G. R. Garbarini, R. F. Eaton, T. K. Kwei, and A. V. Tobolsky Pr~ncetonu n ~ v e r s ~ t y Princeton, New Jersey, 08540
1
Diffusion and Reverse Osmosis through - Polymer Membranes
Although knowledge of osmotic phenomena dates back more than two centuries, osmosis remained a scientific curiosity until recent times. The prospect that the reversal of osmotic flow could prove to be a commercially feasible method of obtaining fresh water from saline sources stimulated new interest in the fields of diusion and osmosis. The process of osmosis involves diffusion of a fluid through a semipermeable membrane separating two solutions of different concentrations. The flow of liquid is from the dilute solution into the more concentrated solution thus tending to equalize the concentration on both sides of the membrane. In a closed system the flow into the concentrated side of the membrane causes an increase in hydrostatic pressure on that side, and osmotic flow continues until the chemical potential of the diffusing component is the same on both sides of the barrier (1). When a pressure higher than the osmotic pressure is applied to a concentrated solution, reverse flow of solvent through the membrane can be obtained, resulting in purified solvent permeating to the opposite side of the membrane. This process is called reverse osmosis or ultrafiltration and is presently developing into a new method of purification and concentration. Brackish uater, sea water, industrial and community waste water, and other contaminated water can be purified by reverse osmosis. To be efficient, membranes should be highly permeable to water as compared to salt. Reverse osmosis experiments are usually carried out in continuous processes under moderate pressures. I n this paper we would like to review the important theoretical aspects of diffusion and osmosis and describe the development of small scale experiments which can be easily carried out in an undergraduate physical chemistry laboratory. The results of diusion of water and salt through reverse osmosis membranes are given and discussed. A comparison is made between the results obtained in a quiescent system and those obtained in large scale flow experiments. General Considerations of Diffusion Fick's lows
The mathematical theory of diffusion in isotropic substances is based upon the hypothesis that the rate of transfer of diffusing substance through unit area of a section is proportional to the concentration gradient measured normal to the section. This hypothesis is expressed as an empirical relation
where J is the rate of transfer per unit area of section, C 226
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is the concentration of diffusing substance, x is the space coordinate measured normal to the section, and D is called the diusion coefficient or diffusivity. Equation (1)is known as Fick's first law. By considering the mass balance of a volume element it can be shown ( 2 , s ) that
where D is a constant. Since diffusion often effectively occurs in one direction only, eqn. (2) can be expressed as
where D varies with concentration. Equation (3) is known as Fick's second law of diffusion (2). Steady State Diffusion
I n diffusion experiments, both the transient state and the steady state are important and yield valuable information especially where diffusion through a membrane of thickness, I, whose surfaces x = 0 and x = I are maintained a t constant concentrations C1 and C2, respectively. At the beginning of the experiment both the rate of flow and the concentration a t any point of the sheet vary with time. At some later time steady state is reached, and the concentration changes linearly from C1 to C2 through the membrane. The rate of transfer of diusing substance is the same across all sections and is given by
If the thickness, 1, and the surface concentrations CI and Ca are known, then D can be obtained from a single measurement of the flux, J . Permeability and Solubility Constants
I n many practical systems, however, the surface concentration is not always known with accuracy, andit is often convenient to express the flux, J , in termsof the concentrations of the external solutions, Cat (or, in the case of gas diffusion, pressure).
I n eqn. (B), P i s known as the permeability coefficient, expressed as gram (or mole) of d i u s a n t per second passing throngh a polymer film per cm2in area and per cm in thickness when the Concentration difference of the
external solution is 1 gram (or mole) per cmt Other suitable units may also he used. If there is a linear relationship between the concentration of the external solution and the corresponding surface concentration, i.e., if Henry's law is obeyed C = ,yce=t
(7)
where S is the solubility constant, then it follows from eqns. (5), (6), and (7) that P
=
DS
(8)
Determination of the diffusion coefficient without a prior knowledge of the surface concentration may he accomplished by the "time lag" method described in the next section. Time Log Method
From the instant the diiusant is first introduced to one side of the polymer sheet and prior to the establishment of a steady state, both the rate of flow and the concentration a t any point of the sheet vary with time. This unsteady state diffusion is expressed mathematically as Fick's second law
where D is constant. Barrer (2) presents the solutions of eqn. (3) for avariety of geometries and boundary conditions. One case very useful in experimental studies is the unsteady state diiusion through a polymer membrane with boundary conditions C=Coatt=OmdO Oandz = O
where Co is the initial concentration of the diffusing species in the membrane and r = 0 and z = 1 refer to the upstream and downstream faces of the membrane. If the diffusion coefficient is constant, the membrane initially free of diffusant, and the diffusant continuously removed from the low concentration side (i.e., C2 = 0), the amount of diiusant, Q, which passes through the membrane in time, t is given by (3)
-
As t a, the steady state is approached and the exponential terms becon~enegligible. Thus a plot of Q as a function oft yields a straight line equation
The intercept, X, on the same axis, the time lag, is A = (12/6D)
(11)
D = lP/6X
(12)
then
Figure 1. Illurtration of the time lag for diffusion through a polymer membrane.
ahility,P, can he calculated from the slopeof the curve a t steady state. Wit,h P and D known, the solubility constant S can be calculated from the relationship, P = DS. Daynes (4) was the first to note the usefulness of this approach. I n cases where the time lag is very small hut an independent measurement of S is possible (6)the steady state method of determining the diiusion coefficient should be used. Desalination
One of the most important applications of diffusion technology is desalination of sea water and brackish water. In many areas of the world the supply of fresh water, about 0.05 wt Oj, salt, cannot meet demand. Therefore it is necessary to he able to purify sea water (3.5 wt yosalt) and brackish water (0.2-0.8 wt % salt). I n reverse osmosis, water is made to diffuse across a polymer membrane from the high salt conceutration side to the low salt concentration side by the application to the salt water of a pressure in excess of the osmotic pressure. I n order to standardize the many experiments which are being conducted on cellulose acetate and other reverse osmosis membranes, a system of evaluation of water and salt permeabilities of candidate membranes has been introduced by Klein (6). The theory on which this system is based was developed by Staverman (7) and Kedem and Katchalsky (8) for composite membranes and Elata (9) for cellulose acetate membranes. The three coefficients which experimentalists determine are: the filtration coefficient, L,, the salt permeability coefficient,G, and the reflection coefficient, a. These coefficients define the behavior of a membrance with respect to water and salt transport under pressure and concentration gradients. Filtration Coefficient, 1,
Equation (12) is used to calculate the diffusion coefficient from the time lag. All that needs to be known is the membrane thickness, 1, and X, the time lag. An illustration of the time lag is shown in Figure 1, for typical experimental diiusion data taken a t constant concentration dierence across the polymer membrane. Thus from one set of data, the diffusion coefficient, D, can he calculated from the time lag, X, and the perme-
The filtration coefficient relates the water flux through a membrane to the applied pressure. In order to measure L, distilled water is charged into a pressure filter. The system is pressurized from a nitrogen cylinder through a reduction valve and effluent collected in a tared flask. L, is then calculated using the expression 'L
=
-J,
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where J , is the water flux and AP is the applied pres- ~ sec-'. sure. L, has the units g ~ m atm-I Solute Permeability Coefficient,
i
Solute permeability measurements are carried out using an osmosis cell of a type which will be described later. Distilled water is placed on one side of the cell (pure water side) and sodium chloride solution on the other side (brine side). The half cells are then closed off so as to produce a condition approximating zero net flux. Diffusion of salt into the pure water side is then measured with a conductivity cell. After converting the conductivity measurements to moles of sodium chloride, salt flux, J,, is obtained by taking the slope of a plot of salt concentration on the solvent side (pure water) versus time. The salt permeability coefficient, i,is then calculated from the expression
where a, is the calculated osmotic pressure for the original sodium chloride concentration used on the brine side in the experiment. The units i are moles cm-%atm-' sec-'. Reflection Coefficient, u
The reflection coefficientgives a measurement of the selectivity of the membrane. Measurements are conducted using an osmosis cell similar to that used for determination of the solute permeability coefficient. The amount of water which flows by osmosis from the pure water side to the brine side is measured by collecting the effluent from the brine side. The volume flux is calculated by
where V is the volume of effluent collected throngh A om2of membrane area in t seconds. From the determiustion of J , in the osmosis experiment g can then be calculated from the expression
where As, is the osmotic pressure of the original brine and L, is the filtration coefficientdetermined previously. The t,hree described transport coefficients are very useful as a means for screening potential membrane materials for reverse osmosis applications. It i s possible to predict reverse osmosis behavior of a membrane from the simple osmosis experiments described. In reverse osmosis experiments the salt rejection coeffcient R, defined as
where C,' and C," are the salt concentrations in the feed solution and product water, respectively, is determined. The rejection coefficient, R, has been shown (eqns. (9), (lo), (11)) to be related to the reflection coefficient,u, by Thus in reverse osmosis experiments, theory predicts that a t high flux rates R will approach the value of u which was determined from direct osmosis experiments. It should he recognized that predictions made from the 228 / Journal of Chemkol Education
application of osmosis data to reverse osmosis are st.rictly valid only when the three basic transport coefficients are the same for both kinds of experiments. It has been found that for cellulose acetate these conditions are often not met. Many polymer membranes may be expected to yield under compressive forces experienced in a reverse osmosis unit. This results in a decrease in the filtration coefficient L,. The solute permeability coefficient is sometimes increased. Laboratory Measurement of Water and Salt Flow Measurement of D,
Cell. For small laboratory diffusion experiments the diffusion cell is of the two-chambered, single membrane type. The cell is usually made of glass but other materials such as stainless steel have been used. In glass cell arrangements the mounted polymer membrane is held in place between the abutting flanges of the cell halves by spring clamps or screw clamps. A seallant type of grease is often used between the membrane and the flanges. Method of Measuring D,
One method of measuring water permeability is described by Peterson and Livingaton (18). They used a glass cell similar to the one described above. A membrane which had been previously equilibrated with water was placed between two halves of the cell. The two cell chambers were filled with water. Pressure of between 0.05 to 0.1 stm was applied ta one of the chambers by means of a water column which was adjustable in height. Increases in the volume of water in the chamber on the other side of the membrane were detected by means of a uniform bore glass capillary attached to the top of the chamber. The capillary was fixed in a horizontal orientat,ion so that the volume of water entering it during the experiment could not generate a hydrostatic pressure difference between the two cell chambers. The rate of flow of water through the membrane under the applied pressure gradient was measured by observing the advance of the meniscus along the capillary as a function of time. The filtrstion coefficient, L,, can be derived from this experiment. Another method of measuring the diffusion of water through less permeable membranes is the radioactive tracer method. Measured quantities of H 2 0 and H 8 0 were added to one side of a glass diffusion cell similar to the one deseribed previously. The initial amount of HaBOadded can be accurately determined by liquid scintillation counting. The side into which the tracer is added is referred to as t,he "hot" side. The other side is the "cold" side. The volume of each half cell was 100 ml. For cellulose acetate membranes it was necessary to add at least 1 microcurie of radiation in order to have meaningful data a t short times (
