Chapter 5. Transport in Membrane

Chapter 5. Transport in Membrane

Contents Contents

5.5 Transport through Nonporous Membranes

5.7 Transport in Ion-exchange Membrane 5.6 Transport through Membrane

5.4 Transport through Porous Membranes 5.3 Non-equilibrium Thermodynamics

5.2 Driving Forces 5.1 Introduction

 On the point of molecular size alone ⇨ P of large molecule < P of simple small gases(N₂)

 In real, P of large organic molecule(Ph-CH₃ or CH₂Cl₂) > 10⁴∼10⁵ × P of small molecule(N₂)

 Difference in interaction ⇨ Difference in solubility ⇨ Difference in permeability

 Solubility↑ ⇨ Segmental motion↑ ⇨ Free volume↑

 Non-linear relationships between concentration ↔ pressure (<Figure 5-25>)

 Strong interaction ⇨ Solubility = non-ideal ⇨ S = f(c) ⇨ not followed by Henry's law

 For high solubility in polymers, c↑→ D↑ ⇨ D = f(c)

 Flory-Huggins thermodynamics

 Convenient method to describe solubility of organic vapor and liquid in polymers

 Activity of the penetrant inside the polymer is given by

Flory-Huggins : (5-118)

where χ = interaction parameter, χ > 2(large) : small interaction

0.5 < χ < 2.0(small) : strong interactions ⇨ high permeability ※ Crosslinked polymer : χ < 0.5

<Figure 5-25> Solubility of CH₂Cl₂(●), CHCl₃(○) and CCl₄(■) in

polydimethylsiloxane(PDMS) as a function of the vapor pressure.

Component Permeability

(Barrel)

N₂ 280

O₂ 600

CH₄ 940

CO₂ 3,200

Ethanol(C₂H₅OH) 53,000 Methylene Chloride(CH₂Cl₂) 193,000 1.2-Dichloroethane(CH₂Cl-CH₂Cl) 248,000 Tetrachloride(CCl₄) 290,000 Chloroform(CHCl₃) 329,000 1,1,2-Trichloroethane(CCl₂CHCl) 530,000 Trichloroethene(CCl₂=CHCl) 740,000 Toluene(Ph-CH₃) 1,106,000 [Table 5-6] Permeability of various components in PDMS at 40°C

 Concentration(c) dependence of D

 D = f(c) but no unique relationship

(∵ it varies from polymer to polymer and from penetrant to penetrant)

 Empirical exponential relationship.

D = D₀ exp (γ·ϕ) (5-119)

where D₀ = diffusion coefficient at c = 0

γ = plasticising constant(plasticising action of penetrant on segmental motion) ϕ = volume fraction of the penetrant

 D₀ dependence of molecular size

 Molecular size↓(water) ⇨ D₀↑

 Molecular size↑(benzene) ⇨ D₀↓(see [Table 5-7])

 D dependence of γ and ϕ

 γ and ϕ appear in the exponent of Eq(5-119) ⇨ highly influence to D ※ Ideal gas ⇨ γ → 0

Component V_m (cm³/mole)

D₀ (cm²/sec)

Water 18 1.2 × 10^-7

Ethanol 41 1.5 × 10^-9 Propanol 76 2.1 × 10^-12

Benzene 91 4.8 × 10^-13 [Table 5-7] Effect of penetrant size on D₀ in poly(vinyl acetate)

 Concentration(c) dependence of the diffusion coefficient(D)

 Describe by free volume theory

<Assume> Penetrant increases the free volume of the polymer.

 Relationship between log D and the volume fraction of the penetrant(ϕ) in the polymer ※ similar to Eq(5-119) : D = D₀ exp (γ·ϕ)

 More quantitative approach than Eq(5-119)

 Large difference in permeability of between glassy ↔ rubbery state

 Glassy state

 mobility of the chain segments is extremely limited

 thermal energy too small to allow rotation around the main chain

 Rubbery state(above T_g)

 Mobility of the chain segments↑

 Frozen micro-voids no longer exist 5.5.2.1 Free volume theory

<Figure 5-26> Specific volume of an amorphous polymer as a function of temperature.

 Free volume (V_f) = V_T – V₀ (5-120)

where V_T = observed volume at a temperature T V₀ = volume occupied by the molecules at 0 K

 Fractional free volume(v_f) = V_f / V_T (5-121)

 v_f ≈ v_f,Tg ≈ 0.025 for most of glassy polymers based on viscosity

 V_f above T_g ∝ T linearly

 v_f ≈ v_f,Tg + Δα(T - T_g) (5-122)

where Δα = difference (thermal expansion coefficient value above T ↔ below T )

 Basic concept of free volume

 Very useful to understand transport of small molecules through polymers

 Molecule can only diffuse from one place to another place if there is sufficient empty space or free volume

 Size of penetrant↑ → amount of free volume↑

 Probability of finding a 'hole' whose size exceeds a critical value ∝ exp(－B/v_f) where B = local free volume needed for a given penetrant

v_f = fractional free volume

 Mobility of penetrant ∝ Probability of a hole of sufficient size for displacement

 Mobility can be related to thermodynamic diffusion coefficient [see Eq(5-100)]

 D_T = (mobility coefficient)·RT = m·RT = RT·A_f exp(－B/v_f) (5 - 123) where D_T = Thermodynamic diffusion coefficient

A_f = dependent on the size and the shape of penetrant molecules

B = related to minimum local free volume necessary to allow a displacement

 D_T = RT·A_f exp(－B/v_f) ⇨ T↑ & Penetrant size(B↓) ⇨ diffusion coefficient(D_T)↑

 Non-interacting systems (polymer with inert gases like He, H₂, O₂, N₂, Ar)

• Polymer morphology is not influenced by the presence of these gases 「Meaning」 There is no extra contribution towards the free volume

• By assuming that A_f and B ≠ f(polymer)

∙ plot of In(D/(RT·A_f)) verse (1/v_f) ⇨ Slope = －B from Eq(5-123)

 Polyimides deviate from this linear behavior

「Meaning」 Assumptions behind Eq(5-123) are not completely correct A_f and B = f(polymer) ⇨ polymer-dependent parameters need

 Interacting systems (e.g organic vapors)

 Free volume = f(temperature, penetrant concentration)

• v_f = f(ϕ,T) = v_f(0,T) + β(T) ϕ (5-124)

where v_f(0,T) = v_f at temperature T and zero penetrant concentration ϕ = volume fraction of penetrant

β(T) = constant(extent to which the penetrant contributes to v_f)

 Diffusion coefficient at zero penetrant concentration : D₀ = D_c→0 at Eq(5-123)

(5-125)

 Combination of Eq(5-123) and (5-125) gives

(5-126)

(5-127)

「Meaning」

 [1/ln (D_T/D₀)] is related linearly to 1/ϕ

 D = D₀ exp (γ· ϕ)[Eq(5-119)] and Eq(5-127) are similar when v_f(0,T) ≫ β(T)

 Plots of ln(D) verse ϕ = linear

 Solubility

 Gas molecules ⇨ apply Henry's law

(solubility of a gas in a polymer ∝ external partial pressure)

 Organic vapor and liquid ⇨ apply Flory-Huggins thermodynamics

(5-118) where χ = interaction parameter

 D(measured diffusion coefficient) ↔ D_T(thermodynamic diffusion coefficient)

(5-128)

 Penetrant concentration↑ ⇨ difference between the two diffusion coefficients↑

 By differentiation of Eq(5-118) with respect to lnϕ_i

(5-129)

 For ideal systems and at low volume fractions(ϕ_i→0) : dlna_i/dlnϕ_i = 1 and D = D_T

 Clustering of penetrant molecules ⇨ Cause deviations from free volume approach

 Component diffuses not as a single molecule but in its dimeric or trimeric form.

 Size of the diffusing components↑ ⇨ Diffusion coefficient↓

(<Ex> water molecules → strong H-bonding ⇨ diffuse by clustered molecules)

 Extent of clustering will also depend on

 Type of polymer

 Other penetrant molecules present

 Zimm-Lundberg theory to describe the clustering ability

 Cluster function : ability or probability of molecules to cluster inside a membrane (5-130) where G₁₁ = cluster integral, V₁ = molar volume of penetrant

ϕ₁ = volume fraction of penetrant

 For ideal system, dlnϕ₁/dlna₁ = 1 ⇨ G₁₁/V₁ = －1 ⇨ no clustering

 G₁₁/V₁ > －1 ⇨ clustering 5.5.2.2 Clustering

 Difference between ternary system and binary system(polymer and liquid) by thermodynamics

 Ternary system (a binary liquid mixture and a polymer)

 Volume and composition of liquid mixture inside the polymer = important parameters

 Composition of liquid mixture inside the polymer ⇨ Sorption selectivity ⇨ Rejection rate

 Concentration of a given component i in the binary liquid mixture in the ternary polymeric phase

(5-131)

 Preferential sorption is then given by ε = u_i – v_i i = 1, 2 (5-132)

 Δμ_f,i = Δμ_m,i + πV_i i = 1,2 (5-133) subscript f (feed) = polymer free phase subscript m (membrane) = ternary phase 5.5.2.3 Solubility of liquid mixtures

<Figure 5-27> Schematic drawing of a binary liquid feed mixture in equilibrium with the polymeric membrane.

 Flory-Huggins thermodynamics ⇨ Expressions for the chemical potentials

 When V₁/V₃ ≈V₂/V₃ ≈ 0 and V₁/V₂ = m,

 Concentration-independent Flory-Huggins interaction parameters and eliminating π gives ;

(5-134)

 Composition of the liquid mixture inside the membrane, can be solved numerically when the interaction parameters and volume fraction of the polymer are known.

 In real, Flory-Huggins parameters for these systems = concentration-dependent ⇨ complex

 Define where α_sorp = sorption (5-135)

「Meaning of Eq(5-134)」

 Difference in molar volume

• If only entropy effects ⇨ Selective sorption of smaller molar volume component preferentially

• Polymer concentration↑ ⇨ These effect↑ ※ Maximum at ϕ₃→1

 Enthalpy of mixing ⇨ Selective sorption of highest affinity component to polymer

 By assuming ideal sorption ⇨ this factor only influences the solubility

(the highest affinity leads to the highest solubility)

 Influence of mutual interaction with the binary liquid mixture on preferential sorption

 depends on the concentration in the binary liquid feed

 and on the value of χ₁₂

 χ₁₂

 For organic liquids, χ₁₂ = f(composition) strongly

 For constant interaction parameter, χ₁₂ should be replaced by a concentration-dependent interaction parameter, g_l2(ϕ).

Solvent V₁/V₂ Methanol 0.44 Ethanol 0.31 Propanol 0.24 Butanol 0.20 Dioxane 0.21 Acetone 0.24 Acetic acid 0.31

DMF 0.23

[Table 5.8] Ratio of molar volumes at 25℃ of various organic solvents with water (V₁ = 18 cm³/mol)

 Concentration-dependent systems

 Apply Fick's law using concentration dependent diffusion coefficients

D_i = D_0,i exp(γ_i·c_i) (5-136)

where D_0,i = diffusion coefficient at c_i→0

γ_i = plasticising constant(plasticising action influence of liquid on segmental motions)

 Substitution of Eq(5-l36) into Fick's law and integration using the BC BC 1 : c_i = c_i,l^m at x = 0

BC 2 : c_i = 0 at x=ℓ

(5-137)

 Parameters in Eq(5-137)

∙ D_0,i , γ and ℓ = constants

∙ main parameter = concentration inside the membrane (c_i,l^m)

∙ c_i,l^m↑ ⇨ permeation rate↑

5.5.2.4 Transport of single liquid

「Meaning of Eq(5-137)」 for single liquid transport

• Interaction between membrane ↔ penetrant ⇨ determine permeation rate

• Affinity between penetrant ↔ polymer↑

⇨ J_i↑ for a given penetrant

 Transport of liquid mixtures through a polymeric membrane

 For a binary liquid mixture, J_i = f (solubility, diffusivity)

 Strong interaction between Solubility ↔ Diffusivity ⇨ Much more complex

 Distinguished phenomena in multi-component transport

 Flow coupling ⇨ described via non-equilibrium thermodynamics

 Thermodynamic interaction ⇨ preferential sorption

 Flux equations for a binary liquid mixture

 J_i = L_ii dμ_i/dx + L_ij dμ_j/dx (5-15) • J_j = L_ji dμ_i/dx + L_jj dμ_j/dx (5-16) Where L_ii dμ_i/dx = flux of component i due to its own gradient

L_ij dμ_i/dx = flux of component i due to the gradient of component j (coupling effect)

 No Coupling (Apply binary system with very low permeability)

 Components permeate through the membrane independently of each other

 L_ij = L_ji = 0 ⇨ reduce to simple linear relationships 5.5.2.5 Transport of liquid mixture

 Crystalline fraction of polymer

 Large number of polymers are semi-crystalline(amorphous + crystalline fraction)

 However, the crystallinity is quite low in most membranes

 Cystallinity < 0.1 → Diffusion resistance by crystalline ⇨ negligible

 Effect of crystallinity on the permeation rate is often fairly small

 Diffusion coefficient = f (crystallinity)

(5-138) where Ψ_cⁿ = fraction of crystalline B = constant

n = exponential factor (n < 1)

<Figure 5-28> The effect of crystallinity on diffusion resistance

 Classification of model

 Based on phenomenological approach • Black box model

• Provide no information as to how the separation actually occurs

 Based on non-equilibrium thermodynamics

 Mechanistic models(pore model and solution-diffusion model) • Relate separation with structural-related membrane parameters in an attempt to describe mixtures.

• Provide information on how separation actually occurs ⇨ factors are important

 Simple model ⇨ starting point

 generalized Fick equation

 generalized Stefan-Maxwell equation

 Flux of component i through a membrane = Velocity × Concentration

 J_i = c_i(v_i + u) (5-139)

 Convective flow(u) : main transport through porous membrane

 Diffusion flow(v_i) : main transport through nonporous

<Figure 5-29> Convective and diffusive flow in membranes.

 Comparison of flux contribution in the case of porous membranes (MF)

 Given conditions

∙ Membrane with a thickness(ℓ) = 100 μm ∙ Average pore diameter = 0.1 μm

∙ Tortuosity(τ) = 1 (capillary membrane) ∙ Porosity(ε) = 0.6

∙ ΔP for water flow = 1 bar

 Convective flux from Poisseuille equation (convective flow)

 Diffusion flux

∙ Driving force : difference in chemical potential = f (Δc or Δa, and ΔP) ∙ Δμ_w = v_w∙ΔP = 1.8× 10^-5∙10⁵ = 1.8 J/mol

 Diffusion flow

 J_i = c_i∙v_i (5-140)

 v_i = X_i / f_i (5-141)

where v_i = mean velocity of a component in the membrane

X_i = driving force acting on the component = gradient(dμ/dx) f_i = frictional resistance = RT/D_T

D_T = thermodynamic diffusion coefficient

 If ideal conditions are assumed(D_T = D_i, observed diffusion coefficient)

Eq(5-140) → (5-142)

Chemical potential : μ_i = μ^o_i + RT ln a_i + V_i∙(P－P^o) (5-6)

Eq(5-6) → Eq(5-142) : (5-143)

<Figure 5-30> Process conditions for transport through nonporous membranes.

(superscripts m = membrane, superscripts s = feed/permeate side)

 <Assume> Thermodynamic equilibrium exists at the membrane interfaces

 μ_i at the feed/membrane interface is equal in both the feed and the membrane

μ_i,1^m = μ_i,1^s ⇨ a_i,1^m = a_i,1^s (5-144)

 Pressure inside membrane = Pressure on feed side at feed interface (phase 1/membrane) P₁ = P_m ≠ P₂

At the permeate interface(membrane/phase 2)

(5-145)

(5-146)

(5-147)

(5-148)

(5-149)

 <Assume> diffusion coefficient ≠ f (concentration)

 Fick's law[Eq(5-83)] can be integrated across the membrane to give

(5-150)

 Eq(5-146), (5-147) and (5-148) → Eq(5-150)

(5-151)

 if α_i = K_i,2 / K_i,l (i.e. the solubility coefficients are similar at both interfaces) and P_i = K_i ∙ D_i , then Eq(5-151) converts into

(5-152)

※ Eq(5-152)

Basic equation used to compare various membrane processes when transport occurs by diffusion.

Process Phase 1 Phase 2

RO L L

Dialysis L L

Gas separation G G

[Table 5-9] Phases involved in diffusion controlled membrane processes

 Application

 Separate a very low MW solute(salt, very small amount of organic)

 Driving force : pressure difference

 total flux = water flux(J_w) + solute flux(J_s)

 Solvent Flux (※ J_s = neglected by high selective)

J_total = J_w +J_s ≈ J_w (5-153)

since Δπ = RT/V_i∙(In c_w,2^s/c_w,1^s) and α₁ =1,

(5-154)

or

(5-155)

For small values of x, the term, 1－exp(－x) ≈ －x (5-156)

and since K_w∙c_w,1^s = c_w,1^m (5-157)

Eq(5-154) → (5-158)

J_w = A_w(ΔP - Δπ) with A_w = D_w∙C_w,1^m∙V_w / RT∙ℓ (5-159) where A_w = called the water permeability coefficient(L_p)

 Solute flux

 Reverse osmosis membranes are generally not completely semipermeable

 From Eq(5-151) with α_j = 1, the solute flux J_s can be written as

(5-151)

(5-160)

and since the exponential term is approximately unity (see section 5.6.4), or J_s = B∙Δc (5-161)&(5-162) where B = permeability coefficient = D_s∙K/ℓ

「Meaning of Eq(5-159) and (5-162)」

 Eq(5-159) → Water flux ∝ applied effective pressure difference in reverse osmosis

 Eq(5-162) → Solute flux ∝ concentration difference in reverse osmosis

 Dialysis

 Liquid phases containing same solvent are present on both sides of membrane

 No pressure difference

 Flux

 Pressure terms = neglected

from Eq(5-152) if α_i = 1 → (5-163)

or

(5-164)

「Meaning of Eq(5-159) and (5-164)」

 Solute flux is proportional to the concentration difference

 Separation arises from differences in permeability coefficients

 D_T and Distribution coefficients of higher MW < Lower MW species

 Gas permeation or Vapor permeation

 both the upstream and downstream sides of a membrane consist of gas or vapor

 However, Eq(5-152) cannot be used directly for gases.

(5-152)

 Concentration of gas in membrane

c_i,1^m = P_i,1^s∙K_i (5-165)

by combining Eq(5-165) with Eq(5-150)

(5-150)

(5-166)

where P_i = K_i∙D_i

「Meaning of Eq(5-166)」

 Rate of gas permeation ∝ ΔP across the membrane

 Pervaporation

 Feed side = liquid, Permeate side = vapor ⇨ very low pressure in downstream

 P₂(downside stream) → 0 (or a₂^s → 0)

 exponential term in Eq(5-152) = 1 and can be neglected

(ΔP ≈ 10⁵ N/m², V_i = 10^－4 m³/mol, RT ≈ 2500 J/mol ⇨ exp(－V_i∙ΔP/RT) ≈ 1)

 lf partial pressure = activity, then:

γ_i^s∙c_i^s = P_i (5-167)

 Eq(5-165) → (5-168)

「Meaning of Eq(5-168)」

 Permeate pressure (p_i,2^s)↑ ⇨ Flux of component i↓

 Permeate pressure (p_i,2^s) = Feed pressure(P_i,l^s) ⇨ Flux of component i = 0

 Donnan exclusion(<Figure 5-31>)

 Use an ion-exchange membrane in contact with an ionic solution

 Same charged ions with the fixed ions in the membrane ⇨ Rejected by membrane

 When an ionic solution is in equilibrium with an ionic membrane

※ Activities(= activity coefficient × molar concentration) are used (not concentrations)

※ Electrolyte solutions = generally behave non-ideal (very low concentrations ⇨ ideal behavior)

 Chemical potential in ionic solution : μ_i = μ_i⁰ + RT In m_i + RT In γ_i + z_i ℱ Ψ (5-169)

 Chemical potential in membrane : μ_i^m = μ^o_i^m + RT ln(m_i^m) + RT ln(γ_i^m) + z_i ℱ Ψ^m (5-170) where subscript m = membrane phase

 At equilibrium,

• μ_i = μ_i^m at interface (5-171)

• μ^o_i = μ^o_i^m in membrane

• Potential difference(E_don) = Ψ_m－Ψ

(5-173) & (5-174)

 For the case of dilute solutions(a_i ≈ c_i),

(5-175)

<Assume> Swelling pressure(π•V_i) = negligible

※ If not, this term should be added to right-hand side of Eq(5-175).

 Calculation example

• Monovalent ionic solute

• Concentration difference = 10

• E_don at the interface

= [(8.314×298)/(96500)] In(1/10) = －59 mV

<Figure 5-31> Schematic drawing of the ionic distribution at the membrane-solution interface(membrane contains

fixed negatively charged groups) and the corresponding potential

<Assume> Solution behaves ideal ⇨ a_i = c_i

 At equilibrium, μ_i = μ_i^m

 Under ideal conditions (γ_i → 0)

(5-176)

 By electrical neutrality,

∑z_i c_i = 0 (5-177)

in membrane (5-178)

in solution (5-179)

 Combination of Eq(5-176) and (5-178) gives

(5-180)

(5-181)

or (5-182)

For a dilute solution, Eq(5-182) reduces to

(5-183)

⇨ Donnan Equilibrium

 Donnan Equilibrium is valid when high concentration of R^– is contacted with dilute solution.

 Feed concentration↑ ⇨ this exclusion↓

<Example> ∙ Brackish water with 590 ppm NaCl (≈ 0.01 eq/L ≈ 10^–5 eq/mL) ∙ Wet-charge density of membrane ≈ 2×10^–3 eq/mL

∙ Co-ion (Cl^－) concentration in the membrane ≈ 5×10^–8 eq/mL ⇨ [C_co-ion]^m ∝ [C_co-ion]² and [fixed charge density in the membrane(R^－)]^－1

 Non-ideal solution ※ Ionic solutions = non-ideal manner in most case

 Using activity coefficients(γ_i)

 Mean ionic activity coefficients(γ±)

For a univalent cation and anion, γ± = (γ⁺•γ^–)^0.5

where γ⁺ and γ^– = activity coefficients of the cation and anion, respectively

 Eq(5-181) → (5-184)

 Ion-exchange membranes in combination with an electrical potential difference

 Forces act on ionic solutes : ΔC and ΔE

 Transport of ion : Fickian diffusion and Ionic conductance

 Nernst-Planck equation : (5-185)

 NF, RO membranes

 Ions are transported across a charged membrane without ΔE

 Convective term has to be included

 3 contributions for ionic transport : an electrical, a diffusive and a convective term

 J_i = J_i,dif + J_i,elec + J_i,conv (5-186)

<Assume> ∙ No coupling phenomena ∙ Ideal conditions

 Extended Nernst-Planck equation : (5-187)