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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