge determiningions and counter ions, which have charge opposite to that of thedetermining ions.Distributions of both ions are mainly controlled by a combinationof the following forces:

(1) Coulombic force or electrostatic force(2) Entropic force or dispersion(3) Brownian motion

The concentration of counter ions is the highest near the solid surface and decreases as the distance from the surface increases.The result of the inhomogeneous distribution of ions is the formation of so-called double layer structure

In the Stern layer the electric potential drops linearly. In the Gouy layer, the counter ions diffuse freely and the electric potential does not reduce linearly:

where h > H, which is the thickness of the Stern layer, 1/ is known asthe Debye-Huckel screening strength.

F is Faraday's constant, 0 is thepermittivity of vacuum, is thedielectric constant of the solvent, andCi and Zi are the concentration andvalence of the counter ions of type i.Double layer thickness is typically ofapproximately 10 nm or larger.Extension to curved surfaces.Electrostatic repulsion between twoparticles, when the double layeroverlaps and repulsive force develops.

Electrostatic repulsion of twoequally sized (r) particles.

van der Waals attraction potentialFor nanoparticles (and microparticles) in a solvent van der Waalsattraction force (weak force significant only at very short distance) andBrownian motion play important roles, whereas the influence of gravityis negligible.The total interaction energy or attraction potential is:

A is a positive constant known as theHamaker constant (magnitude of theorder of 10-19 – 10-20 J) which depends onthe polarization properties of themolecules in the two particles and in themedium which separates them.

The formula assumes a simple form if we consider that the separationdistance between two equal sized spherical particles is significantlysmaller than the particle radius, i.e. S/r << 1:

It should be noted that the interaction between two molecules issignificantly different from that between two particles. Van der Waalsinteraction energy between two molecules can be represented by:

The attraction force between two particles decaysmuch slowly and extends over distances ofnanometers. As a result, a barrier potential mustbe developed to prevent agglomeration. Twomethods are widely applied to prevent agglomerationof particles: electrostatic repulsion and stericexclusion.

The electrostatic stabilization of particles in a suspension issuccessfully described by the DLVO theory, named after Derjaguin,Landau, Venvey and Overbeek.

The total interaction between two particles, which are electrostaticstabilized, is the combination of van der Waals attraction andelectrostatic repulsion:

DLVO Theory

Important assumptions in DLVO theory:- Infinite flat solid surface- Uniform surface charge density- No redistribution of surface charge (i.e. the surface electric potentialremains constant)- No change of concentration profiles of both counter ions and surfacecharge determining ions, i.e. the electric potential remains unchanged- Solvent exerts influences via dielectric constant only, i.e. no chemicalreactions between the particles and solvent.DLVO theory works very well in explaining the interactions between twoapproaching particles, which are electrically charged, and thus is widelyaccepted in the research community of colloidal science.

Near the surface a deep minimum in the potential energy produced by thevan der Waals attraction.A maximum is located a little farther away from the surface, as the electricrepulsion potential dominates the van der Waals attraction potential. Themaximum is also known as repulsive barrier. If the barrier is greaterthan 10kT, where k is Boltzmann constant, the collisions of twoparticles produced by Brownian motion will not overcome the barrierand agglomeration will not occur.

The overall potential is strongly influenced by the concentration andvalence state of counter ions. An increase in concentration andvalence state of counter ions results in a faster decay of the electricpotential and as a result, the repulsive barrier is reduced and itsposition is pushed towards the particle surface.

The secondary minimum is present onlywhen the concentration of counter ionsis higher enough. If secondaryminimum is established, particles arelikely to be associated with each other,which is known as flocculation.

As the distance reduces, the repulsion increases and reaches themaximum when the distance between two particle surfaces equals thedistance between the repulsive barrier and the surface.The repulsion can be understood considering the overlap of electricpotentials of the two particles. It is not directly due to the surfacecharge on solid particles, but to the interaction between two doublelayers.An alternative explanation is theOsmotic flow. When two particlesapproach one another, theconcentrations of ions between the twoparticles increase significantly. Theoriginal equilibrium concentrationprofiles (counter ions and surfacecharge ions) are destroyed. To restorethe original equilibrium, more solventneeds to flow into the region where thetwo double layers overlap. Such anosmotic flow of solvent effectivelyrepels two particles apart.

The DLVO theory is widely applied in practice, as far as the followingconditions are met:

(1) Dispersion is very dilute, so that the charge density and distributionon each particle surface and the electric potential in the proximity next toeach particle surface are not interfered by other particles.

(2) No other force is present besides van der Waals force andelectrostatic potential, i.e. the gravity is negligible or the particle issignificantly small, and there exist no other forces, such as magnetic field.

(3) Geometry of particles is relatively simple, so that the surfaceproperties are the same over the entire particle surface, and, thus surfacecharge density and distribution as well as the electric potential in thesurrounding medium are the same.

(4) The double layer is purely diffusive, so that the distributions ofcounter ions and charge determining ions are determined by all threeforces: electrostatic force, entropic dispersion and Brownian motion.

Electrostatic stabilization conditions

Electrostatic stabilization is limited by :

(1) Electrostatic stabilization is a kinetic stabilization method.

(2) It is only applicable to dilute systems.

(3) It is not applicable to electrolyte sensitive systems.

(4) It is almost not possible to redisperse the agglomeratedparticles.

(5) It is difficult to apply to multiple phase systems, since in agiven condition, different solids develop different surface chargeand electric potential.

Electrostatic stabilization problems

Steric stabilizationSteric stabilization or polymeric stabilization is a method widely usedin stabilization of colloidal dispersions. The main advantages (respectto electrostaic stabilization) are:

(1) It is a thermodynamic stabilization method, so that the particlesare always redispersible.(2) A very high concentration can be accommodated, and thedispersion medium can be completely depleted.(3) It is not electrolyte sensitive.(4) It is suitable to multiple phase systems.

Polymer layer adsorbed on the surface of nanoparticles serves as adiffusion barrier to the growth species, resulting in a diffusion-limited growth in the subsequent growth of nuclei. Therefore, the stericstabilization reduces the size distribution and leads to monosizednanoparticles.

Steric stabilization is widely used in the synthesis of nanoparticles.

Solvent and polymer

When a solvable polymer dissolves into a solvent, the polymer interactswith the solvent and the interaction depends on the temperature. Whena polymer in a solvent tends to expand to reduce the overall Gibbsfree energy of the system, such a solvent is called a “good solvent”.When polymer in a solvent tends to coil up or collapse to reduce theGibbs free energy, the solvent is considered to be a “poor solvent”.

In a system (a given polymer in a given solvent), if the solvent is a “good”or “poor” depends on the temperature. At high temperatures, polymerexpands, whereas at low temperatures, polymer collapses.The temperature, at which a poor solvent transfers to a good solvent, isthe Flory-Huggins theta temperature, or simply the temperature. At T=, the solvent is considered to be at the theta state, at which the Gibbsfree energy does not change whether the polymer expands or collapses.

Depending on the interaction between polymer and solid surface, a polymer can be grouped into:(1) Anchored polymer, which irreversibly binds to solid surface by one end only, and typically are diblock polymer(2) Adsorbing polymer, which adsorbsweakly at random points along thepolymer backbone(3) Non-adsorbing polymer, which doesnot attach to solid surface and thus doesnot contribute to polymer stabilization.

The adsorption can be either by formingchemical bonds between surface ions oratoms on the solid and polymer moleculesor by weak physical adsorption.

Interaction between polymer layerWhen two particles approachone another, the attachedpolymers interact only whenthe separation distance, H,between the surfaces of twoparticles is less than twicethe thickness, L, of polymerlayers. When the distancereduces to less than 2L, butis still larger than L, therewill be interactions between solvent and polymer and between two polymerlayers. But there is no direct interaction between the polymer layer ofone particle and the solid surface of the opposite particle. If thecoverage of polymer on the solid surface is not complete (less than 50%coverage) the two polymer layers tend to interpenetrate to reduce theavailable space between polymers and reducing the freedom of the polymers.In this situation there will be a reduction of entropy (S < 0) and consideringH 0, the final result for the Gibbs free energy is:

So two particles repel one another and the distance between two particlesmust be equal to or larger than twice the thickness of polymer layers.

Anchored polymers

When the coverage of polymer is high, particularly approaching100%, there would be no interpenetration. As a result, the twopolymer layers will be compressed, leading to the coil up ofpolymers in both layers. The overall Gibbs free energy increases, andrepels two particles apart. When the distance between the surfaces oftwo particles is less than the thickness of polymer layers, a furtherreduction of the distance would force polymers to coil up and result inan increase in the Gibbs free energy.

If we consider a poor solvent with low coverage for distance L<H<2Lpolymers adsorbed onto the surface tend to penetrate into the polymerlayer of the other particle. This interpenetration promote a coil up ofthe polymers and the result is a decreasing of the overall Gibbs freeenergy. The particles tend to merge one each other.With high coverage the behaviouris similar to good solvent, nointerpenetration and increase inthe overall free energy.In general, two particles coveredwith polymer layers are preventedfrom agglomeration by the spaceexclusion or steric stabilization.

Adsorbed polymersa) A polymer originally attached to the solid surface of one particlemay interact with and adsorb onto another particle surface, and thusform bridges between two particles.b) Given sufficient time, attached polymer can desorb from thesurface and migrate out of the polymer layer.

When polymer has a strong adsorption and forms a full coverage,the interaction between two polymer layers produces a purelyrepulsive force and results in an increased free energy.For partial coverage and good solvent the interpenetrationhappens and a more ordered polymer arrangement is formed and asa result the overall energy increases.For partial coverage and poor solvent interpenetration promotesfurther coil up of polymers, leads to increased entropy, and thusresults in a reduced free energy (similar to anchored polymers withlow coverage and poor solvent).

Summarizing, the physical basis for the steric stabilization is (i) avolume restriction effect arising from the decrease in possibleconfigurations in the region between the two surfaces when twoparticles approach one another, and (ii) an osmotic effect due to therelatively high concentration of adsorbed polymeric molecules in theregion between the two particles.

Mixed steric and electricinteractions

Electrosteric stabilization is a mixing of steric and electrostaticstabilization.In this case when two particles approach each other, both electrostaticrepulsion and steric restriction would prevent agglomeration.