- Email: [email protected]
Transport coefficients of asymmetric cellulose acetate membranes
Desalination, 16 (1975) 65-83 @ Elscvier Scientific Publishing
TRANSPORT
Company.
COEFFICIENTS
Amsterdam
- Printed in The Netherlands
OF ASYMMETRIC
CELLULOSE
ACETATE
MEMBRANES*
Using the linear relations of thermodynamics
of irreversible processes,
the
transport
coefficients I,,. i,, f,,. and 0 were measured for an asymmetric cellulose acetate membrane with NaCI, Na,SO,. CaCI,. NaF, and snccharosc over the concentration range (O-O.5 mol,A or 1 mol!i) at 2OT or 23’C. The experimental findings manifest a strong dependence of the three transport coeflicients IP, f,, and I,, on solute concentration. This strong dependence on concentration can be attributed to a concentration gradient within the porous sublayer of the asymmetric membrane. Thereby, it is sho\\n that the transport coelkients of an asymmetric membrane depend on the solute concentration on both sides of the membrane instead of the mean concentration C:,, as would be the case for a homogeneous membrane. SYMBOLS
-
-
effective membrane area (cm’) local salt concentration within porous sublnyer (mol/cm’) salt concentration (mol/l) of phase (‘) or (‘). respectively salt concentration at the interface between actike layer and porous sublaycr (mol/cm3 or mol/l) mean salt concentration (mol/l) = (I/Z)(c: + c;) water concentration (mol/cm’) concentration difference across entire membrane = c: - c;’ concentration difference across active layer = c; - c; diffusion coefficient of salt within porous sublayer (cm’/s&) osmotic coefficient = Allrc3,~AlIidrJ, mechanical permeability of entire membrane (cm/set - at) mechanical permeability>f active layer (cm,kec - at)
-__l Paper presented in sections at the Fourth Intern. Symp. on Fresh Water from the Sea. Heidelber& September 9-14, 1973.
66
w. PUSCH .
,, 6
I;
-
1,
-
mechanical permeability of porous sublayer (cm/see . at) intrinsic mechanical permeability of entire membrane (cm/set osmotic permeability of entire membrane (cm/set . at) osmotic permeability of active layer (cmjsec - at)
-
osmotic intrinsic
-
coupling coupling
cocfhcient coehicient
-
pressure pressure pressure
(at) of phase (‘) or (“I. respectively difference across entire membrane difference across active layer (at)
-
pressure difference across porous sublayer (at) = Pm - P’ pressure at the interface between active layer and porous sublayer (at) volume flux (cm/set) mean volume tIuu (cm,kec) = AC’;,4 - Ar volume flow (cmj/sec) = A .q volume (cm3) of phase (‘) or (‘), respectively partial molar volume of water (cm’/mol) volume change during time interval t, to r2 (cm’) = V’(f2) - V’(r,) =
1: ,, ‘* 1:
-
r; I “P I’ f? P” AP APrn AP” Pm 4 14
Q
-
V’, V’
-
VW AV
-
I
-
Al s s
-
I7’. II”
-
l-t+=
-
AI-I hi-l”
- (V”(r,) time (set) time interval
of entire of active
membrane (cmlsec - at) = /,, layer (cm/kc - at) = 0’ - /L (at) = P’ = P’ - Pm
P”
V”(Q)
-
perpendicular to membrane surface (cm) of porous sublayer pressure (at) of phase (‘) or (“), respectively pressure at the interface between active layer (at) pressure difference across entire membrane (at)
=
II’ -
-
osmotic
pressure
= ll’
-
IIm
An”
-
osmotic
pressure
difference
(at)
=
AI-I, Al-r, __._ AI-I
-
osmotic osmotic
pressure pressure
difference difference
-
mean
-
(l/z) (AK + AK) reflection coefficient
@S a, % %‘*%” f
-
osmotic
=
+ z;>
coordinate thickness osmotic osmotic sublayer osmotic
Cr t?‘, c*
-
sublayer (cm/kc . at) of entire membrane (cm,‘sec . at)
permeability of porous osmotic permeability
(/;,)‘/(r;
* at)
difference
pressure
across across
active porous
layer
(at)
sublayer
and
porous
nm
-
II” l-l”
(at) at the beginning of a dialysis experiment (at) at the end of a dialysis experiment
difference
of entire
over time interval
membrane
of measurement ;
reflection coefficient of active layer or porous s_ublXyer, respectively salt flux across the entire membrane (molkm’sec) water flux (moljcm’sec) chemical fiuu (cmjsec) = O,!pS - O,~L;,. chemical flux (cm/set) in phase (‘) or (“), respectively mean chemical flux:(cm/sec) = (l/2)(%’ + x”)
=
TRANSPORT
COEFFICIENTS
OF ASYMMETRIC
%;,x” -
mean value of chemical or (‘). respectively
CA hfEhlERANES
flux over time interval
67
I, -
r2 for phase (‘)
INTRODtiCTIOS
Given an isothermic system, with a membrane separating two salt solutions of different salt concentrations c: and cz (Fig. I) which are kept under different hydrostatic pressures P’ and P’, one can use the linear relaticns of thermodynamics of irreversible processes to describe the transport processes through the membrane. membrane
salt
solution
salt
solution movable piston
Fig. 1. Diagrammatic
presentation
of an isothermal
membrane system.
There exist, as Schlapl (I) has shown, the following linear relations between fluxes and conjugated forces for a system near equilibrium: q = I$P l = l,,AP
+ l,$rI
(1)
+ 1,Al-l
(2)
Thereby. the Onsager relation I_, = I,, was used. The chemical flux z, a measure of the variation of concentration with time within the two phases (‘) and (“), is defined by: (3) With the definition man (2), Eq. (I) yields: q = l,(AP
of the reflection
- 0Al-T)
coefficient
u E -&,/I,
given by StaveriW
Using these linear relations the transport coefficients I,, f,, I,, and u can be determined by measuring the dependence of volume flux 4 and chemical !Iux x on pressure difference AP and osmotic difference All. DESCRIPTIOS
OF EQUIPMENT
With modified equipment (3, 4), shown in Fig. 2, all transport coefficients can be determined using different experimental setups. With two conductivity
EK
/
FIN 2. Cross-sectional
view of ddys~s
ceil.
Fig. 3. Partition wall with membrane holder.
cells (L) immersed in the two solutions, the variatron of salt concentration with time is followed by measuring the conductivity by means of a WayneKerr-Bridge with an accuracy of 0.1 :i_ The volume flux q is measured by the increase of volume AC’ of the corresponding phase using a calibrated capillary (K, 1~1000 ml) or determining the volume of solution passing through a small capillary into an Erlenmeyer flask by weighing. To reduce evaporation the Erlenmeyer flask is sealed by a silicon-rubber stopper (St) containing a perforation for the capillary.
TRhSSl’ORf
Pressure
COEFFICIENTS
equalization
OF ,\SYblW3-RlC
is attnincd
wall (MT) contains the membrane is supported by 3 CONIDUR
CA MEAlBRAbif23
by a canula (M) which stainless
through
69 the stopper.
The
partition
ic &cd cccentricahy. The membrane screen (Hein. Lehmann & Co.,
steel
Diisseldorf, Germany). The screen is 0.35 mm thick and has holes of 0.15 mm diameter. Wtth different partition v\alls, membranes with an effective area of holder is shown -I cm’, h-4 cm’. and - I6 cm’ can be used. The membrane sep;mrcly in Fig. 3. The solutions arc vvoll stirred by means of muanetic stirrers. All surfaces of contact arc sealed by O-rings. Air is used for pressurizing one phase. The air pressure is mointnined constant to +_5 - lo-’ at by means of KEN!\IDAL valves (VIA, Diissctdorf. Germany). The ptcssurc is Ineasured by high accuracy pressure ~iuges ( 2 0. I 0 “) produced by WIKA (Klingcnbcrg am Main. Germany). CELLUI USE ACETr\TE MIXUKhSE
For
most
cured at Y2.5.C. The salt rejection
measurements.
asymmetric
ccllulosc
acetate
membranes
The preparation of these membranes is described of the membranes is about 97 ‘,‘i at 30 at.
The four transport coetlicients - three of which measured u ith the following three dit?krent experimental
are used
elsewhere
are independent setups.
(5).
- \\cre
n)
Deternlittaticw qf I, ut An = 0 Both cell compartments--phase (‘) and (‘)-contain salt solutrons of equal concentration. Measuring the volume flow Q = .+I . q. as a function of AP using four dnhxcnt values of AP. one gets I,, from the slope of the straight line in a Q-AP-diagram if the effective membrane arm is known.
An osmotic difference AIl betvvcen the two cell compartments is established (c: > l-z) and the volume flow Q is measured as a function of AP. From the slope of the straight line in a Q-AP-diagram, one again gets /,,. In addition, from the intersection of the straight line with the abscissa. one gets s = (AP/ATl) at y = 0. Thereby. a mean value All of AIl is determined from AD, at the beginning and AD, at the end of the measurement (arithmetical mean value). The osmotic difference AD, is determined by conductivity titration of the t\vo salt solutions after the experiment is finished. c)
Dctern~imliatt
of I, mid lyp at AP = 0
At t = 0 (t = time) a concentration partments is set up (cc: > c;). The variation
difference between of concentration
the t\vo cell comwith time is then
IV. PUSCH
0
02
Fig. 4.
cl&
Mean
on
0.6
\olumc
ID
an[aml
flux q and mean
chemical
fh.t~ j as P function
of man
osmotic
pressure
difference hII at 25’C using a mean salt concentration C, = 0.1 m NaCl (c,’ = 0.11 m NaCl: c,- = 0.03 nr NaCI) and an as_vmmetricmembrane annealed at 82.5 ‘C (membrane area A :- 3.14cm’). ‘I
followed by measuring the conductivity of both phases at a time interval of about 24 hours. After the measurements are completed (after about ten days) the cell constants of the used conductivity cells are determined once more by conductivity titrations of the salt solutions from both compartments. Assuming a linear relation between conductivity and salt concentration for small concentration differences, one gets the salt concentration from a calibration curve. The volume flow Q is received by determining the volume of solution which flowed into an Erlenmeyer flask during about 24 hours. From the curves of concentrations c:, r,“, and volume increase A V with time a mean value of the chemical flux, 7, and the volume flux, y, as well as a mean value All are estimated by averaging over 24 hours. The mean values are obtained from the measured values for the time interval t, to t2 as follows: Al-I(f,) = l-I’&) - l-I”(t,)
(4)
An(r,)
= I-Of,)
(5)
an
= tCAWrl>
_, %
A;-
-2
= -+-g
- II” + AWt,)l
’ CcXtJ- c:O,)l/At I [c;‘(t,) I
-
4’(t,)J/At
(6)
(7) @I
TRANSPORT
COEFFIClENTS
OF ASYhlhlFTRlC
l7
= AVIA - At
At
= I, -
c:,
= $. cc:u,)
CA BIEhlBRANES
71 (9)
1,
(10) + c:Cr,) + cI(rz)
+ c3f,)]
(11)
as functions of All, one gets straight lines (Fig. 4). The slopes Plotting j * and of the calculated regression lines yield the coefficients f, and &, respectively. If I, is known from a different measurement, one gets once more G = -f&l,, RESULTS
NaF-, CaCI,-, and saccharose-solutions the Using NaCI-, Na2S0,-, mechanical permeabiiity IP was measured as a function of solute concentration (0 ,( c, < 1 moljl) at 20°C. Using NaCI-, Na2SOJ-, NaF-, and CaCl,-solutions the reflection coefficient r~ was also measured. The results are graphically represented in Figs. 5. 6. and 7. In addition, I, and I,, were determined as a function of concentration at 25’C using NaCI- and Na,SO.-solutions. The experimental findings are shown graphically in Figs. S and 9. As one can see from these figures, there exists a strong dependence of I,. I,, and I_ on the salt concentrations c, or (:,, respectively. On the other hand, the reflection coetlicient 0 varies only sliatly with salt concentration. If one calculates u from /,, and I,,, (CT = -/,,/I,) measured by methods (a) and (c), respectively, it is found that these values agree with the Uvalues measured directly using method (b). If o = 1, there exists the following relation betivecn I,,. I,. and In,, (6): I/,,( = [/,I = I/J. As can be seen from Fig. 10, that relation is satisfied for Na,SO, solutions since q - j and (T = 1.
OL
a2
0
04
0.6
0.6
l.0
F,[Md/l]
Fig. 5. Reflection coefficient fl as a function of mean salt concentration & at 20‘C using solutions of different salts (NaCl-(I) curve measured Hith a virgin membrane; NaCI-(II) curve measured with a membrane which has already been used in hypextiltration at 50 atm).
l
j
=
1/‘(;7’
_I
2”).
W. PUSCH
at 20X
0
Fig. 6. Mechanical pcnncabiltty I,, as a function unng solutions of different salts.
0
al
as
u
ln
of salt concentrationC,
=.=
;a k“
=- c.3
w GWll
Fig. 7. Mechanical pernwabiiity I,, as a function of solute concentration c, at 3 -C using saccharose solutions With regard to the concentration dependence of I, in this case, the increase of the viscosity of saccharose solutions with increasing solute concentration has to be taken into a02ount.
The strong dependence concentration,
as manifested
rezxons: I. Concentration
of the three transport coeficients, experimentally,
polarization
to the membrane surfaces.
Gthin
I,,, lx, lx,,, on salt
may occur for the following
an unstirred boundary
three
layer adjacent
TRANSPORT
COEFFICIEKI’S
OF ASYMMETRIC
Fig. 8. Osmotic permcabihty solutions.
CA LlEMBRASES
73
I, as a function of mean salt concentration
F., at 3 ‘C using
I,,, as a function of mean sdt concentration
C; at 25 ‘C using
NaCl and NadOt
Fig. 9. Coupling coeffkicnt NaCl and Na:SOa solutions.
2. Concentration profiles (gradients) within the porous substructure of the asymmetric cetlulose acetate membrane as a consequence of counterbalance of convective flow and <;llt diffusion. 3. Swelling effects of the membrane material due to the influence of salt. TO verify the influence of the unstirred boundary layer over the concentration dependence of the transport coefficients, a second dialysis cell with very effective circulation pumps was use 1. With this modified equipment, the same experimental findings were obtained. This indicates that unstirred boundary layers are not the main reason for the strong concentration dependence of the transport coefficients.
74
W. PUSCH
Fig. 10 Mean bolumr flux q and mean chemial flux 2 as a function of mean osmotic pressure diffcrcnce ifi? ar 25 C using Na=SOa solutions of a m-n salt concentration 2. :: 0.2 1~ NazS0~ (cr’ :-: 0.12 ,n NatSOa: c,* = 0.18 ))I NarSOd: A -= 3.14 cm’). In this special case the wo straight lines ij and 2 as a function of IJZ coincide since the retktion coefficient I; - 1.
On the other hand, if swelling effects are responsible
for the strong concentration dependence of the transport coefficients, one would expect the same experimental results with regard to the concentration dependence with hyperfiltration experiments. But looking at the literature and taking into account our own unpublished results of hypefiltration measurements, it appears that the mechanical permeability of asymmetric cellulose acetate membranes. for instance, does not depend strongly on the salt concentration of the brine. Now, the reason for this difference can be found in the different boundary conditions. Whereas, in hypefiltration
the boundary
condition
of free outflow exists at the low pressure
side, in dialysis experiments the salt concentration on both sides of the membrane is maintained constant. Therefore, one should check the influence of different boundary conditions in dialysis measurements on the value of the mechanical permeability, 1,. To investigate that, measurements with the following boundary conditions have been made. 1. The salt concentration c; was varied between 0 and 0.2 nr NaCl keeping c,” = 0 (pure water in the corresponding cell compartment). Thereby, the active layer of the membrane was juxtaposed with the salt solution_ The volume flow was directed from phase (‘) to phase (“) by adjusting the pressure difference, AP, across the membrane. 2. The same arrangement as in case 1 but the membrane was turned over so that the active layer was juxtaposed with pure water and thus the porous layer was juxtaposed with the salt solution. Thereby, the volume flow, Q, was directed from phase (‘) to phase (‘). 3. The salt concentration c; was varied between 0 and 0.2 IPZNaCl keeping c’ = 0.2 m NaCl. The active layer was again juxtaposed with phase (‘). the solution
0; constant
phase (3_
salt concentration.
The volume
flow was directed
from phase (‘) to
TRANSP0R-f
o+c,-,
COEFFICIENTS OF ASYMMETRiC CA MEMBRANES
a05
Fig. Il. nupectiwly.
am
. an
a20 c;
75
.~;)ccn]
Measured mechanical pcrmcability II, as il function of salt concentration C. or Thcrcby, difTerent boundary condltioss serc chosen with curves I, II. and III.
Fig. II!. Diagrammatic presentation of the concentration layer of an asymmetric cellulose acetate membrane.
profile within the porous sub-
The corresponding experimental results of these measurements are represented graphically in Fig. 11_ As can be seen from this figure, the experimentally determined values off, depend strongly on the salt concentration of that solution which is adjacent to the porous surface of the asymmetric cellulose acetate membrane. It should be already pointed out here that the fact that the measured value of I, depends on the direction of the volume flux and in an asymmetric way on the salt concentrations c; and c; indicates that one is far off from the linear range of the relations of thermodynamics of irreversible processes. Within the linear range there should be no asymmetric behaviour of any kind of membrane. The experimental results demonstrate that the transport coefficients of an
76
W. PUSCH
asymmetric cellulose acetate membrane measured far off the linear range depend strongly on the salt concentrations of both solutions, c: and cz, and on the direction of the volume flow. This statement agrees with conclusions drawn by Jagura double-layer membrane. Here, Grodzinski and Kedem (7) treatin g theoretically this is a coilsequence of the asymmetric structure of the modified cellulose acetate membrane. With a volume flow through the membrane, a concentration profile within the porous sublayer is developed by the interaction of salt rejection at the active layer, volume few, and diffusion of salt within the porous sublayer. Therefore, a concentration profile develops as is shown in Fig. 12. Thereby. an osmotic pressure ditTerence An’” across the active layer is developed. This osmotic pressure difference is responsible for the fluxes through the active layer and thus, for the fluxes through the dniire membrane if one assumes that the reflection coefficient of the porous sublayer is zero or nearly zero. Furthermore, this concentration profile is also responsible for the strong concentration dependence of the transport coefficients within the linear ran_ge where no asymmetric transport behaviour should occur. This will _be deq~onstrated in the following sections. THEORFTICAL MECHANICAL
TREATMiST PERMEABILITY
OF THE IS
INFLUEhCE
DIALYSIS
OF
THE
EkPERllrfENTS
POROUS
SEAR
SUBLAYER
OS
THE
EQUILIBRIUM
Consider a membrane which is composed of two single membrane5 (Fig. 13). One of these is assumed to be a homogeneous cellulose acetate membrane (active
layer) of a thickness of about 2000 A. The other single membrane is assumed to be porous (sublayer) with pore radii r > 100 A and a thickness of 6 ‘u 100 ilrn. This double layer model was proposed by lrlerten CI al.(8) as a consequence of electron microscopic investigations. With this model it should be possible to estimate the influence of the concentration by solving the corresponding
profile on the transport
properties
of the membrane
differential equation for the mass transport in the porous sublayer. Contrary to the treatment of Kedem and Jagur-Grodzinski (7) this kind of treatment uses a membrane model for the porous sublayer of an asymmetric membrane. Therefore, it leads to a deeper insight into the physicochemical reasons for the strong dependence of the transport coefficients on the
solute concentration in that solution which is adjacent to the porous surface of the membrane. Now, the following assumptions are made: a) The transport properties of the active layer can be described using the linear relations of thermodynamics of irreversible processes. The following relationships are used. 4 = f;(APm - ~‘Al-i”)
(21)
z = --*l;APm
(22)
b) The reflection
+ i:Al-Icoefficient
of the porous
sublayer
is assumed
to be zero
TRANSPORT
tn.-=
COEFFICIENTS
0). Therefore,
the following
and pressure difference, q =
OF r\SYMAIEfRIC
CA MEMRRASFS
77
linear relationship
between
volume
tluu, q,
AP”. exists:
l;AP”
(73)
The chemical flu\, x. across the porous sublayer can be determined by the salt and water iluu using Eqs. (I ) and (2). c) The main influence of the porous sublaqcr IS on the salt flux and thereby, on the concentration profile within the supporting layer. Due to this concentration gradient the effective osmotic difference All’” \bhich determines the transport behaviour of the active layer dewates from the osmotic differcncc Ail = II’ - II” across the entire membrane. With the simplifiea Nernst-Planck Equation the following relation for the salt flux is obtained:
a, = - D,(dcJd.x) In the stationary Therefore.
+ c,q
(W
state the salt flux,
one can integrate
9,.
through
is constant
the entire membrane.
boundary condi:ions: c, = c; at s = 5 and cr = CT at x = 0 (d = thickness of porous sublayer). Here, the origin (s = 0) IS chosen to be at the interface active layer/porous sublayer. With these boundary conditions, one arrives at the following relation: the diKerentiA
r; - q - c’:q - cxp( -+5/D,)
= aI{ I -
equation
eup( -q&/D,))
if qS/D, < I one can use a series expansion termiwtmg after one teml, obtaining: c;q(l
-
y&/D,)
After some rearrangement
function,
= ‘Iq./S;D,
(26)
as well as adding and subtracting
AC; = AC, ;c’:qSID,
(23
of the exponential
Now.
cyq -
with the following
-
@,S/D,
c: one arrives at: (27)
where AC: = c: - c,”
AC, = cl - c;
and
If one multiplies this relation by RT - fo, taking coefficient fO. the following relation is obtained: All”’ = AI-I + Wc&D,
a mean kaluc for the osmotic
- RT/,4,6/D,
As can be seen from Eq. (28) the decisive osmotic volume flux, 4, and salt flux, QD,. This relationship
(W
diff’erence Allm depends on the can be rearranged to yield:
Al-I”’ = Al-l + i-I”cj/D,(q - a&‘:) Now,
using the approximations
relationship
9 N V,,.Q, N @,/c,
(2%
and cl ‘v c; the following
is obtained:
Al-I” = Al-I - (i-I”d/D,) - z
(30)
78
W. PUSCH
combines
If one
Eqs. (21), (22), and (23), after
a short
calculation
one
arrives at: q =
I;(AP
-
afAIl”)
%=
--a'l>P +
(31)
r>rr
(32)
where ii = 16 - r;lcf; if one
+ r;,
now substitutes,
1~ = -a’l,AP
and
1: = 1: -
&J’/(r;
All’”
in Eq. (32) and
+ 1;)
solves
for
z, one
arrives
+ I,ALI
at: (33)
where
in = Q(l Substituting after some
+ !:lT’~i/D,)
(35)
also Allm into Eq. (31) and taking into account Eq. (33), one obtains algebraic rearrangements the following relationship:
4 = (1 + (TI”&D,,(/;jf;
-
a”)1;1\
- l,AP
-
a’$ATl
(36)
Above all, the relationships (33) and (36) show that all the transport coefficients of the composite membrane depend on the salt concentration c; (TI”) of that solution which is adjacent to the porous surface of the membrane_ Furthermore, one can see that Onsager relations are again obtained. To begin with the discussion of the Eqs. (33) and (36), two special cases should be considered by which the important effects of the concentration gradient within the porous sublayer are already shown. This are the two special cases with G’ = I and G’ = 0. if 6’ = I. the following relationship between the three transport coefficients I;, /:, and I.& holds (6) as was already mentioned: 1; = 1; =
-r;,
(37)
Using this relationship, and (36) if the discussion y = !,,(AP
z= where
-c$,AP
-
the following is extended
a’ATI) + 1,Ai-I
equations result from to cases with tr’ = 1:
the Eqs.
(33),
(34),
(38) (3%
TRAMPORT
i,
COEFFICIENTS
OF ,\SYXlXlETRIC
79
C.4 MEMI3RANES
z I,
(41)
As can be seen from these equations,
sublayer
influences
esperimental
only
determtnation
the concentration
the mechanical
and
of the reflection
osmotic
coctlicient
gradient within the porous
permeability
the reflection coefficient CT’of the active layer of the composite
already
reported
recently
(9, IO). On the other
hand.
whereas
an
by the use of Eq. (38) yields membrane
the experimentally
as was
determined
mechanical and osmotic permeability depend strongly cn the salt concentration c; (II”). With the help of the rciationships (JO) and (41) it is possible to estimate permesbilities if D, is determined by adjustment to one point of the experimentally dctcrmined
curtes IJL;) and /,(c,).
Second,
if 0’ = 0, the t\vo follctwing
equation%
are obtninxt:
@P
(-2)
1 = 1,Al-l
(43)
q =
and
where I, = /;/(I
Here.
there
+
rg-l”C5,D,)
is no effect
of the concentration
q-W
gradient
within
the porous
sublayer
on the mechanical permeability of the membrane. On the other hand, the apparent influence of the salt concentration c: on the osmotic permeability is accidental. and is only due to the special model used for the porous sublayer. If one considers two ditkrent membranes with the dttferent osmotic permeabilitics /z and c. the corresponding composite membrane would have the osmotic permeability I, = /:I~/(!~ + fc)_ Thus. \vith the model used for the porous sublayer the osmotic already permeability of this layer is 11 = D,!ll”h. This result could be obtained 0 relations and could habe been used as the with Eq. (24) and the correspondin, starting
point of the discussion. The same results would have been obtained. Now, in the zenera case of any reflection coefhcient, G’. (0 c 0’ -C 1) there is only a change in the mechanical permeability coefficient of Eq. (36) vvherens all the other coefftcients depend on the concentration c; (IT-) in the same manner ;1s in the case o’ = 1. The multiplier [I -i (ll”6/DJ(1;/1; - tY’)l;j lowers the etfect of the concentration gradient on the mechanical permeability of the membrane by n factor which depends on the multiplier (Ii/l; - a’“). The magnitude of this multiplier is only determined by th e transport coeffkients of the active layer of the membrane. Thus, for membranes with G’ -C : the ratio of the osmotic permeability at very low salt concentrattons to the osmotic permeability at high salt concentrations (f,/i,) should always be larger than the corresponding ratio of the measured mechanical permeability for pure water to the measured mechanical permeability for a salt solution. Thts conclusion is in agreement with the experimental findings, and the correspondin, - ratio is of the order of 3 for the asymmetric cellulose acetate membrane used and a 0.5 III NaCl solution.
80
IV. On the
also
the
other
hand,
determination
the multiplier of
the
reflection
: I + (Il”d,‘D,) coefficient.
- (l~,//~ -
As can (AP!AIl),,=,
CT”)/;)
be eastly
PUSCH
influences seen
from
Eq. (36) the measured reflection coetlicient (5 = is no longer equnl to the rcflectioti coefficient of the active layer, 0’. with the boundary condition y = 0 Eq. (36) leads to the following relation betlveen CJr\nd a’:
This qundrstic
equation
for o’ can be solved to yield:
With this equation it IS possible to calculstc t’ as ;1 function of a using different parameter values. Some typical results nre graphicall) represented in Fig. 13. P\S
I
a’
0
02
06
06
08
IO
Fig. 13. Calculated reflection coelficlent 6’ of the active laler as a functwn of the mcawrcd rclkction coetficirnt ci using Eq. (46) \\ith D. = 3 - IO-B cm2kc: A .= IO 1 cm; /,“I,,’ -: I: IQ’ --: 2. 10-j cm,sec at: /I’ ;- 0.475 3tm (curre I) corresponding lo c,’ - 0.01 nl NaCl; II’ =. 4.56 atm tcune II) corresponding to c,- -: 0.01 III NaCl, If- := 45.P .ttm (tune 111) corresponding to c.” =2 IOmNaCI.
0
Oi
0.4
0.6
08
Fig. 14. Calculated
10
12
IL
16
18
20 1:/i;
nflcrtion coeffk5ent Q’ of the active layer as a funcuon of the ratio /,‘//,,’ using Eq. (46). a = 0.4, and all the other coeficicnts as with Fig. 13. The calculated curves are somewhat unrralistic smce the measured reflection coetficicnt CI will change Gth l,‘/f,’ changing.
TRANSPORT
COiiFFICIESTS
OF AS\‘MMF’rRIC
CA MEMRRANFS
Sl
can be seen from this figure, thcrc cvist only large differences between (T’ and G at larger salt concentrations cz (cy > 0.2 UJ NaCI. for instance). Thus, with small concentrations one measures nearly the reflection coefficient. G‘, of the active layer of the nsvmmetric cellulose acetate membrdnc. This theoretical result is important with regard to a recent publication (II) on the relation between salt rejection and reflection coefticicnt of asymmetric cellulose acetate membranes. Furthermore. in Fig. IJ the dependence of (T’on the ratio /i//i is graphically sho\\n for a chosen value of ci = 0 -I. 411 the curves originate at G’ = 0.4 since /:;I; has al\\a>s to he larger or equal to G” for theoretical reasons(positive definite form of the entropy production (I)). As can be seen from this graph, the effect off;!/; on the dltkrcncc bct\\tcn measured rektion coefficient. G, and reflection cocfiiclent of the active laber. IT’. is also onI> pronounced at Ltrgcr salt concentrations cz. it should be mentioned that for G’ = 1 the mtxsurcd reflection coefficient, G, is also equ.ll to one This follows from Eq. (45) slncc’ as \\as alrtxdy mentioned for r~’ = 1 the ratio ix:/; has to bc enc. too (6). Thus. this theoretical result is in qrcemcnt u Ith the discussed results in the spcclal case r/ = I. The forqoinc analysis can also be made If the volume flux. 9. is reversed from phase (‘1 to phase i’). In this cae again a concentration profile xbithin the porous sublayer is built up which leads to a larger salt concentration at the interface between the actke
layer and the porous
sublaler
and therefore. creating again an osmotic pressure diffcrenct across the actw layer which opposes the original net driving force AP and;or All reducing the measured volume flux. The \\hole analysis sho\\s that the osmotic pressure difference. developed across the active layer of the mcmbranc. depends. in addition to. the \olumc tlux essentially on the concentration c: of that solution which is adjacent to the porous surface of the asymmetric membrane. The theoretical results allo\\ the folio\! ing conclusions. As Ions as c; and c.1 rare equal or nearly equal the created osmotic pressure difkcnce. 4II’“. \xill be nearly independent on the orientation of the membrane. Thus, the same volume tlux fill be observed if the membrane is turned over. On the other hand. as soon as the diffcrcncc in the tx\o salt concrntratlons. C: and c:. starts to become large enough different values of the volume flux will be measured depending on the orientation of the membrane as \vas demons!rated by the corresponding reported measurements (Fig. 1 I). Therefore. one has to differentiate between three experimental situations. In the first case. no asymmetric behaviour of the membrane system wll be observed. The system is considered to obey the restrictions of the linear range of the thermodynamics of irreversible processes. This will always be true for small concentration differences xross the entire membrane and not too large volume fluxes. The asymmetric structure of the membrane is manifested by the strong concentration dependence of the corresponding transport coefficients. In the second case, one \vill
W.
82
PIJSCH
also find linear relationships to hold between fluxes and forces but the measured fluxes and transport coefiicienrs skill depend upon the orientation This will mostly be true \tith larger concentration differences
of the membrane. but small volume
fiu\cs. One may call this regime of concentrations and fluxes the linear asymmetric regime. In the wo foregoing cases the transport behaviour of the asymmetric cellulose acetate rncrnbrane can be described by the Eqs. (33) to (36) which are symmetric in (I but asymmetric with regard to II’ and ll-. Finally, in the third c.lse. no linear relarionship between fluxes and driving forces will be observed In general and the system will exhibit highly asymmetric beha\ iour. This corresponds to the general case of large fluxes and concentration differences as is well known
from hyperfiltration There
is one
experiments
with asymmetric
point
should
which
membranes
be examined
(5, I).
in somewhat
more detail. With all permeability measurements reported in this paper the asymmerric membranes x\ere supported b_: a porous metal plate. Since concentration polarization in the porous support wili never really be overcome by stirring. the question arises whether the observed effects may be partly or even entirely caused by the porous support instead of beins due mainly to the porous sublayer of the asymmetric membrane. Thus. to make sure that the main reason for the concentration dependence of the fluxes as well as for the asymmetric behaviour is the porous sublayer suitable measurements without any support should be performed. This was done by measuring membrane potentials. As \\as reported recently (I.3), the corresponding experimental findings manifest that the main reason for the concentration dependence and the asymmetric behaviour out of the linear range of the thermodynamics of irreversible processes is a concentration gradient within the porous sublayer of the asymmetric cellulose acetate membrane. The author is indebted to Professor R. Schliigl for his interest in this work and some stimulating discussions. Furthermore, the author \\ould like to thank Mrs. R. Lachmann and Mrs. U. Schaffner for carrying out the time-consuming and difficult measurements and Mr. R. Griipl for his help in designing and producing
the dialysis
author
is also indebted
the
proofs. Forschung
cell as \vell
as for
to Dr. Harold
The work was financially und Technologie”, Bonn.
preparing
Hopfenberg supported Germany.
modified membranes. The and Dr. Peter Bo for reading
the
by the
“Bundesministerium
ftir
REFERESCFS
1.
1. 3. 4. 5. 6.
R. SCHL~~L. Srofiransporl durclr_~lrmbranen.Dr. Dietnch-Steinkopff-Verlag. Damstadt, 1964. A. J. STAVERMA~, REC. Trar. Clrinl. Pup-Bas, 70 (1951) 344. W. PCISCH. Clre.?;ie-IIISeni~,rtr-TEc/miX-,20 (1973) 1216. W. PLXH ADD D. W'OERMAX~. Ser. Blcnsen,Cer.plr?Jili. C/tern., 74(1970)444. R. CiRijPL AVV W. Puscti. Desdinatiun, 8 (1970) 177. N. LAKSUWSARAYASAIAH. Tmmport Phenomena in Xfwdmmes. Academic Press, NC\\ York, N-Y., 1969, p. 314.
TRANSPORT
COEFFICIENTS OF xxhfsmmc
CA
MEMBU~
53
7. J. JAGUR-GRODZINSKI ASD 0. KEDEM, Dedinarion, 1 (1966) 327. 1 (1966) 30. 5. R. L. RILEY, U. MERTES ASD J. 0. GARDNER. Des&u&m. 9. W. PUSCH, Proc. Fourth Intern. Synrp. on Fresh Water fiionr the Sea, Heidelberg, September 9-14. 1973.4 (1973) 321-332. 10. W. WSCH. Proc. Intern. Synp. on Strrtctwe of Nirter and Ayueom Sohrtions, Afurbrtrg, July 19-29, 1973, W. Luck, editor. Verlag Chemie/Physik Vcrlag. Weinheim. 1974. pp. 541562. 11. W. PUSCH ASD R. RII_EY, Desolinutiotr. 14 (1974) 389. 12’. W. BASKS ABD A. SHARPLW. The Mechanism of Desalination by Reverse Osmosis, and Its Relation to Membrane Structure, O@ce of Saline Wafer. Res. Dewlop. Progr. Rcpt. No. 143. June 1965. i 3. W. Puscn. Proc. Nuto Athunccrl Stut& Inst. ~Poiyelecrrofytes II”. Forges-fcs-Euux, 1973, E. Sttiox~ ed., Reidcl Pubhshmg Co.. Dordrecht (in preparation).
TRANSPORT
Company.
COEFFICIENTS
Amsterdam
- Printed in The Netherlands
OF ASYMMETRIC
CELLULOSE
ACETATE
MEMBRANES*
Using the linear relations of thermodynamics
of irreversible processes,
the
transport
coefficients I,,. i,, f,,. and 0 were measured for an asymmetric cellulose acetate membrane with NaCI, Na,SO,. CaCI,. NaF, and snccharosc over the concentration range (O-O.5 mol,A or 1 mol!i) at 2OT or 23’C. The experimental findings manifest a strong dependence of the three transport coeflicients IP, f,, and I,, on solute concentration. This strong dependence on concentration can be attributed to a concentration gradient within the porous sublayer of the asymmetric membrane. Thereby, it is sho\\n that the transport coelkients of an asymmetric membrane depend on the solute concentration on both sides of the membrane instead of the mean concentration C:,, as would be the case for a homogeneous membrane. SYMBOLS
-
-
effective membrane area (cm’) local salt concentration within porous sublnyer (mol/cm’) salt concentration (mol/l) of phase (‘) or (‘). respectively salt concentration at the interface between actike layer and porous sublaycr (mol/cm3 or mol/l) mean salt concentration (mol/l) = (I/Z)(c: + c;) water concentration (mol/cm’) concentration difference across entire membrane = c: - c;’ concentration difference across active layer = c; - c; diffusion coefficient of salt within porous sublayer (cm’/s&) osmotic coefficient = Allrc3,~AlIidrJ, mechanical permeability of entire membrane (cm/set - at) mechanical permeability>f active layer (cm,kec - at)
-__l Paper presented in sections at the Fourth Intern. Symp. on Fresh Water from the Sea. Heidelber& September 9-14, 1973.
66
w. PUSCH .
,, 6
I;
-
1,
-
mechanical permeability of porous sublayer (cm/see . at) intrinsic mechanical permeability of entire membrane (cm/set osmotic permeability of entire membrane (cm/set . at) osmotic permeability of active layer (cmjsec - at)
-
osmotic intrinsic
-
coupling coupling
cocfhcient coehicient
-
pressure pressure pressure
(at) of phase (‘) or (“I. respectively difference across entire membrane difference across active layer (at)
-
pressure difference across porous sublayer (at) = Pm - P’ pressure at the interface between active layer and porous sublayer (at) volume flux (cm/set) mean volume tIuu (cm,kec) = AC’;,4 - Ar volume flow (cmj/sec) = A .q volume (cm3) of phase (‘) or (‘), respectively partial molar volume of water (cm’/mol) volume change during time interval t, to r2 (cm’) = V’(f2) - V’(r,) =
1: ,, ‘* 1:
-
r; I “P I’ f? P” AP APrn AP” Pm 4 14
Q
-
V’, V’
-
VW AV
-
I
-
Al s s
-
I7’. II”
-
l-t+=
-
AI-I hi-l”
- (V”(r,) time (set) time interval
of entire of active
membrane (cmlsec - at) = /,, layer (cm/kc - at) = 0’ - /L (at) = P’ = P’ - Pm
P”
V”(Q)
-
perpendicular to membrane surface (cm) of porous sublayer pressure (at) of phase (‘) or (“), respectively pressure at the interface between active layer (at) pressure difference across entire membrane (at)
=
II’ -
-
osmotic
pressure
= ll’
-
IIm
An”
-
osmotic
pressure
difference
(at)
=
AI-I, Al-r, __._ AI-I
-
osmotic osmotic
pressure pressure
difference difference
-
mean
-
(l/z) (AK + AK) reflection coefficient
@S a, % %‘*%” f
-
osmotic
=
+ z;>
coordinate thickness osmotic osmotic sublayer osmotic
Cr t?‘, c*
-
sublayer (cm/kc . at) of entire membrane (cm,‘sec . at)
permeability of porous osmotic permeability
(/;,)‘/(r;
* at)
difference
pressure
across across
active porous
layer
(at)
sublayer
and
porous
nm
-
II” l-l”
(at) at the beginning of a dialysis experiment (at) at the end of a dialysis experiment
difference
of entire
over time interval
membrane
of measurement ;
reflection coefficient of active layer or porous s_ublXyer, respectively salt flux across the entire membrane (molkm’sec) water flux (moljcm’sec) chemical fiuu (cmjsec) = O,!pS - O,~L;,. chemical flux (cm/set) in phase (‘) or (“), respectively mean chemical flux:(cm/sec) = (l/2)(%’ + x”)
=
TRANSPORT
COEFFICIENTS
OF ASYMMETRIC
%;,x” -
mean value of chemical or (‘). respectively
CA hfEhlERANES
flux over time interval
67
I, -
r2 for phase (‘)
INTRODtiCTIOS
Given an isothermic system, with a membrane separating two salt solutions of different salt concentrations c: and cz (Fig. I) which are kept under different hydrostatic pressures P’ and P’, one can use the linear relaticns of thermodynamics of irreversible processes to describe the transport processes through the membrane. membrane
salt
solution
salt
solution movable piston
Fig. 1. Diagrammatic
presentation
of an isothermal
membrane system.
There exist, as Schlapl (I) has shown, the following linear relations between fluxes and conjugated forces for a system near equilibrium: q = I$P l = l,,AP
+ l,$rI
(1)
+ 1,Al-l
(2)
Thereby. the Onsager relation I_, = I,, was used. The chemical flux z, a measure of the variation of concentration with time within the two phases (‘) and (“), is defined by: (3) With the definition man (2), Eq. (I) yields: q = l,(AP
of the reflection
- 0Al-T)
coefficient
u E -&,/I,
given by StaveriW
Using these linear relations the transport coefficients I,, f,, I,, and u can be determined by measuring the dependence of volume flux 4 and chemical !Iux x on pressure difference AP and osmotic difference All. DESCRIPTIOS
OF EQUIPMENT
With modified equipment (3, 4), shown in Fig. 2, all transport coefficients can be determined using different experimental setups. With two conductivity
EK
/
FIN 2. Cross-sectional
view of ddys~s
ceil.
Fig. 3. Partition wall with membrane holder.
cells (L) immersed in the two solutions, the variatron of salt concentration with time is followed by measuring the conductivity by means of a WayneKerr-Bridge with an accuracy of 0.1 :i_ The volume flux q is measured by the increase of volume AC’ of the corresponding phase using a calibrated capillary (K, 1~1000 ml) or determining the volume of solution passing through a small capillary into an Erlenmeyer flask by weighing. To reduce evaporation the Erlenmeyer flask is sealed by a silicon-rubber stopper (St) containing a perforation for the capillary.
TRhSSl’ORf
Pressure
COEFFICIENTS
equalization
OF ,\SYblW3-RlC
is attnincd
wall (MT) contains the membrane is supported by 3 CONIDUR
CA MEAlBRAbif23
by a canula (M) which stainless
through
69 the stopper.
The
partition
ic &cd cccentricahy. The membrane screen (Hein. Lehmann & Co.,
steel
Diisseldorf, Germany). The screen is 0.35 mm thick and has holes of 0.15 mm diameter. Wtth different partition v\alls, membranes with an effective area of holder is shown -I cm’, h-4 cm’. and - I6 cm’ can be used. The membrane sep;mrcly in Fig. 3. The solutions arc vvoll stirred by means of muanetic stirrers. All surfaces of contact arc sealed by O-rings. Air is used for pressurizing one phase. The air pressure is mointnined constant to +_5 - lo-’ at by means of KEN!\IDAL valves (VIA, Diissctdorf. Germany). The ptcssurc is Ineasured by high accuracy pressure ~iuges ( 2 0. I 0 “) produced by WIKA (Klingcnbcrg am Main. Germany). CELLUI USE ACETr\TE MIXUKhSE
For
most
cured at Y2.5.C. The salt rejection
measurements.
asymmetric
ccllulosc
acetate
membranes
The preparation of these membranes is described of the membranes is about 97 ‘,‘i at 30 at.
The four transport coetlicients - three of which measured u ith the following three dit?krent experimental
are used
elsewhere
are independent setups.
(5).
- \\cre
n)
Deternlittaticw qf I, ut An = 0 Both cell compartments--phase (‘) and (‘)-contain salt solutrons of equal concentration. Measuring the volume flow Q = .+I . q. as a function of AP using four dnhxcnt values of AP. one gets I,, from the slope of the straight line in a Q-AP-diagram if the effective membrane arm is known.
An osmotic difference AIl betvvcen the two cell compartments is established (c: > l-z) and the volume flow Q is measured as a function of AP. From the slope of the straight line in a Q-AP-diagram, one again gets /,,. In addition, from the intersection of the straight line with the abscissa. one gets s = (AP/ATl) at y = 0. Thereby. a mean value All of AIl is determined from AD, at the beginning and AD, at the end of the measurement (arithmetical mean value). The osmotic difference AD, is determined by conductivity titration of the t\vo salt solutions after the experiment is finished. c)
Dctern~imliatt
of I, mid lyp at AP = 0
At t = 0 (t = time) a concentration partments is set up (cc: > c;). The variation
difference between of concentration
the t\vo cell comwith time is then
IV. PUSCH
0
02
Fig. 4.
cl&
Mean
on
0.6
\olumc
ID
an[aml
flux q and mean
chemical
fh.t~ j as P function
of man
osmotic
pressure
difference hII at 25’C using a mean salt concentration C, = 0.1 m NaCl (c,’ = 0.11 m NaCl: c,- = 0.03 nr NaCI) and an as_vmmetricmembrane annealed at 82.5 ‘C (membrane area A :- 3.14cm’). ‘I
followed by measuring the conductivity of both phases at a time interval of about 24 hours. After the measurements are completed (after about ten days) the cell constants of the used conductivity cells are determined once more by conductivity titrations of the salt solutions from both compartments. Assuming a linear relation between conductivity and salt concentration for small concentration differences, one gets the salt concentration from a calibration curve. The volume flow Q is received by determining the volume of solution which flowed into an Erlenmeyer flask during about 24 hours. From the curves of concentrations c:, r,“, and volume increase A V with time a mean value of the chemical flux, 7, and the volume flux, y, as well as a mean value All are estimated by averaging over 24 hours. The mean values are obtained from the measured values for the time interval t, to t2 as follows: Al-I(f,) = l-I’&) - l-I”(t,)
(4)
An(r,)
= I-Of,)
(5)
an
= tCAWrl>
_, %
A;-
-2
= -+-g
- II” + AWt,)l
’ CcXtJ- c:O,)l/At I [c;‘(t,) I
-
4’(t,)J/At
(6)
(7) @I
TRANSPORT
COEFFIClENTS
OF ASYhlhlFTRlC
l7
= AVIA - At
At
= I, -
c:,
= $. cc:u,)
CA BIEhlBRANES
71 (9)
1,
(10) + c:Cr,) + cI(rz)
+ c3f,)]
(11)
as functions of All, one gets straight lines (Fig. 4). The slopes Plotting j * and of the calculated regression lines yield the coefficients f, and &, respectively. If I, is known from a different measurement, one gets once more G = -f&l,, RESULTS
NaF-, CaCI,-, and saccharose-solutions the Using NaCI-, Na2S0,-, mechanical permeabiiity IP was measured as a function of solute concentration (0 ,( c, < 1 moljl) at 20°C. Using NaCI-, Na2SOJ-, NaF-, and CaCl,-solutions the reflection coefficient r~ was also measured. The results are graphically represented in Figs. 5. 6. and 7. In addition, I, and I,, were determined as a function of concentration at 25’C using NaCI- and Na,SO.-solutions. The experimental findings are shown graphically in Figs. S and 9. As one can see from these figures, there exists a strong dependence of I,. I,, and I_ on the salt concentrations c, or (:,, respectively. On the other hand, the reflection coetlicient 0 varies only sliatly with salt concentration. If one calculates u from /,, and I,,, (CT = -/,,/I,) measured by methods (a) and (c), respectively, it is found that these values agree with the Uvalues measured directly using method (b). If o = 1, there exists the following relation betivecn I,,. I,. and In,, (6): I/,,( = [/,I = I/J. As can be seen from Fig. 10, that relation is satisfied for Na,SO, solutions since q - j and (T = 1.
OL
a2
0
04
0.6
0.6
l.0
F,[Md/l]
Fig. 5. Reflection coefficient fl as a function of mean salt concentration & at 20‘C using solutions of different salts (NaCl-(I) curve measured Hith a virgin membrane; NaCI-(II) curve measured with a membrane which has already been used in hypextiltration at 50 atm).
l
j
=
1/‘(;7’
_I
2”).
W. PUSCH
at 20X
0
Fig. 6. Mechanical pcnncabiltty I,, as a function unng solutions of different salts.
0
al
as
u
ln
of salt concentrationC,
=.=
;a k“
=- c.3
w GWll
Fig. 7. Mechanical pernwabiiity I,, as a function of solute concentration c, at 3 -C using saccharose solutions With regard to the concentration dependence of I, in this case, the increase of the viscosity of saccharose solutions with increasing solute concentration has to be taken into a02ount.
The strong dependence concentration,
as manifested
rezxons: I. Concentration
of the three transport coeficients, experimentally,
polarization
to the membrane surfaces.
Gthin
I,,, lx, lx,,, on salt
may occur for the following
an unstirred boundary
three
layer adjacent
TRANSPORT
COEFFICIEKI’S
OF ASYMMETRIC
Fig. 8. Osmotic permcabihty solutions.
CA LlEMBRASES
73
I, as a function of mean salt concentration
F., at 3 ‘C using
I,,, as a function of mean sdt concentration
C; at 25 ‘C using
NaCl and NadOt
Fig. 9. Coupling coeffkicnt NaCl and Na:SOa solutions.
2. Concentration profiles (gradients) within the porous substructure of the asymmetric cetlulose acetate membrane as a consequence of counterbalance of convective flow and <;llt diffusion. 3. Swelling effects of the membrane material due to the influence of salt. TO verify the influence of the unstirred boundary layer over the concentration dependence of the transport coefficients, a second dialysis cell with very effective circulation pumps was use 1. With this modified equipment, the same experimental findings were obtained. This indicates that unstirred boundary layers are not the main reason for the strong concentration dependence of the transport coefficients.
74
W. PUSCH
Fig. 10 Mean bolumr flux q and mean chemial flux 2 as a function of mean osmotic pressure diffcrcnce ifi? ar 25 C using Na=SOa solutions of a m-n salt concentration 2. :: 0.2 1~ NazS0~ (cr’ :-: 0.12 ,n NatSOa: c,* = 0.18 ))I NarSOd: A -= 3.14 cm’). In this special case the wo straight lines ij and 2 as a function of IJZ coincide since the retktion coefficient I; - 1.
On the other hand, if swelling effects are responsible
for the strong concentration dependence of the transport coefficients, one would expect the same experimental results with regard to the concentration dependence with hyperfiltration experiments. But looking at the literature and taking into account our own unpublished results of hypefiltration measurements, it appears that the mechanical permeability of asymmetric cellulose acetate membranes. for instance, does not depend strongly on the salt concentration of the brine. Now, the reason for this difference can be found in the different boundary conditions. Whereas, in hypefiltration
the boundary
condition
of free outflow exists at the low pressure
side, in dialysis experiments the salt concentration on both sides of the membrane is maintained constant. Therefore, one should check the influence of different boundary conditions in dialysis measurements on the value of the mechanical permeability, 1,. To investigate that, measurements with the following boundary conditions have been made. 1. The salt concentration c; was varied between 0 and 0.2 nr NaCl keeping c,” = 0 (pure water in the corresponding cell compartment). Thereby, the active layer of the membrane was juxtaposed with the salt solution_ The volume flow was directed from phase (‘) to phase (“) by adjusting the pressure difference, AP, across the membrane. 2. The same arrangement as in case 1 but the membrane was turned over so that the active layer was juxtaposed with pure water and thus the porous layer was juxtaposed with the salt solution. Thereby, the volume flow, Q, was directed from phase (‘) to phase (‘). 3. The salt concentration c; was varied between 0 and 0.2 IPZNaCl keeping c’ = 0.2 m NaCl. The active layer was again juxtaposed with phase (‘). the solution
0; constant
phase (3_
salt concentration.
The volume
flow was directed
from phase (‘) to
TRANSP0R-f
o+c,-,
COEFFICIENTS OF ASYMMETRiC CA MEMBRANES
a05
Fig. Il. nupectiwly.
am
. an
a20 c;
75
.~;)ccn]
Measured mechanical pcrmcability II, as il function of salt concentration C. or Thcrcby, difTerent boundary condltioss serc chosen with curves I, II. and III.
Fig. II!. Diagrammatic presentation of the concentration layer of an asymmetric cellulose acetate membrane.
profile within the porous sub-
The corresponding experimental results of these measurements are represented graphically in Fig. 11_ As can be seen from this figure, the experimentally determined values off, depend strongly on the salt concentration of that solution which is adjacent to the porous surface of the asymmetric cellulose acetate membrane. It should be already pointed out here that the fact that the measured value of I, depends on the direction of the volume flux and in an asymmetric way on the salt concentrations c; and c; indicates that one is far off from the linear range of the relations of thermodynamics of irreversible processes. Within the linear range there should be no asymmetric behaviour of any kind of membrane. The experimental results demonstrate that the transport coefficients of an
76
W. PUSCH
asymmetric cellulose acetate membrane measured far off the linear range depend strongly on the salt concentrations of both solutions, c: and cz, and on the direction of the volume flow. This statement agrees with conclusions drawn by Jagura double-layer membrane. Here, Grodzinski and Kedem (7) treatin g theoretically this is a coilsequence of the asymmetric structure of the modified cellulose acetate membrane. With a volume flow through the membrane, a concentration profile within the porous sublayer is developed by the interaction of salt rejection at the active layer, volume few, and diffusion of salt within the porous sublayer. Therefore, a concentration profile develops as is shown in Fig. 12. Thereby. an osmotic pressure ditTerence An’” across the active layer is developed. This osmotic pressure difference is responsible for the fluxes through the active layer and thus, for the fluxes through the dniire membrane if one assumes that the reflection coefficient of the porous sublayer is zero or nearly zero. Furthermore, this concentration profile is also responsible for the strong concentration dependence of the transport coefficients within the linear ran_ge where no asymmetric transport behaviour should occur. This will _be deq~onstrated in the following sections. THEORFTICAL MECHANICAL
TREATMiST PERMEABILITY
OF THE IS
INFLUEhCE
DIALYSIS
OF
THE
EkPERllrfENTS
POROUS
SEAR
SUBLAYER
OS
THE
EQUILIBRIUM
Consider a membrane which is composed of two single membrane5 (Fig. 13). One of these is assumed to be a homogeneous cellulose acetate membrane (active
layer) of a thickness of about 2000 A. The other single membrane is assumed to be porous (sublayer) with pore radii r > 100 A and a thickness of 6 ‘u 100 ilrn. This double layer model was proposed by lrlerten CI al.(8) as a consequence of electron microscopic investigations. With this model it should be possible to estimate the influence of the concentration by solving the corresponding
profile on the transport
properties
of the membrane
differential equation for the mass transport in the porous sublayer. Contrary to the treatment of Kedem and Jagur-Grodzinski (7) this kind of treatment uses a membrane model for the porous sublayer of an asymmetric membrane. Therefore, it leads to a deeper insight into the physicochemical reasons for the strong dependence of the transport coefficients on the
solute concentration in that solution which is adjacent to the porous surface of the membrane. Now, the following assumptions are made: a) The transport properties of the active layer can be described using the linear relations of thermodynamics of irreversible processes. The following relationships are used. 4 = f;(APm - ~‘Al-i”)
(21)
z = --*l;APm
(22)
b) The reflection
+ i:Al-Icoefficient
of the porous
sublayer
is assumed
to be zero
TRANSPORT
tn.-=
COEFFICIENTS
0). Therefore,
the following
and pressure difference, q =
OF r\SYMAIEfRIC
CA MEMRRASFS
77
linear relationship
between
volume
tluu, q,
AP”. exists:
l;AP”
(73)
The chemical flu\, x. across the porous sublayer can be determined by the salt and water iluu using Eqs. (I ) and (2). c) The main influence of the porous sublaqcr IS on the salt flux and thereby, on the concentration profile within the supporting layer. Due to this concentration gradient the effective osmotic difference All’” \bhich determines the transport behaviour of the active layer dewates from the osmotic differcncc Ail = II’ - II” across the entire membrane. With the simplifiea Nernst-Planck Equation the following relation for the salt flux is obtained:
a, = - D,(dcJd.x) In the stationary Therefore.
+ c,q
(W
state the salt flux,
one can integrate
9,.
through
is constant
the entire membrane.
boundary condi:ions: c, = c; at s = 5 and cr = CT at x = 0 (d = thickness of porous sublayer). Here, the origin (s = 0) IS chosen to be at the interface active layer/porous sublayer. With these boundary conditions, one arrives at the following relation: the diKerentiA
r; - q - c’:q - cxp( -+5/D,)
= aI{ I -
equation
eup( -q&/D,))
if qS/D, < I one can use a series expansion termiwtmg after one teml, obtaining: c;q(l
-
y&/D,)
After some rearrangement
function,
= ‘Iq./S;D,
(26)
as well as adding and subtracting
AC; = AC, ;c’:qSID,
(23
of the exponential
Now.
cyq -
with the following
-
@,S/D,
c: one arrives at: (27)
where AC: = c: - c,”
AC, = cl - c;
and
If one multiplies this relation by RT - fo, taking coefficient fO. the following relation is obtained: All”’ = AI-I + Wc&D,
a mean kaluc for the osmotic
- RT/,4,6/D,
As can be seen from Eq. (28) the decisive osmotic volume flux, 4, and salt flux, QD,. This relationship
(W
diff’erence Allm depends on the can be rearranged to yield:
Al-I”’ = Al-l + i-I”cj/D,(q - a&‘:) Now,
using the approximations
relationship
9 N V,,.Q, N @,/c,
(2%
and cl ‘v c; the following
is obtained:
Al-I” = Al-I - (i-I”d/D,) - z
(30)
78
W. PUSCH
combines
If one
Eqs. (21), (22), and (23), after
a short
calculation
one
arrives at: q =
I;(AP
-
afAIl”)
%=
--a'l>P +
(31)
r>rr
(32)
where ii = 16 - r;lcf; if one
+ r;,
now substitutes,
1~ = -a’l,AP
and
1: = 1: -
&J’/(r;
All’”
in Eq. (32) and
+ 1;)
solves
for
z, one
arrives
+ I,ALI
at: (33)
where
in = Q(l Substituting after some
+ !:lT’~i/D,)
(35)
also Allm into Eq. (31) and taking into account Eq. (33), one obtains algebraic rearrangements the following relationship:
4 = (1 + (TI”&D,,(/;jf;
-
a”)1;1\
- l,AP
-
a’$ATl
(36)
Above all, the relationships (33) and (36) show that all the transport coefficients of the composite membrane depend on the salt concentration c; (TI”) of that solution which is adjacent to the porous surface of the membrane_ Furthermore, one can see that Onsager relations are again obtained. To begin with the discussion of the Eqs. (33) and (36), two special cases should be considered by which the important effects of the concentration gradient within the porous sublayer are already shown. This are the two special cases with G’ = I and G’ = 0. if 6’ = I. the following relationship between the three transport coefficients I;, /:, and I.& holds (6) as was already mentioned: 1; = 1; =
-r;,
(37)
Using this relationship, and (36) if the discussion y = !,,(AP
z= where
-c$,AP
-
the following is extended
a’ATI) + 1,Ai-I
equations result from to cases with tr’ = 1:
the Eqs.
(33),
(34),
(38) (3%
TRAMPORT
i,
COEFFICIENTS
OF ,\SYXlXlETRIC
79
C.4 MEMI3RANES
z I,
(41)
As can be seen from these equations,
sublayer
influences
esperimental
only
determtnation
the concentration
the mechanical
and
of the reflection
osmotic
coctlicient
gradient within the porous
permeability
the reflection coefficient CT’of the active layer of the composite
already
reported
recently
(9, IO). On the other
hand.
whereas
an
by the use of Eq. (38) yields membrane
the experimentally
as was
determined
mechanical and osmotic permeability depend strongly cn the salt concentration c; (II”). With the help of the rciationships (JO) and (41) it is possible to estimate permesbilities if D, is determined by adjustment to one point of the experimentally dctcrmined
curtes IJL;) and /,(c,).
Second,
if 0’ = 0, the t\vo follctwing
equation%
are obtninxt:
@P
(-2)
1 = 1,Al-l
(43)
q =
and
where I, = /;/(I
Here.
there
+
rg-l”C5,D,)
is no effect
of the concentration
q-W
gradient
within
the porous
sublayer
on the mechanical permeability of the membrane. On the other hand, the apparent influence of the salt concentration c: on the osmotic permeability is accidental. and is only due to the special model used for the porous sublayer. If one considers two ditkrent membranes with the dttferent osmotic permeabilitics /z and c. the corresponding composite membrane would have the osmotic permeability I, = /:I~/(!~ + fc)_ Thus. \vith the model used for the porous sublayer the osmotic already permeability of this layer is 11 = D,!ll”h. This result could be obtained 0 relations and could habe been used as the with Eq. (24) and the correspondin, starting
point of the discussion. The same results would have been obtained. Now, in the zenera case of any reflection coefhcient, G’. (0 c 0’ -C 1) there is only a change in the mechanical permeability coefficient of Eq. (36) vvherens all the other coefftcients depend on the concentration c; (IT-) in the same manner ;1s in the case o’ = 1. The multiplier [I -i (ll”6/DJ(1;/1; - tY’)l;j lowers the etfect of the concentration gradient on the mechanical permeability of the membrane by n factor which depends on the multiplier (Ii/l; - a’“). The magnitude of this multiplier is only determined by th e transport coeffkients of the active layer of the membrane. Thus, for membranes with G’ -C : the ratio of the osmotic permeability at very low salt concentrattons to the osmotic permeability at high salt concentrations (f,/i,) should always be larger than the corresponding ratio of the measured mechanical permeability for pure water to the measured mechanical permeability for a salt solution. Thts conclusion is in agreement with the experimental findings, and the correspondin, - ratio is of the order of 3 for the asymmetric cellulose acetate membrane used and a 0.5 III NaCl solution.
80
IV. On the
also
the
other
hand,
determination
the multiplier of
the
reflection
: I + (Il”d,‘D,) coefficient.
- (l~,//~ -
As can (AP!AIl),,=,
CT”)/;)
be eastly
PUSCH
influences seen
from
Eq. (36) the measured reflection coetlicient (5 = is no longer equnl to the rcflectioti coefficient of the active layer, 0’. with the boundary condition y = 0 Eq. (36) leads to the following relation betlveen CJr\nd a’:
This qundrstic
equation
for o’ can be solved to yield:
With this equation it IS possible to calculstc t’ as ;1 function of a using different parameter values. Some typical results nre graphicall) represented in Fig. 13. P\S
I
a’
0
02
06
06
08
IO
Fig. 13. Calculated reflection coelficlent 6’ of the active laler as a functwn of the mcawrcd rclkction coetficirnt ci using Eq. (46) \\ith D. = 3 - IO-B cm2kc: A .= IO 1 cm; /,“I,,’ -: I: IQ’ --: 2. 10-j cm,sec at: /I’ ;- 0.475 3tm (curre I) corresponding lo c,’ - 0.01 nl NaCl; II’ =. 4.56 atm tcune II) corresponding to c,- -: 0.01 III NaCl, If- := 45.P .ttm (tune 111) corresponding to c.” =2 IOmNaCI.
0
Oi
0.4
0.6
08
Fig. 14. Calculated
10
12
IL
16
18
20 1:/i;
nflcrtion coeffk5ent Q’ of the active layer as a funcuon of the ratio /,‘//,,’ using Eq. (46). a = 0.4, and all the other coeficicnts as with Fig. 13. The calculated curves are somewhat unrralistic smce the measured reflection coetficicnt CI will change Gth l,‘/f,’ changing.
TRANSPORT
COiiFFICIESTS
OF AS\‘MMF’rRIC
CA MEMRRANFS
Sl
can be seen from this figure, thcrc cvist only large differences between (T’ and G at larger salt concentrations cz (cy > 0.2 UJ NaCI. for instance). Thus, with small concentrations one measures nearly the reflection coefficient. G‘, of the active layer of the nsvmmetric cellulose acetate membrdnc. This theoretical result is important with regard to a recent publication (II) on the relation between salt rejection and reflection coefticicnt of asymmetric cellulose acetate membranes. Furthermore. in Fig. IJ the dependence of (T’on the ratio /i//i is graphically sho\\n for a chosen value of ci = 0 -I. 411 the curves originate at G’ = 0.4 since /:;I; has al\\a>s to he larger or equal to G” for theoretical reasons(positive definite form of the entropy production (I)). As can be seen from this graph, the effect off;!/; on the dltkrcncc bct\\tcn measured rektion coefficient. G, and reflection cocfiiclent of the active laber. IT’. is also onI> pronounced at Ltrgcr salt concentrations cz. it should be mentioned that for G’ = 1 the mtxsurcd reflection coefficient, G, is also equ.ll to one This follows from Eq. (45) slncc’ as \\as alrtxdy mentioned for r~’ = 1 the ratio ix:/; has to bc enc. too (6). Thus. this theoretical result is in qrcemcnt u Ith the discussed results in the spcclal case r/ = I. The forqoinc analysis can also be made If the volume flux. 9. is reversed from phase (‘1 to phase i’). In this cae again a concentration profile xbithin the porous sublayer is built up which leads to a larger salt concentration at the interface between the actke
layer and the porous
sublaler
and therefore. creating again an osmotic pressure diffcrenct across the actw layer which opposes the original net driving force AP and;or All reducing the measured volume flux. The \\hole analysis sho\\s that the osmotic pressure difference. developed across the active layer of the mcmbranc. depends. in addition to. the \olumc tlux essentially on the concentration c: of that solution which is adjacent to the porous surface of the asymmetric membrane. The theoretical results allo\\ the folio\! ing conclusions. As Ions as c; and c.1 rare equal or nearly equal the created osmotic pressure difkcnce. 4II’“. \xill be nearly independent on the orientation of the membrane. Thus, the same volume tlux fill be observed if the membrane is turned over. On the other hand. as soon as the diffcrcncc in the tx\o salt concrntratlons. C: and c:. starts to become large enough different values of the volume flux will be measured depending on the orientation of the membrane as \vas demons!rated by the corresponding reported measurements (Fig. 1 I). Therefore. one has to differentiate between three experimental situations. In the first case. no asymmetric behaviour of the membrane system wll be observed. The system is considered to obey the restrictions of the linear range of the thermodynamics of irreversible processes. This will always be true for small concentration differences xross the entire membrane and not too large volume fluxes. The asymmetric structure of the membrane is manifested by the strong concentration dependence of the corresponding transport coefficients. In the second case, one \vill
W.
82
PIJSCH
also find linear relationships to hold between fluxes and forces but the measured fluxes and transport coefiicienrs skill depend upon the orientation This will mostly be true \tith larger concentration differences
of the membrane. but small volume
fiu\cs. One may call this regime of concentrations and fluxes the linear asymmetric regime. In the wo foregoing cases the transport behaviour of the asymmetric cellulose acetate rncrnbrane can be described by the Eqs. (33) to (36) which are symmetric in (I but asymmetric with regard to II’ and ll-. Finally, in the third c.lse. no linear relarionship between fluxes and driving forces will be observed In general and the system will exhibit highly asymmetric beha\ iour. This corresponds to the general case of large fluxes and concentration differences as is well known
from hyperfiltration There
is one
experiments
with asymmetric
point
should
which
membranes
be examined
(5, I).
in somewhat
more detail. With all permeability measurements reported in this paper the asymmerric membranes x\ere supported b_: a porous metal plate. Since concentration polarization in the porous support wili never really be overcome by stirring. the question arises whether the observed effects may be partly or even entirely caused by the porous support instead of beins due mainly to the porous sublayer of the asymmetric membrane. Thus. to make sure that the main reason for the concentration dependence of the fluxes as well as for the asymmetric behaviour is the porous sublayer suitable measurements without any support should be performed. This was done by measuring membrane potentials. As \\as reported recently (I.3), the corresponding experimental findings manifest that the main reason for the concentration dependence and the asymmetric behaviour out of the linear range of the thermodynamics of irreversible processes is a concentration gradient within the porous sublayer of the asymmetric cellulose acetate membrane. The author is indebted to Professor R. Schliigl for his interest in this work and some stimulating discussions. Furthermore, the author \\ould like to thank Mrs. R. Lachmann and Mrs. U. Schaffner for carrying out the time-consuming and difficult measurements and Mr. R. Griipl for his help in designing and producing
the dialysis
author
is also indebted
the
proofs. Forschung
cell as \vell
as for
to Dr. Harold
The work was financially und Technologie”, Bonn.
preparing
Hopfenberg supported Germany.
modified membranes. The and Dr. Peter Bo for reading
the
by the
“Bundesministerium
ftir
REFERESCFS
1.
1. 3. 4. 5. 6.
R. SCHL~~L. Srofiransporl durclr_~lrmbranen.Dr. Dietnch-Steinkopff-Verlag. Damstadt, 1964. A. J. STAVERMA~, REC. Trar. Clrinl. Pup-Bas, 70 (1951) 344. W. PCISCH. Clre.?;ie-IIISeni~,rtr-TEc/miX-,20 (1973) 1216. W. PLXH ADD D. W'OERMAX~. Ser. Blcnsen,Cer.plr?Jili. C/tern., 74(1970)444. R. CiRijPL AVV W. Puscti. Desdinatiun, 8 (1970) 177. N. LAKSUWSARAYASAIAH. Tmmport Phenomena in Xfwdmmes. Academic Press, NC\\ York, N-Y., 1969, p. 314.
TRANSPORT
COEFFICIENTS OF xxhfsmmc
CA
MEMBU~
53
7. J. JAGUR-GRODZINSKI ASD 0. KEDEM, Dedinarion, 1 (1966) 327. 1 (1966) 30. 5. R. L. RILEY, U. MERTES ASD J. 0. GARDNER. Des&u&m. 9. W. PUSCH, Proc. Fourth Intern. Synrp. on Fresh Water fiionr the Sea, Heidelberg, September 9-14. 1973.4 (1973) 321-332. 10. W. WSCH. Proc. Intern. Synp. on Strrtctwe of Nirter and Ayueom Sohrtions, Afurbrtrg, July 19-29, 1973, W. Luck, editor. Verlag Chemie/Physik Vcrlag. Weinheim. 1974. pp. 541562. 11. W. PUSCH ASD R. RII_EY, Desolinutiotr. 14 (1974) 389. 12’. W. BASKS ABD A. SHARPLW. The Mechanism of Desalination by Reverse Osmosis, and Its Relation to Membrane Structure, O@ce of Saline Wafer. Res. Dewlop. Progr. Rcpt. No. 143. June 1965. i 3. W. Puscn. Proc. Nuto Athunccrl Stut& Inst. ~Poiyelecrrofytes II”. Forges-fcs-Euux, 1973, E. Sttiox~ ed., Reidcl Pubhshmg Co.. Dordrecht (in preparation).