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Notes Indian Journal of Chemical Technology Vol. 10, September 2003, pp. 558-560 Osmotic pressure measurement of polymer solutions by dynamic methods: A discussion M A Islam*, M M R Khan & M A Saw pan Department of Chemical Engineering and Polymer Science, Shahjalal University of Science and Technology, 3114 Sylhet, Bangladesh h Received 28 January 2002; revised received 3 March 2003; accepted 7 April 2003 The data treatment procedure of osmotic pressure measurement by dynamic methods is discussed in the light of the theory of pressure relaxation. Some drawbacks of the procedures are noted and finally an improved method is proposed ·for tlie purpose. There are two methods for the determination of osmotic pressure of polymer solutions 1• In the first, called as static method, two reservoirs containing the polymer solution and the pure solvent are separated by a semi-permeable membrane and the difference between liquid levels are registered after the attainment of the equilibrium state. The other one called as the dynamic method is based on a different principle-the flow of the solvent through the membrane is observed. Both the methods have their advantages as well as disadvantages. In the last decades very little is discussed on the theory of the membrane osmometry. The last detailed discussion on the problem was done in 60's and 70's 2 •3 • Important physico-chemical parameters of polymers e.g. molecular weight, interaction parameters, etc. are determined from the osmotic pressure measurements, and hence precise data treatment is important both from theoretical as well as practical viewpoints. This paper discusses mainly the theoretical ground and the data treatment procedures of the dynamic methods. The source of errors is identified and finally an improved method for data treatment is also proposed. Theory The measurement of osmotic pressure is illustrated in Figs (l a-b). The flow of the solvent from the solvent-reservoir to the solution-reservoir is taken to be positive. Applying Darcy's law 4 for the permeation *For correspondence (E-mail: mislam@sust. ed u) (a) (b) Fig. !-Illustrative diagrams for the meas urement of osmotic pressure: (a) constant hydraulic pressure regi me, (b) variable hydraulic pressure regime of a fluid through a porous medium under a pressuregradient, gives dV (~0 -~P) Adt J.1R w=--=---- .. . (1) where w is the flow-rate through the membrane, m/s; V is the volume of the fluid passing through the membrane, m3 ; A is the membrane area, m2 ; t is the time, s; ~p is the hydraulic pressure-difference across the membrane, P.; ~n is the osmotic pressuredifference across the membrane, P. ; J.1 is the fluid viscosity, P•. s; and R is the membrane resistance, m·'. The osmotic pressure and the hydraulic pressuredifferences are given by the following Eqs: (2) (3) where h and h~ are respectively the heights of the solution and the expected equilibrium height measured from the solvent level (taken to be the reference level), pis the density of the solution, kg/m 3 (For low concentration, solvent and solution densities are considered to be equal.), g is the acceleration due to gravity, m/s 2 . Combining Eqs (l-3) Notes Islam et at. : Osmotic pressure measurement of polymer solutions w = _dv_ Adt = ..:..._p~g....:...(h-=~_-_h_:_) J.1R .. . (4) w, rn/s The method of zero flow rate Eq . (4) is the theoretical ground of the method (represented graphically in Fig. 2). Obviously the point of intersection of the curve w versus h with the h-axis gives the value of h~. At a value of h higher than h~, the flow direction changes. If the resistance of the membrane depends on the flow direction (owing to its morphological characteristics), the slope of the curve might change at h= h~ (shown by discrete lines). For high concentrations, the w versus h relationship might not be strictly linear due to the concentration polarization phenomenon 5 . Still the point of intersection indicates the balance between osmotic pressure and the hydraulic one. This method for the determination of osmotic pressure is known as "Method of zero flow rate"'. An illustrative diagram for the determination of osmotic pressure at constant hydraulic pressure regime is as shown in Fig. la. The experimental procedure for data collection for the method of zero flow rate is as follows : A series of flow-rates wi are determined for hi =constant (for hi higher and lower than the probable h~). Then h~ is determined graphically as shown in Fig. 2. The main disadvantage of the method is the variation of the experimental set-up for each measurement. Fig. 2-Fiow rate w (m/s) versus solution level h (m). Illustration of "The method of zero flow rate" h,m ~ 2 c The method of semi-sum An illustrative diagram for the determination of osmotic pressure at variable hydraulic pressure regime is as shown in Fig. 1b. Outing the measurements, the pressure difference varies. The solvent flow (positive or negative) through the membrane is monitored by the movement of the meniscus in the capillary, and is recorded as a function of time. The tangent at any point on the curve h versus t would give the flow rate dh/dt at the corresponding h. Since now onward, the data treatment procedure could be the same as that of "the method of zero flow rate". Instead a more reliable method is applied. Two different experiments are performed with two initial values of h= h 0 , and h02 (hot < h~ and h02 > h~) (curves 1 and 2, Fig. 3). At a given t =t, a vertical line is drawn, which cuts the taxis and the curves respectively at the points 0, A and B. A point C is determined such that OC =(OA +OB)/2 (i.e. C is the mid-point of AB) . A sequence of points <C > is determined in such manner, which are B he! v A 0 1 t, s Fig. }--Pressure relaxation curves h versus t. Illustration of "The method of semi-sum" connected by an averaged line intersecting the h axis at h~ . This method is known as "the method of semisum" . Whether, "The method of semi-sum" as a data treatment procedure is accurate from theoretical viewpoint, can be verified. As h is variable, Eq. (4) may be rewritten as follows: s dh = .:........:::....:.....=._ p g(h~ -h) _ _:_ A. dt .. . (5) 559 Notes Indian J. Che.m. Techno!., September 2003 where sis the cross-sectional area of the capillary, m2 . Solving Eq.(5) for t =0, h =h0 , Eqs (6a) and (6b) are obtained (h~ - (h~ h) -flo) e - a! h,m k - - - - - - - - - - - - , E' .. . (6a) A' -------<;>----1' or, h= h~ - (h~ - flo)e-"1 ... (6b) c A with a =Apg lsJ.lR Eq. (6b) describes the curves in Fig. 4. Let's at a given time t, h 1 and h 2 are respectively the positions of the meniscus in consecutive experiments (curves 1 and 2, Fig. 4). Thus from Eq. (6a), the Eq. (7) is obtained 0 t, s Fig. 4--Pressure relaxation curves h versus t. Illustration of the proposed method ... (7) h - /, + ~ On some algebraic manipulation of Eq. (7) (h~ - ~) (~-flo,) CA AE (~- h~ ) (flo2- ~) CB DB ... (8) Precise estimation of h~ After some algebraic manipulation of Eq. (8), can be expressed as follows: h~ w, tlH - !1H with W, Eq. (8) shows that the point C divides the intercept AB in the ratio AE : DB - A ratio between the displacements of the solution meniscus in the two experiments at a given time. In general, C is not the mid-point of AB. Obviously the assumption of the method of semi-sum that ' the two curves are symmetrical and C is the mid-point of AB is incorrect. Another drawback of the method of semisum is that, the tentative value of h~ is to be known before carrying out the experiment, so th~t the curves lie on the two sides of the probable h = h~ line approximately as the mirror image to each other. Eq. (8) shows that it is not so essential that the curves must lie on different sides of h = h~. If in both the experiments, for example the initial value of h is greater than h~ (curves 1' and 2, Fig. 4), the relationship (8) is still valid. However, in such cases C lies on the extension of A'B. 560 ·~ I 2 (I. - /, ) "2 ·~ .. . (9) = ~ -flo, and tlH 2= ~ -11o2 All the terms in the right hand side of the Eq. (9) are known and the value of h~ can be calculated. Thus a sequence of< h~ i> can be generated applying the Eq. (8) at different values oft. Then an averaged line parallel to the t-axis can be drawn through the points < h~ i> which intersects the h-axis at h~ . Conclusion The semi-sum method for the treatment of osmotic pressure data is not in agreement with the theory of pressure relaxation. An improved method is proposed for the purpose, which does not require that the pressure-relaxation curves must lie on different sides of the curve h = h~ . References I Denev G, Characterization of Polymers (Chemical Institute Publishing House, Bourgas, Bulgaria), 198 1 2 Elias H G, Dynamic Osmometry in Characterization of Macromolecular Structure, Proceedings of a conference, April 5-7, 1967, Warrenton, Virginia (National Academy of Science, Washington), 1968 3 Ulrich R D, Membrane Osmometry in Polymer Molecular Weights, Part I, edited by Slade P E 1 R (Marcel Dekker Inc, New York), 1975 4 Hwang S T & Kammermeyer K, Membranes in Separation (Wiley, New York), 1975 5 Jonsson G, Desalination, 51 (1984) 61