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A new method for measuring diffusion coefficients of electrolytes in nonionic membranes
A NEW METHIOD FOR hnEASURfNG DfFFUSfON COEFFIGIENTS OF ELECTROf..YTES IN NONiONfG MEMBRANES+
it has been shown that the diffusion coeffiient of a salt in a homo~en~us nonionic membrane may rcadity bc c~aluated from the cktrical conductivity of the membrane when it is immersed in an a~ucous utt solution. The diffusion coefficient of sodium chloride in dense isotropic celhrlosc acetate m&nbraws has been measured by this method. and the activatton energy for di&ive satt transport has been abtsincci from the temperarum dependence. The effect of heat-treatment temperature on the ekxtrical conductivity of artisoiotropie get membranes has a&o been examined. and an atimate of the thictncss of the “a&w layer” was obtained from these data.
SYMJJOLS
(cm’) of sample of film or rn~rnbra~~
A
-
area
h
-
ratio of electrolyte concentration
c
-
Cf
-
formal con~ntration of efectrofyle (moles titer- ‘I firm! sotute concentration (molts liter-‘) in a film equilibrated with an aqueous solution solute concentration (mmoies titer- ‘) in an aqueous sofution average solute concentrafion (mmoles liter-‘) in sampfe of polymer film at time r sotute co~~~trat~o~ (mmoles fiter-‘) at distance z from film surface at
to conductivity
:= C/k
time t D,
-
Fickian diffusion
F K
-
-
Faraday constant [esu(g-mof)-‘] ~lt-distribution coeffkient
de
-
43 L
-
cu&kient
(cm* set-t)
apparent ~it~distribution coefficient dissociation constant eicctrical conductivity (mhos)
* &escntcd in part at the 2nd Southwest Regionat Meeting. American Chemical Socety. Albuquerque, New Mczcico, Ueannbcr t , i966, and at the CH%x oFSaline Water (U.S. Dcpiirtment crf t&c interior) Rese-axh Confcrenfc on Re?exwz Osmosis, San !kgo, Califamia, Fcbmary 13-16,
1967.
Besaiina~iun, 0 (1968) 309-327
310
C. N’. SALTOSSTALL,
k,
-
k,
-
k, i.zT
-
k(f) Ettj
-
Q R RI fL Rr) I x s, x, z
-
; 7 d
JR., W. M. KING AKD 0. L. HOERNSCHEMl3ER
final
electrical conductivity (mhos) initial electrical conductivity (mhos) specific conductivity (mhos) product (ergs) of Boltzmann constant and absolute temperature electricat conductivity (mhos) at time t hypothetical electrical conductivity (mhos) assuming X-CLc(r) kinetic factor = (+!x)exp(-+i) eiccrrical resistance (ohms) tinaf electrical resistance {ohms) electrical resistance of mcmbranc (ohms) electrical res’siance at time r (ohms) time (set) thickness (cm) film thickness (cm) active-IaFer thickness, membrane (cm) coordinate axis perpendicular to sheet of film degree of dissociation exponent of C in R = bCw5 mean activity coefficient dimensionless constant ratio of ion mohilitics ratio of the quantity T/( I -i r)’ in-the aqueous solution to that kinetic
factor
-
(x/X)’
in the film
D,z
mobility of ith ion (cm'sec- ’ volt-‘) superscript 0 denotes the value of that quantity
in an aqueous
solution
fn
the original expiontion of desalination by reverse osmosis. Reid and his coworkers {I] proposed a sottition-diffusion mechanism for water and salt transport through work in many laboratories has supported this thin polymer films.* Subsequent hypothesis. In this mechanism, the degree of salt removal is determined by the relative rates of salt and water permeation. Salt-removal effectiveness can be evaluated by reverse osmosis testing of films of the polymer under study. The speed and convcnience of such testing, however. depend on the ability to obtain sufficiently thin films. This is the method of choice for reverse osmosis membranes* of the asymmetric or Loeb type, which have the high fluxes associated with extremely the high fluxes associated with extremely thin active layers (0.1 to 0.2~). The films most easily obtained in the initial screening of new poIymers for possible use in membranes, however, are orders of magnitude thicker (e-g., 25~). In this case, reverse osmosis -We wiil define a film as a homogeneous polymeric sheet. whereas a membranp is considered l
to be anisotropic, consisting of 8 porous substructure surmounted polymer as described by Kering, Barsh. and Vincent (2).
by a region of retatively dense
Desuiinurion, 4 (1968) 309-327
DIff%..6IClN
COEFFICIENTS
OF ELECTROLYTES
311
m~asu~rne~ls
BE-L’ re~~~v~~~sbw and require defect-free specimens. Owing to the high pressures employed in reverse osmosis, strenuous rn~~han~ca? demands are also ptaad on the test specimens, espcciafly when thinner htms are used. A scr~~~n~ method designed to avuid some of these d~~c~~t~es was recently advaneed g’s). It depends on direct meas~~~rnent of the s~~t*d~f~~sju~ ~ue~~jent artd the su~nb~~~t~of salt in refativefy thick specimens of the polymer being studied_ It is not sensiti\e to 3 minor number of specimen im~rf~ctions or pores. Its most difkult step-analysis of very small salt concentrations in Bim spc&nens-is avoided by the simp!iiied method described hrre for detcrm~~at~on of the diffusion coef?i&tats &’ &ctrolytcs in ~~i~rner films through ctcctricat-conductivity mensurcmcnts. The new apftroarh is b&d on an nsJumption that the factors controlling eieetroljlte ~rmeat~on through films also control the etectricaI cond~l~tjv~ty of films. If the ~ondu~ti~~~t~ of a Mm eqnj~~bratcd with a satt solution is lintar with the snltt ~o~centrat~un in the sofurion Gas proved to be the case). it appeared conccivabie that d~~~s~on cocfkients co&d be obtained from c!ocrricaI-condllctivit4’ measure-meats. ft was also hy~tbesj~~~ that the ~rrne~b~~~ty of a Giitm to ~r~t~~~~~~~ is ~ro~rt~o~a~ to the ratio of the ckctricai conductivities of xhe Sitm and she sotution in which it is cqtnlibratrd, The hyporhcsis provides a means for comparing the s&t ~rrneabi~it~es and the thicknesses of the active Iaycrs. of Loch-type membranes. This paper describes svatk conducted to examine the foioregoing h~~othcses and to dcverop for detcrmirtinp diffusion coe%cients of clectro#;)tes in warcr-swollen pot_vmcx films by m~asuremt‘nts of ckctricat conductivity. When $1 film of thickness X is immersed in a strtt solution. the concentration of sait at rime t ak 8 distance z from one side of the ftfm i5 giverx by {4).
where C, is the fkal, or equilibrium concentration of sntt in the fiim and D2 is the diffns~o~ eoefficirnt of saIt in the film. The ~ondit~o~s required far the validity of this equation are: one, Dr ~~de~~dent of z, and twu, that no other rnu~e~n~ar species are simultaneously diI%nsing tvithin the film. The first condition will be met if the sheet of polymer is homo~e~e~ns and if D, is inde~nde~t of the concentration of electrolyte in the polymer. tt wilt be pr~v~s~ona~~~ assumed that the fitms studied are humo~~neous~ the i~d~~~~d~~~ of & and C, for sodium chloride in cetlutose acetate, wifl be demonstrated fater in this paper, With regard to the t~ns~ort of other specks, it is apparent that if the fillm is ~~jt~a~~~drq”, water must diffuse inta the f%m,*artd if it is initially water iterated, water must diffuse ortt of the film due yol its lower activity in the external aqwous sait silXut.km Xn gemral, this would affect the rate of dif%sion of salt into the fifm if the twa Rows were coupled. However, for c&dose acetate ~~~ metyt) this factor xvi81 have a ~e~~~~bIe ~~ffue~~ because the diffusion ~o&kient of water is fW times as large as that of most salts
312
C.
W.
SALl’W%TALL,
JR.,
W.
M. KING
AND
D.
L. HOERSSCHEMEYER
and therefore the film wiil be e-zsentially equilibrated with preciable salt has diffused into the polymer. To relate electrical conductivity. k. to salt conrxntration relation
water
before
any
ap-
in the film we use the
where 6 is the proportionality factor in I;, = C,/b. That conductivity is essentially linear in the concentration of sodium chloride in cellulose aLxtate follows from the demonstration (@I the Experimental Section) that k, is linear in the concentration of salt in the aqueous solution, C,, and that C, is proportional to Co. The latter requirement has been demonstrated to be valid by Lonsdale and coworkers (-3). To facilitate the integration of Eq. (2) we note that for sufficiently long times, for example, for (x/X)’ Dz t 2 0.4, the second term of the sum in Eq. (I) is only I “/
kr -4 lift)_R(f
1,*,,,2 tan-I
_
1 -Q (I
-
Q2)“2
We have here introduced the dimensionless variables CL_ =(xjX)‘D,t and Q = (4/n)exp(-@). As this is a rather awkward expression to use routinely for the determination of D2 values from k(r) data. we seek a simpler, though approximate, relationship. Suppose that at long times k(r) was essentially equal to C?(r)/b, where C(r) is the average concentration of salt in the film. c(t) is obtained from
C(t)
= f
I
X
C(z, 2)d.z
0
and for long times, the first two terms of Eq. (1) to give
Cl - C’(r) =f
8 79
= -eexp
’
- g2D2t
(5)
If the approximation k(r) 2: C(r)/6 is nearly correct at long times, the graph of to the graph of log[l - C&(t)] vs r. )og[t - ~~P(r)l vs r should be assymptotic The data to ascertain the accuracy of this approximation is presented in Table L and plotted in Fig. 1. In terms of Q, C&(r)
is given by
It should be noted that the lower curve in Fig. 1 represents the true time dependence of the diffusive sorption of salt by the polymer fihn while the upper curve represents Resdnation,
4 (M8)
309-327
313 07 0.6
Fig.
1. Tim
dependettcc
THEORmfAt
JI
TtUE
of clwtrical
condwti\itv and sitIt conwntntion of ektrolytr. ‘1; According to Eq. 3. C: According to Eq. 5.
DEPESDFSCE
OF
SALT
Cf%CEhXRATlO>
C,!C(t)
0
._. . _
. .. ._ ._
-
AXD
k-,:/m ___.
for difTusiw
ELECTRICAL
1_._
wrption
COGDtiCnVITY
rmjc,1
t -- rut ,/k,f
0.40
0.853
2.19
3.16
0.543
0.684
0.60
0,699
1.80
2.09
0.445
0.522
0.90
0.518
1 A9
I.57
0.329
0.364
1.20
0.384
I.323
I .353
0.244
0.261
1.50
0.284
I.211
I .zw
0.181
0.190
1.80
0.21 I
l.fSS
I.161
0.133
0.13s
the expected time dependence of the experimental manner. From rhc slopes of the two curves ir is is less than 0.4 the conductimetricalty measured value by less than 8 %. That is, it should be a good coefficients from
conductivity when plotted in this concluded that when 1 -[Ci(t)jk/l value of D2 will exceed the true approximation to obtain diffusion Drsdittation,
4 (1966)
309-327
314
C. W. SALTOSSTALL,
JR..
W. hi. E;ING A.KD D. L. HOERSCHEMEYEiR
ln[l - &(1)/k,] = --;<&I
+ constant,
(7)
paniculary at long times. Consideration must also be given to the cffcct of film “imperfections” or holes on the measured vale of LIZ_If the reasonable assumption is made that these imperfections give rise to a constant addition to the electrical conductivity. it can be shown that this factor foes not change the slope of the curve for In (I - [Mr)ik,]f vs time and thus does not in5uence the derived diffusion coefficient. The same considerations would sppiy to the behavior oi 2 Mm pre-equilibtated in a dilute salt solution and then immerxd ir, a morr concentrarcd salt solution. In either c;iy. the time dependence of I;(f) -at is given (4; by the following equation: long timc-
where ki is the conductivity due to the salt initially present in the film. Because ki and k, arc constants, it is possible to let ki - ck, and to obtain (9) That is, an increased conductivity due to leakage through imperfections. or to salt will have no effect on the derived left in the film from any previous manipulations. dilfusior. coefficient but will change the intercept of the graph of ([k, - k(r)]fk,) vs t from 8jrr:’ to S(l - 0)x2. Conduction through small holes in the film will of course increase k ,; this effect will be discussed further on. A final question of theoretical interest regards the relationship between the conductivity of a Mm equilibrated in an aqueous electrolyte solution and the salt-diffusion coe5icient and salt concentration in the sample. The Nemst formula {-Sa) shows that the diffusion crrefficient of an ion is directiy proportional to ion mobility and also that the electrical conductivity is proportional to ion concentration and mobility. The specific conductivity (k,,) is given by 1000 k,,, = IFC(o,
“i-a,,)
(10)
where F is the Faraday constant. C is the formal concentration of the electrolyte, z is the degree of electrolyte dissociation, and CD, and w2 are the ion mobilities. For a I-1 clcctrolqtc, the diffusion coefficient is given as foIIows f&z):
where 7 is the mean activity coe5icient of the salt, and XrTis the product of BoItzmann’s constant and the absolute temperature. letting c = t$,foz, Eqs. 10 and II canibe combined to give Desahation, 4 (1968)309-327
DIFF’LWON
D, --_--
2kT
COEFFICIEYZS
loo0 1 F2
‘1 -L --c
i (.
315
OF ELECTR0Lf’TE.S
n7
i’ dC, ) (1
T LI 4 T)Z c
(12)
In general. the ion mobilities in the film are not known (i.e.. T is not known). It couId be assumed that T in the sample is the same as in an aqueous solution, but a better procedure is to compare values in the fifm and in an aqueous sofution. ff a superscript is used to denote properties in the aqueous solution and if K denotes the distribution coefficient of salt between the fifm ar.d the solution. Eq. (12) rearranges to five k,, -is-SF
* -l_ --c-0 tip i’o dCO :.:: iAcn7 ‘r’
YD=K ~ti1)
.
(13)
2
dC
Here, z is taken 3s unir~ in the aqueous solution and d(r) denotes the ratio of the quantiry r(i T T)’ in the solution to that in the film. in the fiim can be expressed in terms of its value The quantity 1 - f(Cfi)(&;,tdC)j in the aqueous solution. because efectrolyte activity in the fiim under equilibrium conditions is the same as in the aqueous solution, or Fi = TO. If K is independent of concentration. mathematical manipulation shows that 1 i- [(CP;)(&//ciC)f is the same
in the polymer
as in the solution
so that
Eq. (13) simplifies
to
(14) To obtain an estimate a~
sodium
ximateIy r/(f
of how much &tr) may differ from unity, consider
a salt such
chIoride. In an aqueous solution the ratio of ion mobilities (r) is appro1.5 and ~(1 - z)’ -2 024. Assume that in the polymer T is either 1.0 or 2.0;
-9 T)’ is then 0.25 or 0.22. respectively,
and the ratio of these factors [#(r)] is 0.96 or 1.09. Hence, unless the ratio of the individual ion mohilities changes drastiUW. &r‘, in Eq. (14) will differ from unity by less that IO?;; it will hereafter be assumed that @rl =- 1 and Eq. (14) will be replaced by
With independent measurements of k, D,. and K, it should be possible to obtain an estimate of the degree of electrolyte dissociation in a nonionic, polymeric film. For a rough estimate of II, the case of hydrochloric acid in 82% dioxane/lS” water solutions is examined. Using the following relation (56):
and appropriate data from Harned and Owen (5c.) it is found that. in the hydrochloric acid concentration range from 1.5 to 15 m&f, z ranges between 0.65 and 0.55. Dedhafiott,
4 (B68) 3O!I-377
1
_
316
C. \L‘. SALTOXSTALL,
An apparent further on.
JR..
W.
M. RISG
ASD
D. L. HOERNSCHEMEYER
value of Q for sodium chloride in a film of cciiuiose
APPARATLS
,\SD
MEMRRAGE
acetate
is discussed
PREPARAT’IOS
Ail conductivity
measurements were made with an Industrial Instruments Inc. conductivity bridge (Model RC 1682) having a range from O.,7 to 2,500.QOO ohms. The bridge can hc used for measurements at either 60 or 1000 Hz. but the W-Hz rertdings could be made more precisely and were used throughout the study. A Pyrex-glass conductivity cell with piatinized-platinum electrodes was made in rwo sections to permit insertion of ‘the test specimen, which forms a barrier (arca = 4.36 cm’) between the two electrodes. as shown in Fig. 2.
PLAllNUM
WIRE-.
k-
PLAtlNlZE3 PLAflNUf4
Fig. 3. Conductivity cell. The capacity of the eel1 is approximately 40 ml, In use. the two cell halves were clamp. The entire ccl1 assembfy was maintained at the desired temperature as follows: (a) For film measurements. it was immersed in a constant-temperature bath containing a sodium chloride solution of the same concentration as that in the ceil, and (b) for mcmbrancs, it was held at 25.O’C in an air thermostat. Calibration measurements (shown in Table II) ivere made with the cell filled with various sodium chloride solutions to determine the ceil constant at scvcrai concentrations. The average value obtained for the ceil constant was 1.465 I+_0.008. No trend is apparent in the constant over this concentration range. To obtain the resistance of a fttm or membrane, the resistance of the cell fitted with the salt soIution is subtracted from the resistance of the cell under the same conditions, but with the specimen in place. Fully dense films cf Eastman E 398-3 cxiluiose acetate (degree of acetylation 2.45 were prepared by casting a 20% solution of the polymer in acetone on a glass plate
held together with a bail-joint
Dtwhatimt, 4 (1968) 309-327
DIFFUSION
COEFFICIESTS
317
OF ELECTROLYTFS
TABLE
II
CONDUCTIVITY-CELL CALIBRATION
NaCI rmcetmariotx mm&s liter - f
Solution resistanm (R), ohs meamred .
.
specific’
f
il.900
8.081
I .4-r:!
5
1.410
1.655
I A54
IO
1.225
50
263.0
100
137.8
93.68
I.471
500
31.20
21.36
1.461
Specific-resistance
l
Ceff cumfanr = R m*~*Ylld. fR .&?rr‘,w
valun
843.7
I.472
180.1
1.460
(lib).
and drying inder a close-fitting dust cowx. After drying. the film was equilibrated in deionized water. Sufficiem anisotropic membrane for the study was prepared from a ca&ng formulation of 22.2 parts (by \&eight) of E 395-3 cellulose acetate. 66.7 parts of acetone, 10.0 parts of water, and 3.0 parts of magnesium perchlorate (;,I_ Ir was cast at 10 mils, dried at - 10°C for 2 min. and gelled in water nt ICC. After heat treatment in water. the membranes were stored in deionized water until they were used. EXPERlMtXTAL
Cellrrltse
ucemre film
PROC-EDL’RES
AKD
DISCLSSIOS
OF
RESULTS
tntwrtrretttetm
Experiments were conducted to establish the validity of the assumption that film conductivity is proportional to the concentration of the eiecrrolyte in which the film is immersed. The equilibrium resistance (RI) of a single 94.7p film of E398-3 ~eliulose
acetate in three concentrations
in Fig. 3. The slope
of sodium chloride (0.1,0.5, and 1.0 AI) is shown
of this line is - 1.02. In R,
=
In R,
== InbC-6
Rf
-_PlnC
Expressed
mathematically.
i-inb (17)
=: be-@
A stope of - 1 (i.e., 6 = 1 f permits the fofiowing equation 10 be used (for long times*) in crlcuiating the coefficient of salt diffusion through a polymeric film:
l
As previously shown, for times such thai I --[R,/R(r)] is 2 0.4. Desalination,
4 (1968)
309-327
C. W.
318
SXLTOSSTALL.
JR.. \\‘.
M. K1St.i AND
D. L. HOERKSCHEMEYER
100 80 60
40
IO i
0
SODIUM Fig. 3. Relationship
bctwxn
400
200 CHLORIOE
electrical reswmce
600
CONC..
mhj
of film and salt conccntratmn
of solution.
where R, is the final resistance after the film has reached equilibrium uith the salt solution. and R(r) is the resistance at time t after immersion.* To ascertain the error produced in DZ . computed on the assumption that p = 1-m when it may be slightly different, the slopes of In {I - [RlfR(t)Jp) vs r wertf computed with p =O.SrS, 1.00. and 1.05 and typic& long-time values for R,iR(r) of 0.6 and 03. The variation in slopes (and therefore in Dl) for /3 values in this range is no greater than 17.; of the value obtained by assuming that p = 1.00 (i.e., that the conductivity is exactly linear in C,). If I - [RJR(t)] is plotted against time on a semilogarithmic graph, and the film thickness (x) is measured, the salt-diffusion coefficient (D2) can be calculated from the slope according to Eq. (IS). The 1 -- [RJR(f)] vs I plot shown in Fig. 4 for a film in 0.1 A1 solution is typical of the data obtained in this manner, and the resulting diffusion coefficients are given in Table 11’1. It is seen that Dt of sodium chloride in celhtlose acetate is independent of the concentration of salt. The distribution coefficient can be calculated from the diffusion coeficient and the equilibrium specific electricai conductivity by using Eq. (15). From this equation the -* St is convenient in the experimental measurements to use resistances rather than axtductivities:
therefore,
1 -[k(t)lk,]
is replacedby 1-[RI/R(r)].
Desaharion,
4 (1968) 309-327
DIFFUSION
COEFFlCfENTS
3I9
OF ELECTROLYTES
I.0 0.9
0.6 0.7 0.6
MIN
TIME,
Fig, 4. Wt diffusion
Slblarif_v
-._-_-_-_-
CC~MOSC occtatc.
of
hklcr s0furf-an -_.-c____._-_
in
..-
I&. rm” set- ‘( * WO) _._.
.
__.-.-..___
__...
--.-.-
.-
-__.._-
i? --.-.__.
---
.-.-_--_
0.1
7.16
0.02 I
0.5
6.91 7.29
0.026 WE9
1.0
__.
___,.,
__
foliowing expression for the experimcntal~~ measured resistance of a film of thickness X md area A can be obtained: R =(X~A)(R$,DJD,~), where R, is the specific resistance of the salt solution, and R (which repfaces SK) is the apparent distribution coefficient. Using I.475 x 10s-cm2~-’ for the diffusion coefRcienr of Desalination, 4 (1968) 309-327
320
C. W. SALTOSSTALL,
JR., W. .W. KIXG AND D. L. HOERNSCHEhiEYER
so&urn chloride in water,* the R values given in Table III were calculated for the specimen used in these measurements. It should be pointed out that conduction through any holes present in the film wilt increase the value of the distribution coeffkient, R. However, it is felt that this u-i11 not be an important factor for most dense fiims; the reasoning is as follows. At WC. the final resistance (R,) of the 80p.fiSms given in Table V is - 10s ohms. In order for conduction through holes in the film to increase the value of E by about lo?;. the electrical resistance of the holes would need to be about i06 ohms (assuming parallel conduction paths). This resistance would be equal to the specific resistance of the salt solution (21 ohm cm fo; 0.5 AYsodium chloride), times the interelectrode distance (about 5 cm in our cells), divided by the area of the holes. From this it is calculated that a hole with a diameter of - 0.1 mm would be required to decrease R, by approximately 10%. For thinner or more permeable films (i.e., those with smaller values of R,). the size of the hole wou!d have to be larger to account for a similar error in the value of the distribution coefficient. The measurements were made in ascending order of concentration without leaching the salt from the specimen before immersion in the next higher con~ntrat~on_ This does not affect D, because the salt retained in the Mm only Iouers the intercept of the i - [R,,fR(f)] plot without changing its slope (as shown earlier), nor does it affect R, because the initial amount of salt in the film is insufficient to alter the concentration of the solution in the cell. The values for D2 and R calculated from conductivity measurements on other films of E 398-3 cellulose acetate equilibrated in 1 .%Isodium chloride at 25°C are given in Table IV.
TABLE Ab’ERAGE
&
ntickness. fi
___
ASD
K VALUES
IV
FOR SODItJSS
D?_. cm: xc-
CliLORIDE
IN E
398-3
8( :< 1oq
;I-
_._________~___.-“~---
-.
38.9
6.02
38.1
5.03
0.022
39.4
6.59
0.021
42.2
6.36
0.020
41.2
5.76
0.022
39.9
S.Sl
0.021
0.020
l This is the value for the mutual diffusion coefficient (6~1 in aqueous sodium chloride sotutions, although in &q. IS and prcseding equations should really be the diffusioncocfkicnt of the ckcrrofyte with respect to a fixed reference plane (viz.. the ekctrodes). The latter is usually called the seif-ditTcsion or intrinsic diffusion coefficient and differs from the mutual diffusion coefficient ad from any simple combination of the singleion self-diffusion coefficients (Cd).
Dcsahation.
4(1968)309-327
DIFFUSIOX
These (3)
measurements
arc
COEFFICIEXTS
in reasonable
of 9.4 x lG- lo snd
0.035
for
OF ELECTROLYTES
agreement
D,
and
with
321
previously
reported
values
K. respectillely.
If it is assumed that the true value of K for the cellulose acetate films used in this study is 0.035 (z!), it is deduced, on the basis of the R yalttes reported in Table IV and the definition rK =R, that x r 0.75. This is certainly a reasonable value for the degree of dissociation of sodium chloride in cellulose acetate, which has a dieieetric constant of about 10. However, the trend in K values show‘1 in Table IfI casts doubt on the reliabi?ity of th-. value of 0.75 for z. On .he basis of the data there, one would conclude that z is an increasing function of concentration; this is almost certainly incorrect. There are at least two possible explanations of the
anomalous
result: (1) The true distribution
increasing aqueous-solution concentration: variation in the true self-diffusion coefkicnt changes in the mutual dilfusion coefftcient
cocfhcicnt
could
be increasing
with
(2) the variation in R may be due of the salt, which is not reflected in this concentration range.
to in
The activation energy for sodium chloride diffusion in cellulose acetate (Eastman E 398-3) was calculated from the diffusion coefficients of an 81~ film at various t~rn~r~~u~~s. A specimen (Film Sample A) was mounted in the conductivity cell; the cell was ftlled with a 0.5 .If sodium chIoride solution at the desired temperature, and was immersed in a thcrmostated bath of sodium chloride solution of the same concentration and temperature as that in the cell. The film resistance was measured at various times until equilibrium was achieved: the salt was then leached from the film and the sample was used again at the next temperature, thus allowing the series to be made with a single specimen. The resistance data and calculated values of Dz and D,R are given in Table V. As no data was available for the temperature coefftcient of sodium chloride in water, this was dependence of 04, the ditfusion approximated by the temperature dependence of 0: for potassium chloride at a concentration of 0.01 Af (from Ref. f6), p. 513). These approximate values of D: (T) were used to calculate the D,f? values. Fig. 5 presents a semi-logarithmic plot of 1 - R,/R(r) against time, from which the diffusion coefficients shown in Table V for Sample A were
calculated.
These measurements
thermal-aging
effects.
were made in descending
The
ditfusion
coefficients
order are
of temperature
plotted
to minimize
1jT in Fig. 6.
against
The activation energy for sodium chloride diffusion, calculated from the Arrhenius equation. is 12.3 kcal mole- ‘_ This experiment was repeated with Sample B, a second specimen again essentially
from linear,
the same especially
efficients shown in Table vation energy for sodium moie - t _ The
prior
thermal
history
sheet of at times
film. The plots of I - [RJR(r)] were greater than 1 hour. The diffusion co-
V for Sample B were plotted against l/T, and the actichloride
transport
was
calculated
to
be
13.1
hai
of a cellulose acetate film has an influence on its perby D, values of 5.25 x 10‘ lo and 3.00 x lo- fo
meability to salt, as demonstrated
Desdination, 3 C1968) 309-327
322
C. W. SALTOWXALL,
JR.. \V. ht. KIM? AIW
D. L. HOERNSCHEMEYER
0.6
0.1 0
SO
I!50 200
100
250
300
350
TIME. MIN Fig. 5. Salt diffusion in cekhsc
acetate at sexera temperatures.
TABLE TEUPERATURE
DEPEKDENCE
-
- ._._.----..._--Film
44 3s 2s 15
V
OF SALT
PERMEABILITY
- ._. ---. ----_^
---
_--
_-__ _
skImpIe A
61 .O 73.5 96.5 t 27.0
14.20 9.86 6.10 3.53
11.84 4.99 3.00 I.26
Film xampk B 45 35 25
83.5 113 I40
10.5 6.52 4.28
8.73 3.87 2.05
Film sampks A and B were both 81~ thick.
Desalinarion. 4 (1968) 309-327
DIFFUSION
COEFFICIENTS
RECIPROCAL
323
OF ELECl-ROf..Y-l%S
TEMP
(lO’/f),
OK-’
Fig. 6. Temperature dependence of sodium chloride diffusion coefficient in cellulose acetate.
see-t
at Z’C
smaller
Cellulose
D,
before and after heating Sample A to 44”C, together values for Sample B, which was preheated at 5o’C
acetare
With swollen
membrane anisotropic
with the generally for 10 min.
measureme~ls membranes
having
very thin active
iayers,
it
wouId be
expected hat equilibration with a salt solution would
occur very rapidly so that the final resistance could be measured almost immediately after the conductivity cell was filled with saline water Because of the rapid equilibration, permeability measurements employing various solutes, concentrations, and temperatures should be possible with a minimum expenditure of time. Experiments with swollen membranes have shown that equilibration is indeed established within a few seconds with sodium chloride solutions and that the effect of formative steps ocz the permeability of the active layer may be determined readily by this technique, as shown below. The
resistances of specimens of water-swollen membranes that had been heattreated in water at 70, 75, 85, and 90% were measured at equ~~ibrium with sodium varying in concentration from 0.1 to 1.0 M. As with dried films logarithmic plots of their resistance against solution concentration are linear {Fig. 7). shown Their slopes are -0.97, - 1.06, - 1.01, and 1.00, respectively. As previously with water-swollen, fully dense films ofcellulose acetate, slopes of essentially - 1 establish that the resistance is proportional to the salt concentration of the solution
chloride
in which
solutions
the membrane
is immersed. Desalination, 4 (I 968) 309-327
c. W.
321
SALTOXSTALL,
JR.,
W.
M. RIMi
AND
D.
t.
HOERNSCHEMEYER
too0
SOG
10 SODIUM Fig. 7. Relationship
bclwxn
CHLORIDE
clcctrical rcsistancc
of
CONC..
membrane
rnfl
and salt concentration
of solution.
the membranes exhibited esscntiatly the same resistance as before heat treatment. As shown in (Table VI where only the equilibrium resistance in 0 .Z ,!I sodium chtoride is given). the resistance after 2 hotlrs of reverse osmosis at 1500 psi with 3.5 sodium chloride is less for membranes heated at 76,85. and 9O’C. and is the same or greater for membranes heated to 7OXY or fewer. For
heat-tnxtmcnt
temIxxxturcs
below
60-65’C.
TABLE CHASCE
IS
hUMERASE
RESISTASCE
WITH
VI
NEAT
TREAT%4EXT
AND
REVERSE
OSMOSIS
Membrane resistance, ohms’ ffear-rrectmenl temperatwe, ’ C --.
.--..-
..__ ----..NOW
70 75 85 30
l
Sal: pemreatiou 0’ .‘u
FIUX .dd*’ .-.- --
----
27.3 19.8 12.0 7.6 4.4
before reverse osmosis
---.---_____
85.7 21.9 7.4 1.6 I .o
iZflN
reverse osmosfs e.-
1.7 2.2 5::: IS.7
1.8 2.7 5.Q 32.0 105.8
l Sample area of 4.36 ana. * U.S. gallons per square foot per day.
DesaIinatiott, 4 (1968) 309-327
325
DIFFUSlON COEFFICIENTS OF ELECTROLYTES
Before reverse osmosis, the logarithm
of the ratio of membrane
conductivity
to
salt concentration follows an inverse linear relationship with the heat-treatment temperature. as shown in Fig. 8; after reverse osmosis, the response has changed and is no Ionger linear. The significance of these results has not yet been ascertained. The flux of salt through a sheet. of polymer is usually taken to be proportional elcctricar resistance of a film to D&)X (3) and we have seen that the equilibrium is inversely proportional to D,K/X. It is therefore of interest to test these relations for an anisotropic gel membrane: we expect the salt flux (J2) to be inverseiy proportional to the electrical resistance (R). From the data of Table Vi, the results in Fig. 9 are obtained.
It is seen
that J,
is essentially
linear
in
l/R,
except
for
the rather
open membrane (of ?Zgl, salt permeation). Assuming that the active layer of a highly retentive membrane, free of impcrfcctions, has the same 4t permeability as a water-equilibrated, fully dense film, it would be possible to estimate the thickness of the active lager from resistance measurements if the resistance of the highly Forous substrate is negligible. The validity of the latter approximation is questionable, but its esscrltial correctness is supeested by very small resistance of the unheated membrane, which possesses a very permeable active layer. A swollen membrane, which had been heat-treated 400
10
.I5
HEAT -TREATMENT
85
00
90
TEMPERATURE.
Fig. 8. Effect of heat trcatmcnt and reverse osmosis opcmtion
lC
on electrical conductivity Desalination.
of membrane.
4 ( 1968) 309-327
226
C. W. SALTONSTALL,
i
4
,~,
JR., W. Sf. );I!%
I
i
i
i
AND D. L. H9ERSSCHEhIEYER
;--__
..__.
_ -. -
:
, ’
__--
.
.
-. _&_7
i
;
I -i-e..
,
1
2
3
6
810
20
40
60
100
200
R, okms Fig. 9. Re\rrsc-osmosis salt flux and electrical resistance of anisotropic 9 Before rcvcr% osmosis. h After t-enxse osmosis.
membranes.
for 5 min at WC. had a resistance of 25.4 ohms (over an area of 4.36 cm’) when in equilibrium with 1 1cf sodium chloride. Fully dense film, 134~ thick. with a diffusion coeffcient of 6.66 x lo- l”cm%x-’ and a distribution coeticicnt of 0.021. had an equilibrium resistance of 40.500 ohms (over an area of 4.36 cm’) in 1 M sodium chloride. Assuming that the active lsyer has the same specific resistance as fulIy dense film, ,
Rm
R,=X,
X,
(19)
where the subscripts nr and f represent membrane and fitm. From Eq. (19), the thickness of the active layer (C,) in the membrane tested was calculated to be 845 A. Before this calculation can be accepted as reliable, the basic assumption that the active layer and the “fully dense” film are identical requires further study. It seems clear that the assumption must fail for membranes heated at the lower temperatures, and one is even led to question whether heat treatment at 90°C produces a “fully Lksahation.
4 (1968) 309-327
DIFn’S~O!U COEFFICIENTS OF ELfXTR0LYTF.S
dense”
active layer.
It seems probable
than the value calculated from
Eq.
327
that the active layer will always be thicker l(9).
This work was supported by the Ofice of Saline Water, U.S. Department of the Interior, under Contract IS-01-OOOI-338. The authors thank W. R. White for his assistance in the preparation and examination of the films and membranes used in this study. REFERENCES I.
C. E. REID an E. J. RaEmw. J. Appi. Polvmer Sci.. l(l959) 133: C. E. REID ASD J. R. KUPPERS. ibid.. 2 (1959) 26-l; and REID ASD H. G. SPESCER,ibid., 4 (1960) 3%.
2.
R. E. K~STI~G. M. K. BARSII ASD A. L. VISCEXT. 1. Appl.
3.
H. K. Lo~ALE.c~u/.,~.
4.
W.
5.
H. S. HARMD ,WD R. B. OWES. The Pbpica! Chcmi.srr_vuf Eiccrra&tic Soluticms, 2nd cd., R&hold. New Yorh (i950): (a) pp. S-38. (b) p- 188, tc) pp. 328 and 5%.
6.
R. h. Rwwsos A?+ R. H. S~or;is. Ek-rrulyric Solutions. 2nd rd.. (1959): (a) p. 286. (h) p. 466. (c) p. -115. (df pp_ 314-315.
7.
S. LN.R. Aptxopriate Eletrol>tic hdditivcs in a Casting Solution Uwd for the Productton of High Rzformancc Cclfulo~c hccratc Mcmbrxws Uwd in Kcwtse Osmosis l?csalinittion. Ph.D. Thesis, Dcpartmcnt of Engiwxxing, University of California itt LOS Angclrs (Xlay 1964).
;Ippl_h~~~nrcrSci.,9(t965)
JUST. Difli.rion in .!G.did~,Liquids. Gu.ws. Audcmic
P&mer
Sri.. 9 (1%5)
I Y73.
1341. Press, NC\* York, t 19521p- 37.
Hutter\\orths, London
!ksalinatiurr, 4 ( I968 ) 309-327
it has been shown that the diffusion coeffiient of a salt in a homo~en~us nonionic membrane may rcadity bc c~aluated from the cktrical conductivity of the membrane when it is immersed in an a~ucous utt solution. The diffusion coefficient of sodium chloride in dense isotropic celhrlosc acetate m&nbraws has been measured by this method. and the activatton energy for di&ive satt transport has been abtsincci from the temperarum dependence. The effect of heat-treatment temperature on the ekxtrical conductivity of artisoiotropie get membranes has a&o been examined. and an atimate of the thictncss of the “a&w layer” was obtained from these data.
SYMJJOLS
(cm’) of sample of film or rn~rnbra~~
A
-
area
h
-
ratio of electrolyte concentration
c
-
Cf
-
formal con~ntration of efectrofyle (moles titer- ‘I firm! sotute concentration (molts liter-‘) in a film equilibrated with an aqueous solution solute concentration (mmoies titer- ‘) in an aqueous sofution average solute concentrafion (mmoles liter-‘) in sampfe of polymer film at time r sotute co~~~trat~o~ (mmoles fiter-‘) at distance z from film surface at
to conductivity
:= C/k
time t D,
-
Fickian diffusion
F K
-
-
Faraday constant [esu(g-mof)-‘] ~lt-distribution coeffkient
de
-
43 L
-
cu&kient
(cm* set-t)
apparent ~it~distribution coefficient dissociation constant eicctrical conductivity (mhos)
* &escntcd in part at the 2nd Southwest Regionat Meeting. American Chemical Socety. Albuquerque, New Mczcico, Ueannbcr t , i966, and at the CH%x oFSaline Water (U.S. Dcpiirtment crf t&c interior) Rese-axh Confcrenfc on Re?exwz Osmosis, San !kgo, Califamia, Fcbmary 13-16,
1967.
Besaiina~iun, 0 (1968) 309-327
310
C. N’. SALTOSSTALL,
k,
-
k,
-
k, i.zT
-
k(f) Ettj
-
Q R RI fL Rr) I x s, x, z
-
; 7 d
JR., W. M. KING AKD 0. L. HOERNSCHEMl3ER
final
electrical conductivity (mhos) initial electrical conductivity (mhos) specific conductivity (mhos) product (ergs) of Boltzmann constant and absolute temperature electricat conductivity (mhos) at time t hypothetical electrical conductivity (mhos) assuming X-CLc(r) kinetic factor = (+!x)exp(-+i) eiccrrical resistance (ohms) tinaf electrical resistance {ohms) electrical resistance of mcmbranc (ohms) electrical res’siance at time r (ohms) time (set) thickness (cm) film thickness (cm) active-IaFer thickness, membrane (cm) coordinate axis perpendicular to sheet of film degree of dissociation exponent of C in R = bCw5 mean activity coefficient dimensionless constant ratio of ion mohilitics ratio of the quantity T/( I -i r)’ in-the aqueous solution to that kinetic
factor
-
(x/X)’
in the film
D,z
mobility of ith ion (cm'sec- ’ volt-‘) superscript 0 denotes the value of that quantity
in an aqueous
solution
fn
the original expiontion of desalination by reverse osmosis. Reid and his coworkers {I] proposed a sottition-diffusion mechanism for water and salt transport through work in many laboratories has supported this thin polymer films.* Subsequent hypothesis. In this mechanism, the degree of salt removal is determined by the relative rates of salt and water permeation. Salt-removal effectiveness can be evaluated by reverse osmosis testing of films of the polymer under study. The speed and convcnience of such testing, however. depend on the ability to obtain sufficiently thin films. This is the method of choice for reverse osmosis membranes* of the asymmetric or Loeb type, which have the high fluxes associated with extremely the high fluxes associated with extremely thin active layers (0.1 to 0.2~). The films most easily obtained in the initial screening of new poIymers for possible use in membranes, however, are orders of magnitude thicker (e-g., 25~). In this case, reverse osmosis -We wiil define a film as a homogeneous polymeric sheet. whereas a membranp is considered l
to be anisotropic, consisting of 8 porous substructure surmounted polymer as described by Kering, Barsh. and Vincent (2).
by a region of retatively dense
Desuiinurion, 4 (1968) 309-327
DIff%..6IClN
COEFFICIENTS
OF ELECTROLYTES
311
m~asu~rne~ls
BE-L’ re~~~v~~~sbw and require defect-free specimens. Owing to the high pressures employed in reverse osmosis, strenuous rn~~han~ca? demands are also ptaad on the test specimens, espcciafly when thinner htms are used. A scr~~~n~ method designed to avuid some of these d~~c~~t~es was recently advaneed g’s). It depends on direct meas~~~rnent of the s~~t*d~f~~sju~ ~ue~~jent artd the su~nb~~~t~of salt in refativefy thick specimens of the polymer being studied_ It is not sensiti\e to 3 minor number of specimen im~rf~ctions or pores. Its most difkult step-analysis of very small salt concentrations in Bim spc&nens-is avoided by the simp!iiied method described hrre for detcrm~~at~on of the diffusion coef?i&tats &’ &ctrolytcs in ~~i~rner films through ctcctricat-conductivity mensurcmcnts. The new apftroarh is b&d on an nsJumption that the factors controlling eieetroljlte ~rmeat~on through films also control the etectricaI cond~l~tjv~ty of films. If the ~ondu~ti~~~t~ of a Mm eqnj~~bratcd with a satt solution is lintar with the snltt ~o~centrat~un in the sofurion Gas proved to be the case). it appeared conccivabie that d~~~s~on cocfkients co&d be obtained from c!ocrricaI-condllctivit4’ measure-meats. ft was also hy~tbesj~~~ that the ~rrne~b~~~ty of a Giitm to ~r~t~~~~~~~ is ~ro~rt~o~a~ to the ratio of the ckctricai conductivities of xhe Sitm and she sotution in which it is cqtnlibratrd, The hyporhcsis provides a means for comparing the s&t ~rrneabi~it~es and the thicknesses of the active Iaycrs. of Loch-type membranes. This paper describes svatk conducted to examine the foioregoing h~~othcses and to dcverop for detcrmirtinp diffusion coe%cients of clectro#;)tes in warcr-swollen pot_vmcx films by m~asuremt‘nts of ckctricat conductivity. When $1 film of thickness X is immersed in a strtt solution. the concentration of sait at rime t ak 8 distance z from one side of the ftfm i5 giverx by {4).
where C, is the fkal, or equilibrium concentration of sntt in the fiim and D2 is the diffns~o~ eoefficirnt of saIt in the film. The ~ondit~o~s required far the validity of this equation are: one, Dr ~~de~~dent of z, and twu, that no other rnu~e~n~ar species are simultaneously diI%nsing tvithin the film. The first condition will be met if the sheet of polymer is homo~e~e~ns and if D, is inde~nde~t of the concentration of electrolyte in the polymer. tt wilt be pr~v~s~ona~~~ assumed that the fitms studied are humo~~neous~ the i~d~~~~d~~~ of & and C, for sodium chloride in cetlutose acetate, wifl be demonstrated fater in this paper, With regard to the t~ns~ort of other specks, it is apparent that if the fillm is ~~jt~a~~~drq”, water must diffuse inta the f%m,*artd if it is initially water iterated, water must diffuse ortt of the film due yol its lower activity in the external aqwous sait silXut.km Xn gemral, this would affect the rate of dif%sion of salt into the fifm if the twa Rows were coupled. However, for c&dose acetate ~~~ metyt) this factor xvi81 have a ~e~~~~bIe ~~ffue~~ because the diffusion ~o&kient of water is fW times as large as that of most salts
312
C.
W.
SALl’W%TALL,
JR.,
W.
M. KING
AND
D.
L. HOERSSCHEMEYER
and therefore the film wiil be e-zsentially equilibrated with preciable salt has diffused into the polymer. To relate electrical conductivity. k. to salt conrxntration relation
water
before
any
ap-
in the film we use the
where 6 is the proportionality factor in I;, = C,/b. That conductivity is essentially linear in the concentration of sodium chloride in cellulose aLxtate follows from the demonstration (@I the Experimental Section) that k, is linear in the concentration of salt in the aqueous solution, C,, and that C, is proportional to Co. The latter requirement has been demonstrated to be valid by Lonsdale and coworkers (-3). To facilitate the integration of Eq. (2) we note that for sufficiently long times, for example, for (x/X)’ Dz t 2 0.4, the second term of the sum in Eq. (I) is only I “/
kr -4 lift)_R(f
1,*,,,2 tan-I
_
1 -Q (I
-
Q2)“2
We have here introduced the dimensionless variables CL_ =(xjX)‘D,t and Q = (4/n)exp(-@). As this is a rather awkward expression to use routinely for the determination of D2 values from k(r) data. we seek a simpler, though approximate, relationship. Suppose that at long times k(r) was essentially equal to C?(r)/b, where C(r) is the average concentration of salt in the film. c(t) is obtained from
C(t)
= f
I
X
C(z, 2)d.z
0
and for long times, the first two terms of Eq. (1) to give
Cl - C’(r) =f
8 79
= -eexp
’
- g2D2t
(5)
If the approximation k(r) 2: C(r)/6 is nearly correct at long times, the graph of to the graph of log[l - C&(t)] vs r. )og[t - ~~P(r)l vs r should be assymptotic The data to ascertain the accuracy of this approximation is presented in Table L and plotted in Fig. 1. In terms of Q, C&(r)
is given by
It should be noted that the lower curve in Fig. 1 represents the true time dependence of the diffusive sorption of salt by the polymer fihn while the upper curve represents Resdnation,
4 (M8)
309-327
313 07 0.6
Fig.
1. Tim
dependettcc
THEORmfAt
JI
TtUE
of clwtrical
condwti\itv and sitIt conwntntion of ektrolytr. ‘1; According to Eq. 3. C: According to Eq. 5.
DEPESDFSCE
OF
SALT
Cf%CEhXRATlO>
C,!C(t)
0
._. . _
. .. ._ ._
-
AXD
k-,:/m ___.
for difTusiw
ELECTRICAL
1_._
wrption
COGDtiCnVITY
rmjc,1
t -- rut ,/k,f
0.40
0.853
2.19
3.16
0.543
0.684
0.60
0,699
1.80
2.09
0.445
0.522
0.90
0.518
1 A9
I.57
0.329
0.364
1.20
0.384
I.323
I .353
0.244
0.261
1.50
0.284
I.211
I .zw
0.181
0.190
1.80
0.21 I
l.fSS
I.161
0.133
0.13s
the expected time dependence of the experimental manner. From rhc slopes of the two curves ir is is less than 0.4 the conductimetricalty measured value by less than 8 %. That is, it should be a good coefficients from
conductivity when plotted in this concluded that when 1 -[Ci(t)jk/l value of D2 will exceed the true approximation to obtain diffusion Drsdittation,
4 (1966)
309-327
314
C. W. SALTOSSTALL,
JR..
W. hi. E;ING A.KD D. L. HOERSCHEMEYEiR
ln[l - &(1)/k,] = --;<&I
+ constant,
(7)
paniculary at long times. Consideration must also be given to the cffcct of film “imperfections” or holes on the measured vale of LIZ_If the reasonable assumption is made that these imperfections give rise to a constant addition to the electrical conductivity. it can be shown that this factor foes not change the slope of the curve for In (I - [Mr)ik,]f vs time and thus does not in5uence the derived diffusion coefficient. The same considerations would sppiy to the behavior oi 2 Mm pre-equilibtated in a dilute salt solution and then immerxd ir, a morr concentrarcd salt solution. In either c;iy. the time dependence of I;(f) -at is given (4; by the following equation: long timc-
where ki is the conductivity due to the salt initially present in the film. Because ki and k, arc constants, it is possible to let ki - ck, and to obtain (9) That is, an increased conductivity due to leakage through imperfections. or to salt will have no effect on the derived left in the film from any previous manipulations. dilfusior. coefficient but will change the intercept of the graph of ([k, - k(r)]fk,) vs t from 8jrr:’ to S(l - 0)x2. Conduction through small holes in the film will of course increase k ,; this effect will be discussed further on. A final question of theoretical interest regards the relationship between the conductivity of a Mm equilibrated in an aqueous electrolyte solution and the salt-diffusion coe5icient and salt concentration in the sample. The Nemst formula {-Sa) shows that the diffusion crrefficient of an ion is directiy proportional to ion mobility and also that the electrical conductivity is proportional to ion concentration and mobility. The specific conductivity (k,,) is given by 1000 k,,, = IFC(o,
“i-a,,)
(10)
where F is the Faraday constant. C is the formal concentration of the electrolyte, z is the degree of electrolyte dissociation, and CD, and w2 are the ion mobilities. For a I-1 clcctrolqtc, the diffusion coefficient is given as foIIows f&z):
where 7 is the mean activity coe5icient of the salt, and XrTis the product of BoItzmann’s constant and the absolute temperature. letting c = t$,foz, Eqs. 10 and II canibe combined to give Desahation, 4 (1968)309-327
DIFF’LWON
D, --_--
2kT
COEFFICIEYZS
loo0 1 F2
‘1 -L --c
i (.
315
OF ELECTR0Lf’TE.S
n7
i’ dC, ) (1
T LI 4 T)Z c
(12)
In general. the ion mobilities in the film are not known (i.e.. T is not known). It couId be assumed that T in the sample is the same as in an aqueous solution, but a better procedure is to compare values in the fifm and in an aqueous sofution. ff a superscript is used to denote properties in the aqueous solution and if K denotes the distribution coefficient of salt between the fifm ar.d the solution. Eq. (12) rearranges to five k,, -is-SF
* -l_ --c-0 tip i’o dCO :.:: iAcn7 ‘r’
YD=K ~ti1)
.
(13)
2
dC
Here, z is taken 3s unir~ in the aqueous solution and d(r) denotes the ratio of the quantiry r(i T T)’ in the solution to that in the film. in the fiim can be expressed in terms of its value The quantity 1 - f(Cfi)(&;,tdC)j in the aqueous solution. because efectrolyte activity in the fiim under equilibrium conditions is the same as in the aqueous solution, or Fi = TO. If K is independent of concentration. mathematical manipulation shows that 1 i- [(CP;)(&//ciC)f is the same
in the polymer
as in the solution
so that
Eq. (13) simplifies
to
(14) To obtain an estimate a~
sodium
ximateIy r/(f
of how much &tr) may differ from unity, consider
a salt such
chIoride. In an aqueous solution the ratio of ion mobilities (r) is appro1.5 and ~(1 - z)’ -2 024. Assume that in the polymer T is either 1.0 or 2.0;
-9 T)’ is then 0.25 or 0.22. respectively,
and the ratio of these factors [#(r)] is 0.96 or 1.09. Hence, unless the ratio of the individual ion mohilities changes drastiUW. &r‘, in Eq. (14) will differ from unity by less that IO?;; it will hereafter be assumed that @rl =- 1 and Eq. (14) will be replaced by
With independent measurements of k, D,. and K, it should be possible to obtain an estimate of the degree of electrolyte dissociation in a nonionic, polymeric film. For a rough estimate of II, the case of hydrochloric acid in 82% dioxane/lS” water solutions is examined. Using the following relation (56):
and appropriate data from Harned and Owen (5c.) it is found that. in the hydrochloric acid concentration range from 1.5 to 15 m&f, z ranges between 0.65 and 0.55. Dedhafiott,
4 (B68) 3O!I-377
1
_
316
C. \L‘. SALTOXSTALL,
An apparent further on.
JR..
W.
M. RISG
ASD
D. L. HOERNSCHEMEYER
value of Q for sodium chloride in a film of cciiuiose
APPARATLS
,\SD
MEMRRAGE
acetate
is discussed
PREPARAT’IOS
Ail conductivity
measurements were made with an Industrial Instruments Inc. conductivity bridge (Model RC 1682) having a range from O.,7 to 2,500.QOO ohms. The bridge can hc used for measurements at either 60 or 1000 Hz. but the W-Hz rertdings could be made more precisely and were used throughout the study. A Pyrex-glass conductivity cell with piatinized-platinum electrodes was made in rwo sections to permit insertion of ‘the test specimen, which forms a barrier (arca = 4.36 cm’) between the two electrodes. as shown in Fig. 2.
PLAllNUM
WIRE-.
k-
PLAtlNlZE3 PLAflNUf4
Fig. 3. Conductivity cell. The capacity of the eel1 is approximately 40 ml, In use. the two cell halves were clamp. The entire ccl1 assembfy was maintained at the desired temperature as follows: (a) For film measurements. it was immersed in a constant-temperature bath containing a sodium chloride solution of the same concentration as that in the ceil, and (b) for mcmbrancs, it was held at 25.O’C in an air thermostat. Calibration measurements (shown in Table II) ivere made with the cell filled with various sodium chloride solutions to determine the ceil constant at scvcrai concentrations. The average value obtained for the ceil constant was 1.465 I+_0.008. No trend is apparent in the constant over this concentration range. To obtain the resistance of a fttm or membrane, the resistance of the cell fitted with the salt soIution is subtracted from the resistance of the cell under the same conditions, but with the specimen in place. Fully dense films cf Eastman E 398-3 cxiluiose acetate (degree of acetylation 2.45 were prepared by casting a 20% solution of the polymer in acetone on a glass plate
held together with a bail-joint
Dtwhatimt, 4 (1968) 309-327
DIFFUSION
COEFFICIESTS
317
OF ELECTROLYTFS
TABLE
II
CONDUCTIVITY-CELL CALIBRATION
NaCI rmcetmariotx mm&s liter - f
Solution resistanm (R), ohs meamred .
.
specific’
f
il.900
8.081
I .4-r:!
5
1.410
1.655
I A54
IO
1.225
50
263.0
100
137.8
93.68
I.471
500
31.20
21.36
1.461
Specific-resistance
l
Ceff cumfanr = R m*~*Ylld. fR .&?rr‘,w
valun
843.7
I.472
180.1
1.460
(lib).
and drying inder a close-fitting dust cowx. After drying. the film was equilibrated in deionized water. Sufficiem anisotropic membrane for the study was prepared from a ca&ng formulation of 22.2 parts (by \&eight) of E 395-3 cellulose acetate. 66.7 parts of acetone, 10.0 parts of water, and 3.0 parts of magnesium perchlorate (;,I_ Ir was cast at 10 mils, dried at - 10°C for 2 min. and gelled in water nt ICC. After heat treatment in water. the membranes were stored in deionized water until they were used. EXPERlMtXTAL
Cellrrltse
ucemre film
PROC-EDL’RES
AKD
DISCLSSIOS
OF
RESULTS
tntwrtrretttetm
Experiments were conducted to establish the validity of the assumption that film conductivity is proportional to the concentration of the eiecrrolyte in which the film is immersed. The equilibrium resistance (RI) of a single 94.7p film of E398-3 ~eliulose
acetate in three concentrations
in Fig. 3. The slope
of sodium chloride (0.1,0.5, and 1.0 AI) is shown
of this line is - 1.02. In R,
=
In R,
== InbC-6
Rf
-_PlnC
Expressed
mathematically.
i-inb (17)
=: be-@
A stope of - 1 (i.e., 6 = 1 f permits the fofiowing equation 10 be used (for long times*) in crlcuiating the coefficient of salt diffusion through a polymeric film:
l
As previously shown, for times such thai I --[R,/R(r)] is 2 0.4. Desalination,
4 (1968)
309-327
C. W.
318
SXLTOSSTALL.
JR.. \\‘.
M. K1St.i AND
D. L. HOERKSCHEMEYER
100 80 60
40
IO i
0
SODIUM Fig. 3. Relationship
bctwxn
400
200 CHLORIOE
electrical reswmce
600
CONC..
mhj
of film and salt conccntratmn
of solution.
where R, is the final resistance after the film has reached equilibrium uith the salt solution. and R(r) is the resistance at time t after immersion.* To ascertain the error produced in DZ . computed on the assumption that p = 1-m when it may be slightly different, the slopes of In {I - [RlfR(t)Jp) vs r wertf computed with p =O.SrS, 1.00. and 1.05 and typic& long-time values for R,iR(r) of 0.6 and 03. The variation in slopes (and therefore in Dl) for /3 values in this range is no greater than 17.; of the value obtained by assuming that p = 1.00 (i.e., that the conductivity is exactly linear in C,). If I - [RJR(t)] is plotted against time on a semilogarithmic graph, and the film thickness (x) is measured, the salt-diffusion coefficient (D2) can be calculated from the slope according to Eq. (IS). The 1 -- [RJR(f)] vs I plot shown in Fig. 4 for a film in 0.1 A1 solution is typical of the data obtained in this manner, and the resulting diffusion coefficients are given in Table 11’1. It is seen that Dt of sodium chloride in celhtlose acetate is independent of the concentration of salt. The distribution coefficient can be calculated from the diffusion coeficient and the equilibrium specific electricai conductivity by using Eq. (15). From this equation the -* St is convenient in the experimental measurements to use resistances rather than axtductivities:
therefore,
1 -[k(t)lk,]
is replacedby 1-[RI/R(r)].
Desaharion,
4 (1968) 309-327
DIFFUSION
COEFFlCfENTS
3I9
OF ELECTROLYTES
I.0 0.9
0.6 0.7 0.6
MIN
TIME,
Fig, 4. Wt diffusion
Slblarif_v
-._-_-_-_-
CC~MOSC occtatc.
of
hklcr s0furf-an -_.-c____._-_
in
..-
I&. rm” set- ‘( * WO) _._.
.
__.-.-..___
__...
--.-.-
.-
-__.._-
i? --.-.__.
---
.-.-_--_
0.1
7.16
0.02 I
0.5
6.91 7.29
0.026 WE9
1.0
__.
___,.,
__
foliowing expression for the experimcntal~~ measured resistance of a film of thickness X md area A can be obtained: R =(X~A)(R$,DJD,~), where R, is the specific resistance of the salt solution, and R (which repfaces SK) is the apparent distribution coefficient. Using I.475 x 10s-cm2~-’ for the diffusion coefRcienr of Desalination, 4 (1968) 309-327
320
C. W. SALTOSSTALL,
JR., W. .W. KIXG AND D. L. HOERNSCHEhiEYER
so&urn chloride in water,* the R values given in Table III were calculated for the specimen used in these measurements. It should be pointed out that conduction through any holes present in the film wilt increase the value of the distribution coeffkient, R. However, it is felt that this u-i11 not be an important factor for most dense fiims; the reasoning is as follows. At WC. the final resistance (R,) of the 80p.fiSms given in Table V is - 10s ohms. In order for conduction through holes in the film to increase the value of E by about lo?;. the electrical resistance of the holes would need to be about i06 ohms (assuming parallel conduction paths). This resistance would be equal to the specific resistance of the salt solution (21 ohm cm fo; 0.5 AYsodium chloride), times the interelectrode distance (about 5 cm in our cells), divided by the area of the holes. From this it is calculated that a hole with a diameter of - 0.1 mm would be required to decrease R, by approximately 10%. For thinner or more permeable films (i.e., those with smaller values of R,). the size of the hole wou!d have to be larger to account for a similar error in the value of the distribution coefficient. The measurements were made in ascending order of concentration without leaching the salt from the specimen before immersion in the next higher con~ntrat~on_ This does not affect D, because the salt retained in the Mm only Iouers the intercept of the i - [R,,fR(f)] plot without changing its slope (as shown earlier), nor does it affect R, because the initial amount of salt in the film is insufficient to alter the concentration of the solution in the cell. The values for D2 and R calculated from conductivity measurements on other films of E 398-3 cellulose acetate equilibrated in 1 .%Isodium chloride at 25°C are given in Table IV.
TABLE Ab’ERAGE
&
ntickness. fi
___
ASD
K VALUES
IV
FOR SODItJSS
D?_. cm: xc-
CliLORIDE
IN E
398-3
8( :< 1oq
;I-
_._________~___.-“~---
-.
38.9
6.02
38.1
5.03
0.022
39.4
6.59
0.021
42.2
6.36
0.020
41.2
5.76
0.022
39.9
S.Sl
0.021
0.020
l This is the value for the mutual diffusion coefficient (6~1 in aqueous sodium chloride sotutions, although in &q. IS and prcseding equations should really be the diffusioncocfkicnt of the ckcrrofyte with respect to a fixed reference plane (viz.. the ekctrodes). The latter is usually called the seif-ditTcsion or intrinsic diffusion coefficient and differs from the mutual diffusion coefficient ad from any simple combination of the singleion self-diffusion coefficients (Cd).
Dcsahation.
4(1968)309-327
DIFFUSIOX
These (3)
measurements
arc
COEFFICIEXTS
in reasonable
of 9.4 x lG- lo snd
0.035
for
OF ELECTROLYTES
agreement
D,
and
with
321
previously
reported
values
K. respectillely.
If it is assumed that the true value of K for the cellulose acetate films used in this study is 0.035 (z!), it is deduced, on the basis of the R yalttes reported in Table IV and the definition rK =R, that x r 0.75. This is certainly a reasonable value for the degree of dissociation of sodium chloride in cellulose acetate, which has a dieieetric constant of about 10. However, the trend in K values show‘1 in Table IfI casts doubt on the reliabi?ity of th-. value of 0.75 for z. On .he basis of the data there, one would conclude that z is an increasing function of concentration; this is almost certainly incorrect. There are at least two possible explanations of the
anomalous
result: (1) The true distribution
increasing aqueous-solution concentration: variation in the true self-diffusion coefkicnt changes in the mutual dilfusion coefftcient
cocfhcicnt
could
be increasing
with
(2) the variation in R may be due of the salt, which is not reflected in this concentration range.
to in
The activation energy for sodium chloride diffusion in cellulose acetate (Eastman E 398-3) was calculated from the diffusion coefficients of an 81~ film at various t~rn~r~~u~~s. A specimen (Film Sample A) was mounted in the conductivity cell; the cell was ftlled with a 0.5 .If sodium chIoride solution at the desired temperature, and was immersed in a thcrmostated bath of sodium chloride solution of the same concentration and temperature as that in the cell. The film resistance was measured at various times until equilibrium was achieved: the salt was then leached from the film and the sample was used again at the next temperature, thus allowing the series to be made with a single specimen. The resistance data and calculated values of Dz and D,R are given in Table V. As no data was available for the temperature coefftcient of sodium chloride in water, this was dependence of 04, the ditfusion approximated by the temperature dependence of 0: for potassium chloride at a concentration of 0.01 Af (from Ref. f6), p. 513). These approximate values of D: (T) were used to calculate the D,f? values. Fig. 5 presents a semi-logarithmic plot of 1 - R,/R(r) against time, from which the diffusion coefficients shown in Table V for Sample A were
calculated.
These measurements
thermal-aging
effects.
were made in descending
The
ditfusion
coefficients
order are
of temperature
plotted
to minimize
1jT in Fig. 6.
against
The activation energy for sodium chloride diffusion, calculated from the Arrhenius equation. is 12.3 kcal mole- ‘_ This experiment was repeated with Sample B, a second specimen again essentially
from linear,
the same especially
efficients shown in Table vation energy for sodium moie - t _ The
prior
thermal
history
sheet of at times
film. The plots of I - [RJR(r)] were greater than 1 hour. The diffusion co-
V for Sample B were plotted against l/T, and the actichloride
transport
was
calculated
to
be
13.1
hai
of a cellulose acetate film has an influence on its perby D, values of 5.25 x 10‘ lo and 3.00 x lo- fo
meability to salt, as demonstrated
Desdination, 3 C1968) 309-327
322
C. W. SALTOWXALL,
JR.. \V. ht. KIM? AIW
D. L. HOERNSCHEMEYER
0.6
0.1 0
SO
I!50 200
100
250
300
350
TIME. MIN Fig. 5. Salt diffusion in cekhsc
acetate at sexera temperatures.
TABLE TEUPERATURE
DEPEKDENCE
-
- ._._.----..._--Film
44 3s 2s 15
V
OF SALT
PERMEABILITY
- ._. ---. ----_^
---
_--
_-__ _
skImpIe A
61 .O 73.5 96.5 t 27.0
14.20 9.86 6.10 3.53
11.84 4.99 3.00 I.26
Film xampk B 45 35 25
83.5 113 I40
10.5 6.52 4.28
8.73 3.87 2.05
Film sampks A and B were both 81~ thick.
Desalinarion. 4 (1968) 309-327
DIFFUSION
COEFFICIENTS
RECIPROCAL
323
OF ELECl-ROf..Y-l%S
TEMP
(lO’/f),
OK-’
Fig. 6. Temperature dependence of sodium chloride diffusion coefficient in cellulose acetate.
see-t
at Z’C
smaller
Cellulose
D,
before and after heating Sample A to 44”C, together values for Sample B, which was preheated at 5o’C
acetare
With swollen
membrane anisotropic
with the generally for 10 min.
measureme~ls membranes
having
very thin active
iayers,
it
wouId be
expected hat equilibration with a salt solution would
occur very rapidly so that the final resistance could be measured almost immediately after the conductivity cell was filled with saline water Because of the rapid equilibration, permeability measurements employing various solutes, concentrations, and temperatures should be possible with a minimum expenditure of time. Experiments with swollen membranes have shown that equilibration is indeed established within a few seconds with sodium chloride solutions and that the effect of formative steps ocz the permeability of the active layer may be determined readily by this technique, as shown below. The
resistances of specimens of water-swollen membranes that had been heattreated in water at 70, 75, 85, and 90% were measured at equ~~ibrium with sodium varying in concentration from 0.1 to 1.0 M. As with dried films logarithmic plots of their resistance against solution concentration are linear {Fig. 7). shown Their slopes are -0.97, - 1.06, - 1.01, and 1.00, respectively. As previously with water-swollen, fully dense films ofcellulose acetate, slopes of essentially - 1 establish that the resistance is proportional to the salt concentration of the solution
chloride
in which
solutions
the membrane
is immersed. Desalination, 4 (I 968) 309-327
c. W.
321
SALTOXSTALL,
JR.,
W.
M. RIMi
AND
D.
t.
HOERNSCHEMEYER
too0
SOG
10 SODIUM Fig. 7. Relationship
bclwxn
CHLORIDE
clcctrical rcsistancc
of
CONC..
membrane
rnfl
and salt concentration
of solution.
the membranes exhibited esscntiatly the same resistance as before heat treatment. As shown in (Table VI where only the equilibrium resistance in 0 .Z ,!I sodium chtoride is given). the resistance after 2 hotlrs of reverse osmosis at 1500 psi with 3.5 sodium chloride is less for membranes heated at 76,85. and 9O’C. and is the same or greater for membranes heated to 7OXY or fewer. For
heat-tnxtmcnt
temIxxxturcs
below
60-65’C.
TABLE CHASCE
IS
hUMERASE
RESISTASCE
WITH
VI
NEAT
TREAT%4EXT
AND
REVERSE
OSMOSIS
Membrane resistance, ohms’ ffear-rrectmenl temperatwe, ’ C --.
.--..-
..__ ----..NOW
70 75 85 30
l
Sal: pemreatiou 0’ .‘u
FIUX .dd*’ .-.- --
----
27.3 19.8 12.0 7.6 4.4
before reverse osmosis
---.---_____
85.7 21.9 7.4 1.6 I .o
iZflN
reverse osmosfs e.-
1.7 2.2 5::: IS.7
1.8 2.7 5.Q 32.0 105.8
l Sample area of 4.36 ana. * U.S. gallons per square foot per day.
DesaIinatiott, 4 (1968) 309-327
325
DIFFUSlON COEFFICIENTS OF ELECTROLYTES
Before reverse osmosis, the logarithm
of the ratio of membrane
conductivity
to
salt concentration follows an inverse linear relationship with the heat-treatment temperature. as shown in Fig. 8; after reverse osmosis, the response has changed and is no Ionger linear. The significance of these results has not yet been ascertained. The flux of salt through a sheet. of polymer is usually taken to be proportional elcctricar resistance of a film to D&)X (3) and we have seen that the equilibrium is inversely proportional to D,K/X. It is therefore of interest to test these relations for an anisotropic gel membrane: we expect the salt flux (J2) to be inverseiy proportional to the electrical resistance (R). From the data of Table Vi, the results in Fig. 9 are obtained.
It is seen
that J,
is essentially
linear
in
l/R,
except
for
the rather
open membrane (of ?Zgl, salt permeation). Assuming that the active layer of a highly retentive membrane, free of impcrfcctions, has the same 4t permeability as a water-equilibrated, fully dense film, it would be possible to estimate the thickness of the active lager from resistance measurements if the resistance of the highly Forous substrate is negligible. The validity of the latter approximation is questionable, but its esscrltial correctness is supeested by very small resistance of the unheated membrane, which possesses a very permeable active layer. A swollen membrane, which had been heat-treated 400
10
.I5
HEAT -TREATMENT
85
00
90
TEMPERATURE.
Fig. 8. Effect of heat trcatmcnt and reverse osmosis opcmtion
lC
on electrical conductivity Desalination.
of membrane.
4 ( 1968) 309-327
226
C. W. SALTONSTALL,
i
4
,~,
JR., W. Sf. );I!%
I
i
i
i
AND D. L. H9ERSSCHEhIEYER
;--__
..__.
_ -. -
:
, ’
__--
.
.
-. _&_7
i
;
I -i-e..
,
1
2
3
6
810
20
40
60
100
200
R, okms Fig. 9. Re\rrsc-osmosis salt flux and electrical resistance of anisotropic 9 Before rcvcr% osmosis. h After t-enxse osmosis.
membranes.
for 5 min at WC. had a resistance of 25.4 ohms (over an area of 4.36 cm’) when in equilibrium with 1 1cf sodium chloride. Fully dense film, 134~ thick. with a diffusion coeffcient of 6.66 x lo- l”cm%x-’ and a distribution coeticicnt of 0.021. had an equilibrium resistance of 40.500 ohms (over an area of 4.36 cm’) in 1 M sodium chloride. Assuming that the active lsyer has the same specific resistance as fulIy dense film, ,
Rm
R,=X,
X,
(19)
where the subscripts nr and f represent membrane and fitm. From Eq. (19), the thickness of the active layer (C,) in the membrane tested was calculated to be 845 A. Before this calculation can be accepted as reliable, the basic assumption that the active layer and the “fully dense” film are identical requires further study. It seems clear that the assumption must fail for membranes heated at the lower temperatures, and one is even led to question whether heat treatment at 90°C produces a “fully Lksahation.
4 (1968) 309-327
DIFn’S~O!U COEFFICIENTS OF ELfXTR0LYTF.S
dense”
active layer.
It seems probable
than the value calculated from
Eq.
327
that the active layer will always be thicker l(9).
This work was supported by the Ofice of Saline Water, U.S. Department of the Interior, under Contract IS-01-OOOI-338. The authors thank W. R. White for his assistance in the preparation and examination of the films and membranes used in this study. REFERENCES I.
C. E. REID an E. J. RaEmw. J. Appi. Polvmer Sci.. l(l959) 133: C. E. REID ASD J. R. KUPPERS. ibid.. 2 (1959) 26-l; and REID ASD H. G. SPESCER,ibid., 4 (1960) 3%.
2.
R. E. K~STI~G. M. K. BARSII ASD A. L. VISCEXT. 1. Appl.
3.
H. K. Lo~ALE.c~u/.,~.
4.
W.
5.
H. S. HARMD ,WD R. B. OWES. The Pbpica! Chcmi.srr_vuf Eiccrra&tic Soluticms, 2nd cd., R&hold. New Yorh (i950): (a) pp. S-38. (b) p- 188, tc) pp. 328 and 5%.
6.
R. h. Rwwsos A?+ R. H. S~or;is. Ek-rrulyric Solutions. 2nd rd.. (1959): (a) p. 286. (h) p. 466. (c) p. -115. (df pp_ 314-315.
7.
S. LN.R. Aptxopriate Eletrol>tic hdditivcs in a Casting Solution Uwd for the Productton of High Rzformancc Cclfulo~c hccratc Mcmbrxws Uwd in Kcwtse Osmosis l?csalinittion. Ph.D. Thesis, Dcpartmcnt of Engiwxxing, University of California itt LOS Angclrs (Xlay 1964).
;Ippl_h~~~nrcrSci.,9(t965)
JUST. Difli.rion in .!G.did~,Liquids. Gu.ws. Audcmic
P&mer
Sri.. 9 (1%5)
I Y73.
1341. Press, NC\* York, t 19521p- 37.
Hutter\\orths, London
!ksalinatiurr, 4 ( I968 ) 309-327