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The Physical Mechanism of Osmosis and Osmotic Pressure The Physical Mechanism of Osmosis and Osmotic Pressure; A Hydrodynamic Theory for Calculating the Osmotic Reflection Coefficient by David Charles Guell B.S., Chemical Engineering Iowa State University (1985) S.M., Ciemnical Engineering Massachusetts Institute of Technology (1987) SUBMITTED TO TIfE DEPARTMENT OF CHEMICAL ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHIIYIN CIIEMICAL ENGINEERING at the MASSACHUSETTS INSTITUTE OF TECIINOLOGY February 1991 ( 1991 Massachusetts Institute of Technology Signature of Author: --- - Department of Chemlical Engineering November 20, 1990 Certified by: HIoward Brenner Professor, Department of Chemical Engineering Thesis Supervisor Accepted by: NY~~ -- - William Deen Department of Chemica.l Engineering MASSACHUTS :STrl UTE Chairman, Collmmitteeon Graduate Students OFTECHIVfI nGY FEB 08 1991 UBAIES 1 The Physical Mechanism of Osmosis and Osmotic Pressure; A Hydrodynamic Theory for Calculating the Osmotic Reflection Coefficient by David Charles Cuell Submitted to the Department of Chemical Engineering on November 20, 1990 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Chemical Engineering ABSTRACT The inicroscale, mechanical cause of osmosis and oslmotic pressure in membrane systems is not well understood, and no fully satisfactory mechanism has been of- fered to explain these phenomena. Presented here is a general theory demonstrating that the force the membrane exerts on the solute particles causes osmosis and osmotic pressure. At equilibrium, the greater force exerted by the membrane on the more con- centrated solution is balanced by the macroscopically observable pressure difference known as osmotic pressure. Under nonequilibrium conditions, the membrane-solute force is transmitted to the solvent which is thereby accelerated from rest. The result- ing convective flow of solvent is the macroscopic phenomenon of osmosis. While these ideas have been proposed previously, the general proof of this mechanism is novel. Beyond the academic interest in establishing a mechanical understanding of osmotic pressure, this work lays the foundation on which a general, quantitative theory of osmosis is constructed in Chapter 2 of this thesis. Osmosis in dilute solutions is usually quantified in terms of the osmotic reflection coefficient o-, relating small pressure and concentration differences across a membrane at zero total volume flow according to AP = aoRLTAc. Existing theories for comput- ing o rely upon implicitly averaged momentum transport equations on the length scale of the membrane pores (on which scale the individual solute particles can also be resolved) as well as upon thermodynamic concepts and equations which are not ap- plicable on such scales. The work presented provides a general theory for computing ao which does not appeal to thermodynamic or pre-averaged hydrodynamic equa- tions. Instead, the theory combines 'instantaneous' microscalemomentum-transport equations with recent approaches for modelling the hlydrodynamniceffects arising from Brownian particles. Computing the long-time average behavior of these equations al- lows macroscopically observable quantities such as ao, to be calculated. The theory applies for membranes consisting of uniform pores of arbitrary cross-sectional shape and solute particles possessing any geometry. Qualitative structure property rela- tionships and general bounding results are also derived. Finally, important insights 2 into previous theories are obtained, thereby illuminating the physicochemical hydro- dynamics underlying those approaches. Thesis Advisor: Howard Brenner Professor, Department of Chemical Engineering 3 Acknowledgements I wish to thank Dr. Howard Brenner for his wisdom and guidance during the course of this research. These qualities and many others have permitted me to enjoy the best experience that MIT has to offer. The philosophy he has imparted and the independence he has granted have combined to enable me to achieve goals far greater than my modest abilities would have otherwise allowed. Reflecting on my study with Dr. Brenner, I find that I have learned from him by a subtle (and in retrospect, surprising) process occuring against an existing natural resistance, a process one might accurately describe as "osmotic". I also wish to thank the members of my thesis advisory committee, Dr. William Deen, Dr. T. Alan Hatton, and Dr. Ali Nadimn. Without their guidance and sug- gestions, this research would not have been as successful and satisfying, and I am grateful to have had their input. I also offer thanks to all office mates who have provided feedback on various issues arising during this research. Daily access to such sharp minds has been a valuable resource. Gretchen Mavrovouniotis and Ludwig and Johannes Nitsche deserve special recognition in this regard. I want to thank Bob and Margaret, whose love and esteem have long been a great source of comfort. I also thank Mom and Dad for the support, guidance, encouragement and love which have enabled me to forge the abilities by which I have attained a few, humble accomplishments. Of course, receiving a slide rule for my 4 ninth birthday didn't hurt either Finally, I offer my heart-felt gratitude to mnywife Karen. Her support, acceptance, companionship and love have made the last seven years what I had always hoped life would be. I am fortunate to have married such a wonderful woman. This research was supported, in part, by a National Science Foundation Graduate Fellowship. 5 Contents 1 The Physical Mechanism of Osmosis and Osmotic Pressure 12 1.1 Introdluction . ....... .12 1.2 Literature Review ........... ....... ..13 1.3 The Theory of Osmotic Phenomena . ....... ..25 1.3.1 Introduction. ....... ..25 1.3.2 Foundation of the Theory . ....... ..28 1.3.3 Analysis of Osmotic Pressure ....... ..30 1.4 Discussion ............... ....... .52 1.5 Conclusion. .............. .59 2 A Hydrodynamic Theory for Calculating the Osmotic Reflection Co- efficient 61 2.1 Introduction . ....... ... .61 2.2 Literature Review ........... ...... .. .63 2.3 Motivating Analysis .......... .......... ..71 2.4 A Hydrodynamic Theory of Osmosis . 79 2.4.1 General Equations .......... .. 81 6 2.4.2 Membranes Consisting of Long Uniform Pores . .... 87 2.4.3 Short Range Interaction Potentials ......... ... 100 2.4.4 Implicit Assumptions of the Theory ......... ... 102 2.5 Results. ............................ ... 106 2.5.1 Slit Pores ....................... ... 106 2.5.2 Cylindrical Pores ................... ... 117 2.6 Conclusions .......................... ... 120 2.6.1 Bounds on o and Structure-Property Relationships ... 130 2.7 Summary ........................... ... 134 3 Supporting Calculations 135 3.1 Slit Pores ...... 136 3.1.1 Hydrodynamic Weight Function .......136 .......... 3.1.2 Spherical Solute Particles . 139.... .......... 3.1.3 Needles ............. .....142 .......... 3.1.4 Thin Axisymmnetric Particles .. .....149 .......... 3.2 Cylindrical Pores ............ ......156 .......... 3.2.1 Hydrodynamic Weight Function . 156 ... 3.2.2 Spherical Particles ....... ......159 .......... 3.2.3 Needles ............. 167 ... 3.3 Approximate Representations of the Solute Particle . 171 ... 3.4 Correspondence with the Theory of Anderson et al. ... 173 ... 3.5 Sensitivity to the Choice of Force Distribution Function 182 7 4 Computer Output and Source Code 189 8 List of Figures 1.1 Hypothetical osmotic pressure experiment. ............... 31 1.2 One piston osmotic pressure experiment ................. 44 2.1 Data of Durbin (diamonds), Schultz et al. (triangles) and Long & An- derson (squares). ............................. 67 2.2 A membrane showing the separate reservoir and pore regions. .... 73 2.3 A long, thin pore with uniform cross-section ............... 88 2.4 Pore cross-section showing contour traced out by particle locator point when the particle contacts the pores wall ................. 101 2.5 A slit pore with spherical and needle-like particles .......... 107 2.6 Theoretical curves for X for short range potentials .......... 113 2.7 Theoretical curves for O,, for hard potentials ............... 114 2.8 Comparison of theory with experimental data of Long & Anderson. 121 3.1 Four equivalent orientations for a needle in a slit pore. ........ 144 3.2 Cross-sectional view of spherical particle in contact with a cylindrical pore with A < 1/2. ............................. 160 9 3.3 Cross-sectional view of a spherical particle in contact with a cylindrical pore with A > 1/2 ............................ 165 3.4 Cross-sectional view of a needle in a cylindrical pore. ......... 168 10 List of Tables 2.1 Results assuming different force distributions. ............. 127 11 Chapter 1 The Physical Mechanism of Osmosis and Osmotic Pressure 1.1 Introduction Membranes are of major biological, physiological and industrial importance. They allow varied organisms to exchange waste products for food, physicians to administer drugs with great precision via transdermal delivery devices, and water to be purified simply and inexpensively. Osmotic phenomena remain worthy of study because "no physical phenomenon has any greater import in biology than does osmosis" [102] yet, "despite the fundamental significance for biological fluid transport, an understanding of osmosis at the molecular level has been lacking" [54]. In particular, the microscale, mechanical cause of osmotic pressure in equilibrium membrane systems is not well characterized
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