DIFFUSION_IN_POLYMERS drug delivery systems

DIFFUSION_IN_POLYMERS drug delivery systems

1. Diffusion in Polymers 2020 The Diffusion Equation can model: Absorption of solvent Desorption of solvent Permeation of solvent Charles M. Hansen

2. OUTLINE Laws of Diffusion Find correct diffusion coefficients Concentration dependent coefficients Surface effects can be significant Combine these in diffusion equation to model Film formation by solvent evaporation and ” Anomalies ” of absorption (S- shaped , Case II). Reliability of the modeling : The software in Hansen Solubility Parameters in Practice ( HSPiP ) used here gives the same results as are found in Crank (The Mathematics of Diffusion) and analytical solutions.

3. FICK’S FIRST AND SECOND LAWS Law 1: F = - D (  c/  x) For constant D in the x Direction, and Law 2:  c/  t =  /  x (D  c/  x) This is also called the Diffusion Equation. (Accumulation equals flux in minus flux out) Calculations are referred to dry polymer. Initial and 2 boundary conditions are required. D can be replaced by exponential D(c). Finding D(c) curve requires iteration for different c.

4. DIMENSIONLESS VARIABLES Dimensionless time: T = D t/L 2 (cm 2 /s)(s/cm 2 ) Dimensionless distance: X = x/L Dimensionless concentration: C = (c – c )/(c  - c ) L is the thickness of a free film

5. Absorption with a Constant Diffusion Coefficient Straight line absorption - square root of time (Diffusion coefficients upper right, Concentration gradients lower left , absorption curve lower right)

6. Absorption with Exponential D(c) Gives Advancing Front Straight line absorption - square root of time

7. SURFACE CONDITION F s = h( C eq – C s ) = - D s  C s / x Flux through surface to(from) external phase , F s , equals flux through surface from(to) the bulk. External Flux to/from surface, F s , equals mass transfer coefficient, h, (cm/s) times concentration difference, g/cm 3 giving g/cm 2 s Flux to/from bulk equals diffusion coefficient ( cm 2 /s) times concentration gradient (g/cm 3 cm) h can be found from h = F s /( C eq – C s ) @ t = 0

8. MEASURING DIFFUSION COEFFICIENTS The Old Way : Correction Factors The New Way : Curve Fitting Half-time (t ½ ) equation for measuring constant D D = 0.049 L 2 /t ½ For Concentration dependence multiply this by F a , for absorption, or F d for desorption For significant surface resistance multiply by F B See also Nordtest POLY 188

9. CORRECTIONS FOR CONCENTRATION DEPENDENCE ALONE Note huge corrections for desorption Desorption Absorption D max /D ( F d ) 1/2 ( F d ) 1/4 (F a ) 1/2 1 1.00 1.00 1.00 2 1.56 1.55 1.30 5 2.70 2.61 1.70 10 1 4.00 3.84 2.01 10 2 13.40 10.20 3.30 10 3 43.30 23.10 4.85 10 4 138.7 47.40 6.14 10 5 443.0 89.0 7.63 10 6 1,370.0 160.5 8.97 10 7 4,300.0 290.0 10.60 10 8 13,670.0 506.0 12.10 F B is a function of dimensionless surface parameter B = hL /D 1 See Hansen, C.M., J.Appl Poly Sci , 26,3311-3315 (1981)

10. EXPONENTIAL DIFFUSION COEFFICIENTS FOR CHLOROBENZENE IN POLY(VINYL ACETATE) The system chlorobenzene in poly(vinyl acetate) has been studied extensively with all relevant data reported in my dissertation and subsequent journal articles. The experiments were for: Absorption from one equilibrium to another, Desorption from different equilibrium values to vacuum, and film drying (years). Coherent understanding is found only by accounting for concentration dependence and significant surface effects when present.

11. CORRECTIONS: ABSORPTION 0.22 V f to 0.27 V f (F a ) FOR D(c) AND F B FOR SURFACE EFFECTS

12. DESORPTION AND ABSORPTION GIVE SAME D(c) WHEN CORRECTED (HANSEN 1967, 2007) (Iteration Required )

13. D(c) FOR CHLOROBENZENE IN PVAc FOR ALL CONCENTRATIONS (HANSEN, 1967)

14. DRYING OF A LACQUER FILM (Hansen, 1963, 1967, 1968)

15. RELATIVE SOLVENT RETENTION DIFFUSION CONTROLED - SIZE/SHAPE NOT HYDROGEN BONDING OR HSP

16. WHOLE EQUALS SUM OF PARTS E = COHESION ENERGY = Δ E vap E = E D + E P + E H D - Dispersion ( Hydrocarbon ) P - Polar ( Dipolar ) H - Hydrogen Bonds ( Electron Interchange) V - Molar Volume E/V = E D /V + E P /V + E H /V  2 =  2 D +  2 P +  2 H HANSEN SOLUBILITY PARAMETERS (HSP)  = Square Root of Cohesion Energy Density

17. POTENTIALLY SIGNIFICANT SURFACE EFFECTS IN VAPOR ABSORPTION External phase diffusion from source to film Diffusion in stagnant boundary layer at film Heat removal on condensation Adsorption (How well do HSP match?) Orientation (Does n-hexane enter sideways?) Number of absorption sites, hole size and shape, n-hexane sidewise? Transport into bulk (Diffusion coefficient, molecular size and shape)

18. POTENTIALLY SIGNIFICANT SURFACE EFFECTS IN ( LIQUID ) ABSORPTION Adsorption (How well do HSP match?) Polymer rotation to “match” HSP of external phase: reason for success with a constant h? Orientation (Does n-hexane enter sideways?) Absorption site (hole size and shape) Number of absorption sites (h depends on equilibrium uptake and similarity of HSP) Transport into bulk (Diffusion coefficient, molecular size and shape)

19. ABSORPTION DELAY FOR LIQUID CONTACT COC POLYMER TOPAS ® 6013 TICONA (NIELSEN, HANSEN 2005)

20. Apparent h and Equilibrium Uptake for COC Topas ® 6013 on Liquid Contact Solvent Apparent h, cm/s Equilibrium uptake, vol. fraction Tetrahydrofuran 1.89(10) -4 0.676 Hexane 7.78(10) -6 0.351 Diethyl ether 1.21(10) -6 0.268 Propylamine 1.49(10) -7 0.181 Ethylene dichloride 1.18(10) -7 0.176 Ethyl acetate 1.46(10) -8 0.076 n-Butyl acetate 8.30(10) -10 0.202 Phenyl acetate 0 0 Acetophenone 0 0 1,4-Dioxane 0 0 Tetrahydrofuran: apparent h is too low since diffusion controls. n-Butyl acetate: apparent h is strongly lowered by size and shape.

21. Surface Mass Transfer COC ( Topas ® 6013) For Given Size Range, log(h) Depends on Saturation Absorption, i.e. ( Δ HSP)

22. MAJOR REFERENCES EXPLAINING “ANOMALIES” USING DIFFUSION EQUATION Chapter 16 of Second Edition of Hansen Solubility Parameters: A User’s Handbook , CRC Press, 2007. Hansen CM. The significance of the surface condition in solutions to the diffusion equation: explaining "anomalous" sigmoidal , Case II, and Super Case II absorption behavior. Eur Polym J 2010;46;651-662. Abbott S, Hansen CM, Yamamoto H. Hansen Solubility Parameters in Practice ( HSPiP ) , www.hansen-solubility.com . (The HSPiP package includes validated software for absorption, desorption and permeation.) One can curve fit data for films, cylinders, and spheres. Several downloads on www.hansen-solubility.com .

23. Thomas and Windle Case II Example Methanol/PMMA with Iodine Tracer Straight line absorption with linear time cited as excellent example of Case II behavior. This result is duplicated: Diffusion equation with iteration of both h and exponential D(c). Stress Explanation Not Correct

24. Methanol/PMMA Absorption at 30ºC Diffusion Equation Simulation “ Breakthrough” 9.1 h, Gradients Flat at about 11 h

25. ” Experimental ” Entry Coefficient , h Methanol in PMMA “Experimental” h is found from the straight line (Case II) absorption data of Thomas and Windle h = F s /( C eq – C s ) @ t = 0 in cm/s C eq = 0.24 % wgt ; 0.373 g s /cm 3 dry PMMA, C s = 0 C eq = 0.265 vol fraction, 2x4 cm 2 = 16cm 2 surface 23 hours gives F = 2.09(10) -7 g/cm 2 s ” Experimental ” h = 5.6(10) -7 cm/s ”Topas” figure h = 7.0(10) -7 cm/s Iterative modeling h = 11.0(10) -7 cm/s

26. Thomas and Windle Case II Example Windle , “Case II Sorption” in Comyn , Polymer Permeability (1985) Iodine tracer lags methanol in PMMA at 30°C showing apparent step-like gradient. Methanol does not have this “advancing sharp front”. Iodine tracer is far too slow as shown in the following. Methanol gradients become horizontal, not vertical. Thickness: 930 microns

27. Effect of Molecular Properties on D Compare Methanol with Iodine

28. The Next Figure was Reproduced from the Reference Below Uptake of Methanol by Poly ( methyl methacrylate ): An Old Problem Addressed by a Novel Raman Technique Jakob Nixdorf, Giuseppe Di Florio , Lars Bröckers , Carolin Borbeck , Helen E. Hermes, Stefan U. Egelhaaf, and Peter Gilch Macromolecules 2019, 52, 13, 4997-5005 Heinrich-Heine University, Düsseldorf FSRM ( Femtosecond Stimulated Raman Microscopy ) Beam Focus at Middle of Disk Sample With Sides Being Exposed

29. FSRM Raman Analysis of Concentration Profiles Methanol in PMMA ” Breakthrough (BT)” at about 18+ hours

30. FSRM Raman Analysis of Concentration Profiles Methanol in PMMA with Leakage at end Surface Concentration about Doubled Personal Communication from Peter Gilch

31. Prediction for FSRM Study , 1mm, Cylinder D(c) and h from ”Thomas and Windle ” BT 21.1 h BT ca. 18+ h

32. For Calculated Horizontal Gradients at Left to be found in Practice by (FSRM) at Right, C eq MUST be about 0.265 volume fraction. The FSRM Experiment on the Right was NOT continued to equilibrium

33. Hopfenberg et.al. Super Case II n-Hexane/Polystyrene (retained styrene) C. H. M. Jacques, H. B. Hopfenberg , V. T. Stannet , Vapor sorption and liquid interactions with glassy polyblends of Polystyrene and Poly(2,6-Dimethyl-1,4-Phenylene Oxide), Polym . Eng. Sci. 1973 , 13(2)(March), 81-87. R. H. Holley, H. B. Hopfenberg , V. Stannet , Anomalous transport of hydrocarbons in Polystyrene. Polym . Eng. Sci. 1970 , 10(6) (November), 376-382.

34. Hopfenberg and Coworkers Super Case II “Closely” Modeled Absorption

35. PETROPOULOS et.al Hansen cannot explain these data! Stress mechanism given as cause of “anomalous” behavior. Petropoulos JH Sanopoulou M Papadokostaki KG. Physically insightful modeling of non- Fickian kinetic energy regimes encountered in fundamental studies of isothermal sorption of swelling agents in polymeric media. Eur Polym J 2011;47:2053-2062.

36. Hansen cannot explain these data! These slides do explain the data for liquid DCM absorption into stretched, confined Cellulose Acetate F = - D (  c/  x) confirms leakage D several times higher than for liquid DCM Concentration gradients Stretched direction – step Perpendicular due to leakage

37. CALCULATED ABSORPTION CURVE AND GRADIENTS MATCH EXPERIMENTAL DATA FOR ABSORPTION PERPENDICULAR TO STRETCH DIRECTION: METHYLENE CHLORIDE IN CELLULOSE ACETATE

38. METHYLENE CHLORIDE IN STRETCH DIRECTION CALCULATED ABSORPTION CURVE IS PERFECT, FRONT NOT A SHARP STEP, BUT APPROACHES EXPERIMENTAL. LACK OF ADHESION BETWEEN GLASS AND SAMPLE IMPOSSIBLE D(c) TO APPROACH LEAKAGE RESULT!

39. Data: Hasimi et al. Eur.Polym.J . 2008;44:4098-4107 Diffusion Equation matches absorption of water into bone dry PVAlc to 0.748 volume fraction Stress Control is Unreasonable in liquid state!

40. Supercritical Carbon Dioxide 1µ PMMA V. Carlà , et. al. Ind. Eng. Chem. Res. 2009 , 48(8), 3844-3854 Total Surface Control Bulk Explanation not Plausible

41. PERMEATION WITH SURFACE AND/OR EXTERNAL RESISTANCES F =  p/(L/P app ) =  p/(L/P  + R 1 + R 2 + R 3 …) L/P app = L/P  + R 1 + R 2 + R 3 …. 1/P app = 1/P  + (R 1 + R 2 + R 3 ….)/L Use Plot of 1/P  Versus 1/L

42. TRUE PERMEATION COEFFICIENT (P ∞ ) BY EXTRAPOLATION (ACRYLIC FILMS)

43. Permeation of Methylene Chloride in Viton ® ASTM BT 38.4 min matched SS Permeation in g/cm 2 s, 1.15 vs.1.13 E-06 Calc. Data from: K. M. Evans, J. K. Hardy, Predicting solubility and permeation properties of organic solvents in Viton ® glove material using Hansen’s solubility parameters, J. App. Polym . Sci . 2004 , 93, 2688-2698.

44. THE SCIENTIFIC CONCLUSION: THE DIFFUSION EQUATION CAN MODEL DIFFUSION IN POLYMERS The diffusion equation has been shown to fully model diffusion in many situations where other explanations have been presented. An advancing front concentration gradient in absorption is caused by concentration dependent diffusion coefficients. A significant surface condition causes absorption curves that are not linear with square root of time.

45. THE PRACTICAL CONCLUSION Mismatch Hansen solubility parameters to get 1. Lower equilibrium absorption, and therefore: A. Lower concentration gradients B. Lower diffusion coefficients C. Lower surface entry coefficients 2. In practice, for drug and cosmetics packaging , body suits , gloves , geomembranes , etc. Just - Mismatch HSP for Better Barriers

46. SUMMARY Laws of Diffusion Valid for Diffusion in Polymers Exponential Diffusion Coefficients (D(c)) Surface Condition (h) explains ” Anomalies ” Combining D(c) and h - Complete Descriptions Estimate Behavior at Different Conditions Improved understanding and modeling of absorption, desorption, and permeation Improve Barriers with (HSP p ≠ ≠ HSP s )

47. Thank you for your attention! For further contact please visit: www.hansen-solubility.com