Colloid Chemistry. La chimica moderna e la sua comunicazione Silvia Gross.

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1 Colloid Chemistry La chimica moderna e la sua comunicazione Silvia Gross Istituto Dipartimento di Scienze di e Scienze Tecnologie Chimiche Molecolari ISTM-CNR, Università Università degli Studi degli Studi di Padova di Padova silvia.gross@unipd.it silvia.gross@unipd.it

2 Electrical double layer when ions present in a system containing an interface variation in the ion density near that interface described by the left profile: Gedankenexperiment: separate the two surfaces Ions close to an interface: equal and opposite charge Charged portions: EDL

3 Electrical double layer: aims - Describe the electrical double layer (EDL) through models - Refine and make more realtistic models - Obtain equations for the distribution of charges in EDL - Obtain equations for the potential in EDL - Expression for overlapping EDL of different surfaces - Implications on colloidal stability

4 Electrical double layer: develop the concept. 1. Origin of charge at colloidal surface 2. Distribution of charge near the surface: capacitor model 3. More realtistic approximations: Debye Hückel and Gouy-Chapman models 4. Several assumptions/semplifications: In a first step: a. Planar surface b. Isolated surface c. Constant potential surface In a second step: a. Variation of the potential with distance from surface b. Effect of added electrolyte on the potential c. Effect of spherical or cylindrical surfaces 5. Application of the model(s) to coagulation phenomena (effects of electrostatic forces and van der Walls interactions): two surfaces approaching each other 6. Structure of the Stern layer (inner edge of the double layer) accounting for preferential ion adsorption

5 to understand coagulation phenomena and stability - when two similar surfaces approaches each other - coagulation phenomena in terms of potential energy curves - Schulz-Hardy rule

Electrical double layer 1. quantitative treatment of the electrical double layer: extremely difficult problem 2. several assumptions and simplifications 3.

6 Electrical double layer 1. quantitative treatment of the electrical double layer: extremely difficult problem 2. several assumptions and simplifications 3. case of an infinite, flat, charged planar surface 4. x is the distance normal to the surface. x see Hunter, R Foundations of Colloid Science I & II, Oxford, 1989

7 Electrical double layer E = δψ δx negative since potential decreases as distance increases Y 0 Y ( x) Y 0 is the electric potential at the surface Y (x) is the electric potential at a distance x from the surface

8 Electrical double layer Aim: Understand the potential variation between x 1 (small distance into phase a) and x 2 (small distance into phase b; (distances in the order of molecular dimensions) Variation of electrochemical potential in the vicinity of the interface between two phases a and b: (a) according to a schematic profile and (b) to the parallel plate capacitor model x =molecular distances Source of the picture: Paul C. Hiemenz, Raj Rajagopalan, Principles of Colloid and Surface Chemistry, CRC Press Book

9 Electric double layer 1. Origin of charge at colloidal surface 2. Distribution of charge near the surface: capacitor model 3. More realtistic approximations: Debye Hückel and Gouy-Chapman models 4. Several assumptions/semplifications: a. Planar surface b. Isolated surface c. Constant potential surface In a second step: a. Variation of the potential with distance from surface b. Effect of added electrolyte on the potential c. Effect of spherical or cylindrical surfaces 5. Application of the model(s) to coagulation phenomena (effects of electrostatic forces and van der Walls interactions) 6. Structure of the Stern layer (inner edge of the double layer) accounting for ion adsorption

The relationship between the charge on a capacitor and the voltage across the capacitor is linear with a constant, C, called the

10 Simple capacitor model Q = C V where: V is the voltage across the plates. You will need to define a polarity for that voltage. Q is the charge on the plate with the "+" on the voltage polarity definition. C is a constant - the capacitance of the capacitor. The relationship between the charge on a capacitor and the voltage across the capacitor is linear with a constant, C, called the capacitance. Q = C V When V is measured in volts, and Q is measured in couloumbs, then C has the units of farads. Farads are coulombs/volt.

11 Staring point: Coulomb law F C = q 1q 2 4πε 0 ε r 1 r 2 e 0 = 8.85*10-12 C 2 * J -1 * m -1 e r = dielectric constant of medium (e.g. 78 for water) e e e o r

12 Electric field close to colloid surface strenght of electric field E: force per unit charge E = 2.26 * 10 7 Vm -1 Very strong field closed to a charged interface s = surface charge density in C/m 2 Conditions: Areas per molecule 10 nm 2 monovalent ion (1.6 *10-19 C) Charge of a molecule having an area of 10 nm 2 and monovalent ion s* = (ion/10 nm 2 )=(1.60*10-19 C/ion)/(10*10-18 m 2 ) = 1.6*10-2 Cm -2

13 Potential drop between two phases E = δψ δx = Ψ δ = σ ε 0 ε r ignore the directional (sign) aspect here Ψ potential drop between plates separates by a distance δ Determine distance over which potential drop occurs from the initial value of E (assume to have plate separation d in equivalent capacitor) δ = Ψ E = 0.1 V Vm 1 = m = 4. 4 nm distance over which surface charge neutralisation is accomplished

Introduce the concept of diffuse double layer: charge density changes with distance from the interface

14 Limits of the capacitor model We have assumed the parallel plates of a capacitor as preliminarly model, but: If one of the phases is an aqueous electrolyte solution one: inadequacy of the model Introduce the concept of diffuse double layer: charge density changes with distance from the interface Or Variation of potential with distance from a charged surface of arbitrary shape through classical electrostatics

15 Poisson equation For a charged interface of arbitrary shape the potential is related to the charge density by Poisson s equation (variation of potential with distance from an interface: fundamental law of electrostatics) r = q/v = volume charge density in C/m 3 (x, y, z) δ 2 Ψ δx 2 + δ2 Ψ δy 2 + δ2 Ψ δz 2 = ρ ε e e e o r (1)

Poisson equation interaction between the surface charge and the counter-ions is quantitatively described by the Poisson equation, which relates the electrical potential Y(x, y, z) due

16 Poisson equation interaction between the surface charge and the counter-ions is quantitatively described by the Poisson equation, which relates the electrical potential Y(x, y, z) due to the charge density at any position x, y, z away from the surface and the charge density r(x,y, z) at this position. (1) Assumption: isolated surface (no overlaps of EDL, TBD later on)

17 Poisson equation: physical meaning Interpretation of field as force per unit charge field at distance r from a charge +q q = 4 3 πρ r 3 e e 0 e r E c = rρ 3ε

18 Poisson equation: physical meaning E c = rρ 3ε Multiply by r 2 and differentiate with respect to r (dr) d(r 2 E c ) dr = d r3 ρ 3εdr = 3r2 ρ 3ε = r2 ρ ε E = Ψ r negative since potential decreases as distance increases

19 Laplacian operator (1)

20 Laplacian in spherical coordinates

21 Poisson equation: physical meaning E c = rρ 3ε Multiply by r 2 and differentiate with respect to r (dr) d(r 2 E c ) dr = d r3 ρ 3εdr = 3r2 ρ 3ε = r2 ρ ε E = Ψ r negative since potential decreases as distance increases

22 Poisson equation: physical meaning 1 r 2 r r 2 Ψ r = ρ ε = 2 Ψ Poisson equation 2 Ψ can be written in spherical coordinates (q, f) radial dependence of potential identifies with 2 Ψ for spherical symmetry (spherical distribution of charge) potential Ψ independent on q, f d(r 2 E c ) dr = d r3 ρ 3εdr = 3r2 ρ 3ε = r2 ρ ε

23 Electric double layer 1. Origin of charge at colloidal surface 2. Distribution of charge near the surface: capacitor model 3. More realtistic approximations: Gouy-Chapman and Stern models 4. Several assumptions/semplifications: a. Planar surface b. Isolated surface c. Constant potential surface In a second step: a. Variation of the potential with distance from surface b. Effect of added electrolyte on the potential c. Effect of spherical or cylindrical surfaces 5. Application of the model(s) to coagulation phenomena (effects of electrostatic forces and van der Walls interactions) 6. Structure of the Stern layer (inner edge of the double layer) accounting for ion adsorption

Inner/Outer Helmholtz planes Inner Helmholtz Plane (IHP) plane cutting through the center of the adsorbed species. IUPAC: The locus of the electrical centres of specifically adsorbed ions.

24 Inner/Outer Helmholtz planes Inner Helmholtz Plane (IHP) plane cutting through the center of the adsorbed species. IUPAC: The locus of the electrical centres of specifically adsorbed ions. Outer Helmholtz plane (OHP) plane cutting through the counter ions at their position of closest approach. IUPAC: At an electrified interface, the locus of the electrical centres of non-specifically adsorbed ions in their position of closest approach.

Inner/Outer Helmholtz planes loose solvation sheaths and approach surface very closely Electrochemistry 2nd, Completely Revised and Updated Edition, Wiley VCH Carl H.

25 Inner/Outer Helmholtz planes loose solvation sheaths and approach surface very closely Electrochemistry 2nd, Completely Revised and Updated Edition, Wiley VCH Carl H. Hamann, Andrew Hamnett, Wolf Vielstich Souce of picture: Chem. Commun., 2011, 47, only approach within a given distance determined by the extent of solvation sheats

26 Shear (or slipping plane) As the particle moves through solution, due to gravity or an applied voltage, the ions move with it. At some distance from the particle there exists a boundary, beyond which ions do not move with the particle. This is known as the surface of hydrodynamic shear, or the slipping plane, and exists somewhere within the diffuse layer. It is the potential that exists at the slipping plane that is defined as the zeta potential

27 Models for charge distribution Molecular capacitor as basic model Difference with respect to bulk capacitor as microelectronic real device: the capacity of the molecular capacitor depends from potential Example: metallic surface in an electrolyte solution

28 Models for charge distribution Simple capacitor model Helmholtz: counterions form a plate capacitor Gouy-Chapman: thermal movement creates diffuse double layer Stern: linear potential function continues exponentially after outer Helmholtz plane

29 Helmhotlz model of EDL Helmholtz, 1879 Helmholtz, H., Studien über elektrische Grenzschichten, Ann. Phys., (1879) Electrostatic interaction between the surface and counter-ions of the solution Electric double layer forms (capacitor) Rigid picture, valid only at T = 0 K Linear potential drop from the surface to the outer Helmholtz plane (OHP): in reality is exponential This model does not take into account: - thermal motion - ion diffusion - adsorption onto the surface of ions - interactions away from OHP - solvent/surface interactions - nature, origin and amount of charge

30 Gouy-Chapman model of EDL Gouy & Chapman, Flat, infinite, uniformly charged surface Ions considered as point charges Boltzmann statistical distribution near the surface Accounts for thermal motion Not IHL considered: only diffuse layer (+ and ions) Counter-ion concentration decreases, co-ion concentration increases from the surface (in the bulk solution the two concentrations become equal) Exponential potential decrease Debye length (1/κ): thickness of the diffuse double layer or thickness of an equivalent plane capacitor Chapman, D. L., A contribution to the theory of electrocapillarity, Philos. Mag. 6, (1913) Gouy, M. G., Sur la constitution de la charge electrique a la surface d'un electrolyte, J. Phys. Radium, (1910)

31 Electrical double layer (Stern model) Electric double-layer according to the Stern model. The inner and outer Helmholtz planes are indicated as IHP and OHP, respectively. The slipping plane is denoted by S and its charge is characterized by the z potential

32 Stern model of EDL Stern, 1924 Stern, O., Z. Elektrochem., 30, 508 (1924) First layer of solvated ions of finite size, tightly adsorbed onto the surface (OHP varies with ionic radius) Subsequent layers as point charges like in the Gouy-Chapman model Slipping (shear) plane: at the boundary of the diffuse layer Potential at the shear plane: ζ or electrokinetic potential If shear plane and Stern-plane close enough: Ψ St ζ The model can deal with specific ion sorption

33 Stern model of EDL Charge reversal If polyvalent or surface-active co-ions are adsorbed, charge reversal can occur In this case, Ψ 0 and Ψ St have different sign The elektrokinetic potential (ζ) changes also its sign

34 Stern model of EDL Overcharge If surfactant co-ions adsorb to the interface, ΨSt can become bigger than Ψ0 charge increase If the Stern plane and the shear planes are close enough, elektrokinetic potential also increase

Summary of the EDL models Models of the electrical double layer at a positively charged surface: (a) the Helmholtz model, (b) the Gouy Chapman model, and (c) the Stern model, showing the inner

35 Summary of the EDL models Models of the electrical double layer at a positively charged surface: (a) the Helmholtz model, (b) the Gouy Chapman model, and (c) the Stern model, showing the inner Helmholtz plane (IHP) and outer Helmholtz plane (OHP). The IHP refers to the distance of closest approach of specifically adsorbed ions (generally anions) and OHP refers to that of the non-specifically adsorbed ions. The OHP is also the plane where the diffuse layer begins. d is the double layer distance described by the Helmholtz model. φ 0 and φ are the potentials at the electrode surface and the electrode/electrolyte interface, respectively., Chem. Soc. Rev., 2009, 38,

36 Electrical double layer: phenomenological description 1. Most colloidal particles acquire a surface charge in a polar liquid, often negative, via a number of mechanisms (see Origin of charge, Lecture Charge 1) 2. A key prerequisite for the formations/presence of this surface charge is the high dielectric constant e r of a polar medium which reduces the Coulomb energy between oppositely charged ions. The Coulomb energy favours combining the surface charge and the opposite charges (called counter-ions) to maintain charge neutrality, whereas the entropic thermal agitation of the counter-ions favours smearing them out throughout the polar medium. 3. The balance between these two effects results in the formation of the electric double layer: the surface charge layer and the diffuse layer of opposite charges.

V. Electrostatics. MIT Student

V. Electrostatics. MIT Student V. Electrostatics Lecture 26: Compact Part of the Double Layer MIT Student 1 Double-layer Capacitance 1.1 Stern Layer As was discussed in the previous lecture, the Gouy-Chapman model predicts unphysically