Colloids: Dilute Dispersions and Charged Interfaces

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1 Colloids: Dilute Dispersions and Chared Interaces Andreas B. Dahlin ecture 1/6 Jones: , Hamley: Sot Matter Physics 1 Continuous and Dispersed Phases Colloidal system: The dispersed phase appears in the size rane rom a ew nm to a ew μm. May be inluenced but not dominated by ravity! the dispersed phase is not continuous Continuous phase Dispersed phase Gas iquid Solid Gas Not possible! iquid aerosol Solid aerosol iquid Foam Emulsion Sol Solid Solid oam Solid emulsion Very stable called dispersions here Sot Matter Physics 1

d 17-9-15 Repetition: Viscosity Consider a material o thickness d sandwiched between two ininite plates.

2 d Repetition: Viscosity Consider a material o thickness d sandwiched between two ininite plates. We apply a shear stress σ s, which is the orce that the upper plate is pulled with divided by its area (like a pressure). The plates move with a velocity v relative to each other with the material perectly attached to the plates. In an elastic solid this leads to a shear strain e = Δx/d (dimensionless) which eventually balances the orce. In a Newtonian liquid, the velocity radient (strain rate) e/ t = v/d will be linearly related to the stress. The proportionality constant is the dynamic viscosity η: Δx strain top plate (movin) low stress v s v e d t bottom plate (stationary) Sot Matter Physics 3 Reynold s Number What orces act on a colloid? We irst need to understand the role o the liquid. The Reynold s number is a dimensionless parameter which is a measure o inertial orces normalized to viscous orces: v Re is the characteristic lenth o the system (e.. the diameter o a channel or a colloid). The low velocity is v, the luid density ρ and viscosity η. When Re is lower than ~1 the low is laminar (in contrast to turbulent). Colloids have very small compared to low situations in everyday lie (e.. swimmin in a lake). In the nanoworld low is deterministic! Sot Matter Physics 4

17-9-15 Stokes Friction aminar low means dra orce is proportional to velocity: F v The Stokes riction coeicient (rom luid dynamics): 6πR Questionable assumptions: Spherical particle (many colloids

Note that riction orce or turbulent low (oten in air) diers: F Av 17-9-15 Sot Matter Physics 5 Terminal Velocity Gravity, buoyancy and liquid riction or a colloid: F m V Force balance will ive

3 Stokes Friction aminar low means dra orce is proportional to velocity: F v The Stokes riction coeicient (rom luid dynamics): 6πR Questionable assumptions: Spherical particle (many colloids are not spheres). Hard material (many colloids are lexible and solvated). Wikipedia: Stokes law Smooth surace (no surace is perectly smooth). Note that riction orce or turbulent low (oten in air) diers: F Av Sot Matter Physics 5 Terminal Velocity Gravity, buoyancy and liquid riction or a colloid: F m V Force balance will ive terminal velocity: For a sphere: v t V vt πr 3 R 3 4 6πR F V 9 Sedimentation i ρ > ρ and creamin i ρ < ρ. b F v joy o cookin v t can be chaned and colloid size analyzed by centriuation! Suests that all colloids will move either up or down, althouh with dierent speed Sot Matter Physics 6 3

17-9-15 Brownian Motion Microscopic picture: By chance there will be more molecular collisions on one side o a small object eneratin a orce F B in a random direction.

4 Brownian Motion Microscopic picture: By chance there will be more molecular collisions on one side o a small object eneratin a orce F B in a random direction. Principle o Brownian motion with a displacement vector r = (x, y, z): r x The averae movement must be zero and movement in each dimension is independent. Consider Newtonian mechanics (inorin ravity) in one dimension with laminar low: x x FB m t t y z We can use two mathematical identities: x t x x t x x x x x t t t t Sot Matter Physics 7 Kinetic Enery We rewrite the dierential equation with the math tricks: x x FB x m x m t t t Next, perorm an averain o each side and remember that x must be zero on averae: FB x m t From thermodynamics the kinetic enery o a particle is by the equipartition theorem: E k zero I the molecules cannot rotate, vibrate etc. (monoatomic ideal as) this is also the internal enery (U). Assumin all particles move independently in x, y and z: 3 B k T m v x k B T x x t x t x m t zero not zero x t Sot Matter Physics 8 not zero 4

5 Diusion Coeicient Usin m<v x > = k B T we are let with: x t kbt For three dimensions: r 6kBT r t 6 kbt t eneral ormula So we can deine the diusion coeicient (same as in Fick s laws): kbt D Note that D enerally depends only on and T! For Stokes dra we et the amous: kbt D 6πR Sot Matter Physics 9 Stable Dispersions Macroscopic picture: Can the diusive lux overcome ravity to prevent sedimentation (or creamin)? Potential enery chane with heiht or = 9.8 ms - : z F z V z E z Boltzmann statistics ives the probability that a colloid appears at a heiht z relative to the bottom where z = and concentration C : C z V C exp z k T B For creamin, just start rom the surace and reverse the direction o z! C Sot Matter Physics 1 5

6 Mass Balance Consider the diusive lux J (many colloids) at heiht z usin Fick s diusion: J z C V D DC exp z kbt The lux due to sedimentation (or creamin) is the concentration multiplied with the terminal velocity: J sed z Czv Cz t At equilibrium the luxes are equal: DC z V C k T B V V V z V terminal velocity k T z D k T C So it is veriied that Einstein s relation D = k B T is recovered. (Good sanity check!) B B z rom exponential Sot Matter Physics 11 Equilibrium Distribution In a typical 1 cm test tube, the concentration distribution is only interestin when the densities ρ and ρ (1 cm -3 or water) are very similar or when the colloid is very small. Usually one ets either a (almost) homoenous mixture or (almost) ull sedimentation/creamin as the equilibrium state. 1 1 C/C R = 1 nm ρ (cm -3 ) C/C ρ (cm -3 ) always sedimentation R = 1 nm z (m) test tube bottom z (m) Sot Matter Physics 1 6

7 Demonstration: Sedimentation Suspension o 8 nm polystyrene-sulphate colloids (ρ = 1.5 cm -3 ) in water Sot Matter Physics 13 Exercise 1.1 Determine sedimentation rate or a spherical lass (ρ = cm -3 ) particle with R = 5 nm assumin it starts in rest! How lon time does it take to reach the terminal velocity and how hih is it? Sot Matter Physics 14 7

8 Exercise 1.1 Use ordinary Newtonian mechanics we can write the velocity as: v m V t v This irst order ordinary dierential equation has the eneral solution: v m t A1 exp t A The constants A 1 and A can be determined rom the two known values o v. First, the initial velocity is zero: v t A A1 Second, or t we must et the terminal velocity: v t V V A Sot Matter Physics 15 Exercise 1.1 The unction v(t) is thus: v t V 1 exp m t Consider the characteristic time in the exponential. We have = 6πηR and with η = 1-3 Pas this ives = ks -1. The mass is m = 4πR 3 /3 ρ = k. The characteristic time is thus m/ = s. (Only a nanosecond!) The terminal velocity is in the preactor, as time oes to ininity: v t R 9 We et v t = ms -1. This happens instantly, but the velocity is only a ew nm per second! Sedimentation in test tube takes several months! Sot Matter Physics 16 8

9 Chared Interaces Colloidal systems have hih surace to volume ratio and much interacial area. We have already talked about interacial eneries in relation to phase transitions and sel assembly. One thin we have not talked about is chared interaces! Unless we are doin electrochemistry, chares come rom various chemical roups at the interace. Example: Glass in water is neatively chared. SiO - SiO The chare normally varies with ph, solvent and temperature! Sot Matter Physics 17 Ions In a liquid environment there are always (at least some) ions present. Example o carbon dioxide dissolvin in water: CO () H O HCO 3 - (aq) H Even in the absence o CO there is sel-protonation o water: H O OH - H 3 O Ions are mobile chares just like the conduction band electrons in a metal. How do they respond to a chared interace? The electric potential close to a chared interace will be screened by ions Sot Matter Physics 18 9

10 The Electric Double ayer The standard theory or the chared interace is a diuse Gouy-Chapman layer and/or a Helmholtz-Stern layer with physically adsorbed ions. Adsorbed layer only is not realistic and diuse layer does not work or hiher potentials. Helmholtz-Stern model, adsorbed ions. Gouy-Chapman model, diuse layer Sot Matter Physics 19 The Diuse ayer Unortunately we must reduce the problem to one dimension by assumin a planar surace. We want to know the potential ψ and the ion concentration C as a unction o distance z. The potential enery chane when movin an ion a distance z rom the location where the diusive layer starts (z = ) is: E z z Q Here ψ is the potential at z = and Q is the chare o the ion, which is determined by the valency ν (, -, -1, 1,, ) by Q = νe. (The elementary chare is e = C.) z ψ ψ = Sot Matter Physics 1

11 Poisson-Boltzmann Equation To et ψ(z) at equilibrium, we use Poisson s equation rom electrostatics: e ici z total chare density z i Here ε = Fm -1 is the permittivity o ree space and ε is the relative permittivity o the medium (or a static ield). We use Boltzmann statistics or ion concentration (as or sedimentation/creamin): e z Cz C exp or each ionic species kbt Note that C is the concentration in the bulk (not at the surace). We can now combine these into the (complicated) Poisson-Boltzmann equation with boundary conditions: e z i C i i ie exp kbt z z z z z Sot Matter Physics 1 Approximate Solution For low potentials the equation has a very simple approximate solution (no details here): z expz Clearly, a very important parameter or the solution is κ which is iven by: e k T One reers to κ -1 as the Debye lenth. It shows how ar into a solution a surace eect extends! For a solution containin only a monovalent salt: C e k T B B i 1/ i C i 1/ κ -1 bulk solution, bulk properties chanes in ion concentration, potential and all kinds o weird thins chared interace Ionic strenth inluences κ -1 but the surace properties do not! Sot Matter Physics 11

12 Model imits Now we can model the diuse layer. However, the exponential decay solution o ψ is only valid or low potentials (deinitely ψ < 1 mv). This still means the model is quite accurate in many practical situations, but this is just by luck. It has many problems: Ions are treated as ininitely small. Continuous chare distributions rather than point chares. Hydration o ions nelected. Perectly smooth surace. Even i the model ives ood results it does not mean there are no adsorbed ions! Sot Matter Physics 3 Exercise 1. Assume we have a water solution with 15 mmol -1 NaCl (physioloical) at room temperature. Calculate the concentration o Cl -.5 nm rom a surace with a potential o mv usin the Gouy-Chapman model (no adsorbed ions). Comment on the result! Sot Matter Physics 4 1

13 Exercise 1. First calculate the Debye lenth, or monovalent salt: C = 15 mmol -1 = 15 molm -3 = m -3 e = C, k B = JK -1, ε = Fm -1 Water means ε = 8, room temperature is T = 3 K. 1/ C e 9 m k T B The potential at z =.5 nm is then:.5 nm.exp V z The souht ion concentration is thus: 1 e z.5 nm C.15exp mol kbt So we et C = 9.3 mol -1, but the maximum solubility o NaCl in water is 6. mol -1 at room temperature, so the model is not realistic or this surace potential Sot Matter Physics 5 Grahame Equation How can we relate surace potential to chare density σ (Cm - ). A relation can be derived rom the arument that the chares inducin the diusive layer must compensate the net chare o the ions inside it. This ives the Grahame equation: Ci i k BT z C i i Remember that we know C i we know ψ! For low potentials (<5 mv) an approximate relation is: Very important: We are still only considerin the diuse layer! The chare density you et will enerally not be that at the actual surace. σ = σ σ s??? Sot Matter Physics 6 13

14 Adsorbed Ions The Helmholtz-Stern layer can be thouht o as a plate capacitor. The ield between two chared plates is E = σ/[εε ] = V/d and thus: dγione s Here Γ ion is the surace coverae o adsorbed ions (inverse area). Simple ormula, but the values are very hard to know. The distance d can be approximated with the radius o the adsorbed ion. However, the permittivity will be very dierent rom that o the bulk liquid because the water molecules are hihly oriented. Aain very important: Only a part o the surace chares are compensated by ions in the adsorbed layer! ψ ψ s d Sot Matter Physics 7 Zeta Potential The zeta potential ζ is deined as the potential at the no slip position or the shear plane within the electric double layer. This is the distance at which ions and water molecules no loner are stuck. When the particle moves, water molecules and ions closer than the point o the zeta potential will move with the particle. The zeta potential is sometimes assumed to be equal to the potential at which the ζ ψ diusive layer starts (ζ = ψ ). ψ s low no low Sot Matter Physics 8 14

17-9-15 Electrophoresis The chared interace makes colloids move in electric ields. Some ions and water molecules will be stuck and ollow the colloid.

15 Electrophoresis The chared interace makes colloids move in electric ields. Some ions and water molecules will be stuck and ollow the colloid. Only a ractions o the total chare interacts with the external ield. It is the zeta potential which determines how ast the colloid moves. Experimentally we can thus et inormation about ζ but it is much harder to measure ψ or ψ s! E movement Sot Matter Physics 9 Exercise 1.3 A polystyrene colloid has sulphate roups (-SO 4 - ) on its surace. The zeta potential is - mv in 1 mmol -1 NaCl in water. Assume there is only a diuse layer and estimate how many sulphate roups there are on the colloid i it has a radius o nm Sot Matter Physics 3 15

17-9-15 Exercise 1.3 I there are no adsorbed ions, or an estimate we can assume the zeta potential is equal to the surace potential (ζ = ψ ).

16 Exercise 1.3 I there are no adsorbed ions, or an estimate we can assume the zeta potential is equal to the surace potential (ζ = ψ ). We irst calculate the Debye lenth assumin 3 K: We can use the simpliied Grahame equation to et σ: / 8-1 1/ 3 C e m 1 3 kbt Cm Note the unit o C per m. The chare o each -SO 4 - roup is C (neative but this is cancelled by the sin o ψ ). This ives a number o sulphate roups o.45 per nm. The area o a colloid is ~5 nm, which ives about 3 sulphate roups per colloid Sot Matter Physics 31 Relections and Questions? Sot Matter Physics 3 16

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