Chapter 5. Transport in Membrane

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Evangeline Haynes
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1 National October 7, 2015 (Wed) Chang-Han Yun / Ph.D.

2 Contents 5.1 Introduction 5.2 Driving Forces Contents Contents 5.3 Non-equilibrium Thermodynamics 5.4 Transport through Porous Membranes 5.5 Transport through Nonporous Membranes 5.6 Transport through Membrane 5.7 Transport in Ion-exchange Membrane 2

3 5.5 Transport through Nonporous Membranes interactive Systems On the point of molecular size alone P of large molecule < P of simple small gases(n 2 ) In real, P of large organic molecule(ph-ch 3 or CH 2 Cl 2 ) > P of small molecule(n 2 ) 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 3

4 5.5 Transport through Nonporous Membranes interactive Systems [Table 5-6] Permeability of various components in PDMS at 40 C Component Permeability (Barrel) N O CH CO 2 3,200 Ethanol(C 2 H 5 OH) 53,000 Methylene Chloride(CH 2 Cl 2 ) 193,000 <Figure 5-25> Solubility of CH 2 Cl 2 ( ), CHCl 3 ( ) and CCl 4 ( ) in polydimethylsiloxane(pdms) as a function of the vapor pressure. 1.2-Dichloroethane(CH 2 Cl-CH 2 Cl) 248,000 Tetrachloride(CCl 4 ) 290,000 Chloroform(CHCl 3 ) 329,000 1,1,2-Trichloroethane(CCl 2 CHCl) 530,000 Trichloroethene(CCl 2 =CHCl) 740,000 Toluene(Ph-CH 3 ) 1,106,000 4

5 5.5 Transport through Nonporous Membranes interactive Systems 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 = D 0 exp (γ ϕ) (5-119) where D 0 = diffusion coefficient at c = 0 D 0 dependence of molecular size γ = plasticising constant(plasticising action of penetrant on segmental motion) ϕ = volume fraction of the penetrant Molecular size (water) D 0 Molecular size (benzene) D 0 (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 [Table 5-7] Effect of penetrant size on D 0 in poly(vinyl acetate) Component V m (cm 3 /mole) D 0 (cm 2 /sec) Water Ethanol Propanol Benzene

6 5.5 Transport through Nonporous Membranes interactive Systems 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 = D 0 exp (γ ϕ) Free volume theory 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 6

7 5.5 Transport through Nonporous Membranes interactive Systems <Figure 5-26> Specific volume of an amorphous polymer as a function of temperature. Free volume (V f ) = V T V 0 (5-120) where V T = observed volume at a temperature T V 0 = volume occupied by the molecules at 0 K Fractional free volume(v f ) = V f / V T (5-121) v f v f,tg 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 g below T g ) 7

8 5.5 Transport through Nonporous Membranes interactive Systems 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 8

9 5.5 Transport through Nonporous Membranes interactive Systems 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, H 2, O 2, N 2, 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 ) 9

10 5.5 Transport through Nonporous Membranes interactive Systems Diffusion coefficient at zero penetrant concentration : D 0 = 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 /D 0 )] is related linearly to 1/ϕ D = D 0 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 10

11 5.5 Transport through Nonporous Membranes interactive Systems 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 11

12 5.5 Transport through Nonporous Membranes Clustering interactive Systems 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 G 11 = cluster integral, V 1 = molar volume of penetrant ϕ 1 = volume fraction of penetrant For ideal system, dlnϕ 1 /dlna 1 = 1 G 11 /V 1 = -1 no clustering G 11 /V 1 > -1 clustering 12

13 5.5 Transport through Nonporous Membranes Solubility of liquid mixtures interactive Systems 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 <Figure 5-27> Schematic drawing of a binary liquid feed mixture in equilibrium with the polymeric membrane. 13

14 5.5 Transport through Nonporous Membranes Flory-Huggins thermodynamics Expressions for the chemical potentials interactive Systems When V 1 /V 3 V 2 /V 3 0 and V 1 /V 2 = 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 ϕ

15 5.5 Transport through Nonporous Membranes interactive Systems 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 χ 12 χ 12 For organic liquids, χ 12 = f(composition) strongly For constant interaction parameter, χ 12 should be replaced by a concentration-dependent interaction parameter, g l2 (ϕ). [Table 5.8] Ratio of molar volumes at 25 of various organic solvents with water (V 1 = 18 cm 3 /mol) Solvent V 1 /V 2 Methanol 0.44 Ethanol 0.31 Propanol 0.24 Butanol 0.20 Dioxane 0.21 Acetone 0.24 Acetic acid 0.31 DMF

16 5.5 Transport through Nonporous Membranes Transport of single liquid interactive Systems 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 m i,l at x = 0 Meaning of Eq(5-137) for single liquid transport BC 2 : c i = 0 at x=l Interaction between membrane penetrant (5-137) determine permeation rate Parameters in Eq(5-137) Affinity between penetrant polymer D 0,i, γ and l = constants J i for a given penetrant main parameter = concentration inside the membrane (c i,lm ) c i,lm permeation rate 16

17 5.5 Transport through Nonporous Membranes interactive Systems Transport of liquid mixture 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 17

18 5.5 Transport through Nonporous Membranes Effect of Crystallinity 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 n c = fraction of crystalline B = constant n = exponential factor (n < 1) <Figure 5-28> The effect of crystallinity on diffusion resistance 18

19 5.6 Transport through Membrane A unified approach 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 19

20 5.6 Transport through Membrane A unified approach 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. 20

21 5.6 Transport through Membrane A unified approach 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 J/mol 21

22 5.6 Transport through Membrane A unified approach 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) 22

23 5.6 Transport through Membrane A unified approach <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) P 1 = P m P 2 23

24 5.6 Transport through Membrane A unified approach At the permeate interface(membrane/phase 2) (5-145) (5-146) (5-147) (5-148) (5-149) 24

25 5.6 Transport through Membrane A unified approach <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. [Table 5-9] Phases involved in diffusion controlled membrane processes Process Phase 1 Phase 2 RO L L Dialysis L L Gas separation G G Pervaporation L G 25

26 5.6 Transport through Membrane A unified approach 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 ) Reverse Osmosis 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,2s /c w,1s ) and α 1 =1, (5-154) or (5-155) For small values of x, the term, 1-exp(-x) -x (5-156) and since K w c s w,1 = c m w,1 (5-157) Eq(5-154) (5-158) J w = A w (ΔP - Δπ) with A w = D w C w,1m V w / RT l (5-159) where A w = called the water permeability coefficient(l p ) 26

27 5.6 Transport through Membrane A unified approach Reverse Osmosis 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 27

28 5.6 Transport through Membrane A unified approach Dialysis 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 28

29 5.6 Transport through Membrane A unified approach Gas Permeation 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 m i,1 = P i,1s 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 29

30 5.6 Transport through Membrane A unified approach Pervaporation Pervaporation Feed side = liquid, Permeate side = vapor very low pressure in downstream P 2 (downside stream) 0 (or a s 2 0) exponential term in Eq(5-152) = 1 and can be neglected (ΔP 10 5 N/m 2, V i = 10-4 m 3 /mol, RT 2500 J/mol exp(-v i ΔP/RT) 1) lf partial pressure = activity, then: γ is c s i = P i (5-167) Eq(5-165) (5-168) Meaning of Eq(5-168) Permeate pressure (p i,2s ) Flux of component i Permeate pressure (p i,2s ) = Feed pressure(p i,ls ) Flux of component i = 0 30

31 5.7 Transport in Ion-exchange Membrane Basic theory on principles : Nernst-Planck equation and Donnan equilibrium 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 0 i + RT In m i + RT In γ i + z i F Ψ (5-169) Chemical potential in membrane : μ m i = μ o i m + RT ln(m im ) + RT ln(γ im ) + z i F Ψ m (5-170) where subscript m = membrane phase At equilibrium, μ i = μ m i at interface (5-171) μ o i = μ o i m in membrane Potential difference(e don ) = Ψ m -Ψ 31

32 5.7 Transport in Ion-exchange Membrane (5-172) (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 = [( )/(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 32

33 5.7 Transport in Ion-exchange Membrane Ion-exchange membrane with a fixed negative charge (R - ) with NaCl solution <Assume> Solution behaves ideal a i = c i At equilibrium, μ i = μ m i 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) 33

34 5.7 Transport in Ion-exchange Membrane 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 eq/ml Co-ion (Cl - ) concentration in the membrane 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 34

5.7 Transport in Ion-exchange Membrane 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 :

35 5.7 Transport in Ion-exchange Membrane 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) 35

CH.8 Polymers: Solutions, Blends, Membranes, and Gels

CH.8 Polymers: Solutions, Blends, embranes, and Gels 8. Properties of Polymers Polymers are chain-like molecules. Linear polymer Branched polymer Cross-linked polymer Polymers show little tendency to crystallize.