Gravimetric analysis of membrane casting

Journal of Membrane Science, 11(1982) 11-25 Elsevier Scientific Publishing Company, Amsterdam

GRAVIMETRIC

ANALYSIS

I. CELLULOSE

ACETATE

11 -Printed

OF MEMBRANE

in The Netherlands

CASTING

- ACETONE BINARY CASTING SOLUTIONS

MITSUO ATAKA and KANJI BASAKI Research Institute for Polymers and Textiles, l-l-4,

fJap@

(Received February

10,198l;

Yatabe-Higaehi, Tsukuba, Ibamki 305

accepted in revised form October 29,198l)

Solvent evaporation from cellulose acetate solutions cast to a thicknees of 100-500 Irm is followed gravimetrically at 12-28”C. The desolvation curves include a region in which w (solvent/polymer weight ratio) decreases almost exponentially with time. The exponential decrease lasts, at 28°C. until 75-85% solvent has evaporated, whereas below 19°C. upward bending in the log w vs. t relation is observed after evaporation of about half the solvent. The time constant of desolvation, 7, increases with the initial acetone weight per unit area, d, as 7 a d7, where 7 = 1.4-1.6. About a threefold increase in + is observed on decreasing temperature from 28 to 12°C. Numerical Bolutions of the diffusion equation are obtained by taking time-dependence of surface solvent concentration and movement of the solution/air interface into account. The experimental facts are shown to be explicable by comparison with calculations. The effectiveness of the gravimetric approach to an under&anding, characterization, and control of membrane casting process is dlluesed.

Introduction Solution casting is one of the most convenient methods for preparing polymer membranes. Dense membranes are obtained if the solvent is evaporated from a binary solution of a polymer and a good solvent, while porous membranes can be formed if phase separation takes place in the course of casting as a result of the presence of a poor solvent or a swelling agent [l]. Higher-order structures and density of polymer molecules change gradually as the solvent evaporates, so that immersion of casting solution into a nonsolvent is sometimes carried out in order to rapidly precipitate the polymer in the solution as evaporation proceeds. Studies on cellulose acetate (CA) membranes for reverse osmosis have revealed that the membrane structure depends to a high degree on the time and rate of solvent evaporation, and that even the slightest variations in these parameters may induce great changes in membrane performance [2-71. Understanding the dynamic process of casting is, therefore, essential for the preparation of desirable and reproducible membranes. The purpose of

0376-7388/82/0000-0000/$02.75

0 1982 Elsevler Scientific Publishing Company

12

this study is to follow gravimetricallythe solvent evaporationwhich is observedwhen CA is cast from its acetone solutions. CA was chosen partly because it is one of the most frequently used membranematerials,and also because membranesof quite different porosities can be made simply by changingcastingconditions. Precisegravimetricmeasurementshave been extensivelycarriedout with organic vapors sorbed to, or desorbed from, CA films [8-lo]. Such measurements, together with their comparison with typical solutions of the diffusion equation, proceeded to yield much information on the diffusion mechanism, its kinetics, per&rant concentration distribution, and polymer structure. However, the desolvationprocess of organic CA solutions, which corresponds to membranecasting,has been measuredgravimetricallyonly with certain kinds of ternarycastingsolutions, and within a limited time range [ 2,5,11-131. In this paper, the shape of desolvation curvesand their changeswith temperature,castingthickness,and solvent/polymer ratio are reported. The diffusion equation is also solved, under appropriateboundary conditions, to interpret the experimentalresults.The evaporationof ternaryand quaternary systemswill be compared with the presentresultsin our next paper. Experimental Binarysolutions of CA and acetone (weight ratios, 1:2.9,1:3.0, or 1:5.6) were prepared. The CA was purchasedfrom Eastman(No. 4644 or 398-3 type, Lot No. A6A). Reagentgradeacetone was used as received. These solutions were cast onto a glassplate, 7 X 20 cm2 in size and around 69 g in weight, with the aid of a membrane-castinginstrument.This instrument had a motor-drivenbar which could push the glassplate at a constant speed of 2.4 cm/set, and a precisely manufacturedstainlesssteel blade, by which the solution could be spreaduniformly to a thicknessof 100,150,200,300, or 500 pm. The weight changecaused by acetone evaporationwas followed by an electronic balance (Metler,PT-150), which had an integraltime of 0.5 set, to an accuracy of +O.OOlg. In order to avoid end effects, castingwas performed on three successivepieces of glassplate, and the middle piece was used for the measurements[12]. The atmospherewas open air. The weight at t set after casting, Wt , was recorded at an intervalof 5-60 sec. The origin of time was set at the time when the middle glassplate startedto pass under the castingblade. Since it took 2.9 set for the whole plate to pass this position, the determinationof t = 0 is subject to an unavoidableuncertainty of around *3 sec. When the weight became unchanged,this weight was considered W_ . The plate was then vacuum dried at 100°C for 30 min, after which the weight became Wvn . The weight of the bare glassplate, measuredbefore casting,was Wg. The measurementswere carriedout at four temperatures:28.1 f 0.2% (relativehumidity, 65-77!%), 19.2 f 0.2% (40-58%), 16.2 f 0.2% (3244%),

13

and 12.1 f 0.3’C (36-43%). The temperatureof the castii solution and the glassplate was the same as that of the atmosphere.The thicknessof the membranesafter vacuumdrying was determinedby a micrometer to *l pm. Results Wt decreasedmonotonically with time and approached a constant value, - W, by 3-4s. The densitiesof the W_ .w,,-wwgwassmallerthanw_ resultingtransparentmembranes,calculated from the area,thickness,and W VD - W,,were 1.28 + 0.02 g/cm9. Since wvn - W, corresponds to the weight of the non-volatilemembrane material,the ratio of the solvent weight to this quantity (g solvent/g polymer), or, w = (wt -

wVD)/(wVD

-

wg)

(1)

,

used to analyse the experimentalresults.A similarequation was adopted by Kunst and Sourirajan[2] for the evaporationof ternary solutions. The value of log w, obtained at 28°C by castinga solution (CA:acetone ratio = 1:2.9) to five thicknesses,is plotted as a function of time in Fig. 1. The desolvation curvesshowed a lineartime dependence between 2.7 > w > 0.4 (except for the initialcastingthicknessof 600 pm, for which the linearity was observed between 2.7 > w > 0.8). The slope was dependent on the casting thickness. The extrapolation of this linearportion to t = 0 did not necessarilyagree with the initialacetone:CA ratio of 2.9. This discrepancy may partly come from the uncertaintyin determiningt = 0, but, in some cases, exceeds estimatedexperimentalerrors. was

0

1

2 T

1

3 M

E

4

(m

5

6

in)

Fig. 1. Desolvation curves obtained at 28.1% by carting a solution with cellulose acetate (CA):acetone weight ratio of 1:2.9. Casting thickness: 100 (e), 160 (v), 200 ( 0), 300 (A), 500 (0) pm. The quantity w b defined by eqn. (1) in the text.

0

2

1

4

3

TIME

5

6

(min)

Fig. 2. Desolvation curves obtained at 16.2% by casting a solution in which CA:acetone ratio is 1:S.O. Symbols are the same as in Fig. 1.

The desolvation curves, obtained at 16°C (with a solution whose CA:acetone ratio is 1:3.0), are shown in Fig. 2. Although linear log w vs. t relation was obtained between 2.7 > w > 1.5, the curves showed upward bending between 1.6 > w Z 0.2. This bending appeared at 19,16, and 12%. At these temperatures, the log w vs. t relation exhibited the steepest slope at around w = 0.5. After w reached 0.2, the curves once again showed a linear time dependence with much lower slope at all temperatures. The slope of this part did not vary appreciably with cast thickness. Although the time needed for solvent evaporation varied markedly with the casting thickness, the overall shapes of the curves were similar to each other at a given temperature; consequently, we could separate the curves into several parts, as shown above, by using characteristic values of w . The time constant of evaporation of the first linear part, defined as r (in set) in the expression, w = const. X exp (-t/r)

,

(2)

is plotted in Fig. 3 as a function of initial casting thickness. The thickness in this figure is expressed by the weight of acetone present initially per unit area, calculated from WvD - W,,and from the known composition of the casting solution. The reasons for expressing the thickness in this way are as follows: (1) It was impossible to control the casting thickness itself accurately and reproducibly, even by the use of the same blade and glass plate; the thickness of the resulting membranes varied by as much as +5%. Therefore, the use of the blade gap (“nominal thickness”) was inadequate. (2) The tin$.constant obtained at’“l6”C by casting a solution whose CA: acetone ratio is 1:5.6 is included in $Ii’ig.3. Since this solution contained more

0

I 20

10 WEIGHT OF ACETONE (mg/cm2)

TEMPERATURE

30 (‘C

1

Fig. 3. Time constant of desolvation (T), defined by eqn. (2), plotted against initial acetone amount per unit area. Casting temperature and acetone:CA weight ratio in the casting solution are: v (28.1% and 2.9), 0 (19.2% and 3.0), 0 (16.2% and 3.0). l (16.2% and 6.6), A (12.1% and 3.0). Fig. 4. Temperature dependence acetone weight 10.0 mg/cml.

of evaporation

rate (r-l), obtained

from Fig. 3 for initial

solvent, the evaporation was slower when cast to the same thickness. However, if an initial acetone amount was taken on the abscissa as in Fig. 3, the time constants obtained with the 1: 3 .O and the 1: 5.6 solutions fell almost on the same line. Therefore, we considered that the evaporation rate (l/7) is determined by the initial acetone weight, regardless of the solvent:polymer ratio, as long as the other conditions are kept constant. The time constant of the first linear region at 19” and 12°C is also included in Fig. 3. At all of the temperatures, the evaporation rate was strongly dependent on the casting thickness. This implies that evaporation is not governed by the solvent amount evaporable per unit time at the surface, but by the rate at which acetone is supplied to the surface by diffusion within the solution. Note that diffusioncontrolled evaporation appears only as a final stage in the drying of moist materials [14] . This observation on the nature of the rate-limiting step, that of weight decrease, is particularly interesting, because the membrane structure has been related to solvent volatility [ 51, or the surface solvent concentration gradient [ 151. The relation between 7 and the initial acetone weight, cl, was ro:dr,

where y = 1.4 (at 12”(Z)-1.6 (at 28°C).

(3)

16

The evaporation rate increased about three times as the temperature was increased from 12” to 28%. The temperature-dependence of 7-l) obtained from Fig. 3 for a virtual initial weight of 10 mg acetone/cm2, is shown in Fig. 4. We could not observe any systematic change in 7 with the relative humidity of the laboratory. The dew point temperatures, calculated from the relative humidity, were lower by more than ten degrees for the measurements at 12”) 16”) and 19” C. For the measurements at 28”C, the dew points were lower by 4.9,4.9, 7.2, 7.0, and 4.4 degrees, respectively, for thickness of 500,300, 200,150, and 100 ,um. We consider that the effect of absorbing moisture from the atmosphere on the results reported above, if any, is negligible. Model Calculation

Formulation of the problem The diffusion equation which solvent concentration in the casting solution, c, satisfies is

a c(x,t) _AL(lq!)_

(4)

at

The diffusion coefficient, D, may depend on temperature, concentration, and the state and history of the polymer--solvent system. The distance x is measured perpendicularly from the glass surface. The solution/air interface at x = s moves as evaporation proceeds (8 = s(t)). Initially, c will be equal to the composition of the prepared casting solution, co: c(x,O) = co,

Os;x-Gs(O).

(5)

Atx=O,

szo

(f-3

’

ax

for any t > 0, because no solvent is supplied from the glass plate. If c(x,t) which satisfies these equations can be found, the solvent weight we are concerned with will be

W(t) = A j@’

c(x,t) dx ,

0

where A is the area of the casting plate. From eqns. (7), (4) and (6),

dWV(t) dt The first term on the right hand side of this equation expresses the weight

(7)

17 decrease due to diffusion, while the second term corresponds to the decrease caused by boundary movement. We can derive the equation to describe the boundary movement from the additivity of the volumes of polymer and solvent: upW, + uSW(t) = As(t) ,

(9)

where up and u8 are the partial specific volumes of the polymer and solvent, respectively, and W, the weight of the polymer, all of which may be regarded as time invariants. By differentiating eqn. (9) we have dW -= dt

Ads -u, dt

(10)

’

In order to compare the experimental results with that which can be predicted from these equations, we first assume that D does not vary with time. This will not be an unrealistic approximation as long as c is large [16], and temperature of the casting solution is kept almost constant. Next, we assume that

ac

axI

=

-bc(s,t)

x=s( t)

(11)

.

This is equivalent to the condition that the surface solvent flux is determined by, and is proportional to, the surface concentration. The constant, b, is a measure of the surface evaporation rate, and is determined by the vapor pressure and thermal balance. Then, by combining eqns. (8), (10) and (ll), ds -=dt

bDc(sJ)

_

(12)

l/u, - c(s,t)

Also, from eqns. (10) and (12), dW -=dt

AbDc(s,t) 1 -u,c(s,t)

(13)

-

Analytical solution with a fixed boundary An analytical solution of eqn. (4) has been obtained under the conditions of eqns. (5), (6) and (11) (in a heat-flow problem) [17] ifs = s(t) is fixed. In this case, ds/dt = 0, and dW/dt becomes proportional to at/ax at x = s, and hence, to c(s,t). We will look into the analytical solutions first, and then study the influence of boundary movement as a deviation. In this subsection, s(0) is denoted as s. The analytical solution in an integrated form (Eq. 7) is [17] -WI

W(O)

i=CWJ =l-

c izl

2Sh2#i

4; + #@l#iCOS#i

[l - exp(+@t/s2)]

(14)

18

where & values aH2 the roots Of Cot $i = @i/b8 (0 < #I< $2 < $3 s s * )mIt is known that the “desolvation curves” are character&xl by two dimensionless parameters, a = Dt/s2 and 0 = be. The value of log W(t)/W(O) is plotted in Fig. 6 as a function of (Yfor various values of 0. The terms higher than the first one in the summation of eqn. (14)

Fig. 5. The value of eqn. (14). plotted as function of ar= Dt/s( 0)‘, for various @ = bs( 0).

Fig. 6. (a) Log-Iog plot of time con&ant of desolvation vs. @. Time constant in this figure is measured in reduced time scale, i.e. unit of time is D/s( 0)‘. Solid line (fixed boundary) and broken line (moving boundary). (b) The value of -y = 6 +2 as function of 8, where 6 is tangent of curves in Fig. 6(a).

19

contribute only when a is small; for 01> 0.2, the “desolvation curves” can be considered to take a simple exponential form (linear log w vs. t relation), with the time constant of s2/.l&p12.However, due to the terms i > 2, the extrapolation of this part to t = 0 does not agree with zero. The value of l/4 12is the time constant if time is measured in units of D/s2 sec. In Fig. 6(a), l/@~~~[18] is plotted as a function of /I in a log-log scale. When 0 is small, l/# 12decreases in almost inverse proportion with p. However, as P becomes larger, a change in l/gi 12becomes progressively less sensitive to that inp. At a fixed temperature, b and D will be constant. The dependence of the slope of the evaporation curves on casting thickness is then obtained as follows. The tangent of the curve in Fig. 6(a) is the power 6 when l/#,’ is considered to be proportional to p6 , or to Ss.6 takes the values between -1 and 0, depending on p. Since the actual time constant is obtained if the time variable is transformed from Dt/s2 to t, its dependence on 8 will be sb+2. The value of 6+2 (= y in eqn. (3)) is plotted in Fig. 6(b). It is seen that for 0 I, 2.5,~ takes the values around 1.5, which is close to the experimentally observed y. Effect of boundary movement The effect of boundary movement was evaluated numerically. For the sake of direct comparison with the above result, it is necessary first to transform the variables r, 8, t, and c into X = x/s(O), S = s/s(O), T = Dt/8(o)2, and C = c/c,. Then, eqns. (4), (6), (11) and (12) become

ac/aT=a2c/aX

(4’)

aclax=o

atX=O

(6’)

at/ax= -gc(s,T)

atX=S

(11’)

- C(S,T)).

(12’)

and dWdT = -0W,T)/(W,c,

The concentrations at T = T + AT were calculated from those at T = T by use of the difference equations corresponding to eqn. (4’). The concentrations at both boundaries were determined so as to satisfy eqns. (6’) and (11’). The boundary movement during A T (AS) was also determined from eqn. (12’). If the densities of CA and acetone are taken to be 1.28 and 0.77 g/cm3, respectively, and if their volumes are additive on mixing, then vsco of their 1:3 mixture will be 0.833. This value was used in eqn. (12’). On determining AS, the intervals between the lattice points along the Xaxis at T = T + A T were reduced by a small amount which was proportional to AS, and the concentrations at these new points were calculated by interpolation from those at the old lattice points. The new concentrations were used to calculate C and AS at T = T + 2A T. These procedures were used repeatedly from T = 0. AT was 5 X lo-‘ or 5 X lo-’ and AX was S/100. Details of the calculation will be reported separately.

20

Fig. 7. Desolvation curves, obtained by numerical calculation, for case of moving boundary. Note fivefold expansion of scale of abscissa as compared with Fig. 6.

Some results are presented in Fig. 7. Compared with the results with a fixed boundary for the same value of p, the effect of boundary movement is to accelerate a decrease in W(t)/I+‘(O). The desolvation curves obtained contained a region in which log w vs. t was almost linear. The time constant of this part is included in Fig. 6(a). The characteristic dependence of 7 on /I was similar to the result with a fixed boundary. The dependence of the tangent of this curve on p is shown in Fig. 6(b) by a broken line. The desolvation curves also showed upward bending after W(t)/W(O) reached 0.5-0.6. Solution of the diffusion equation has been obtained in connection with membrane casting by Anderson and Ullman [16], and most recently by Castellari and Ottani [19] . The former authors took into account the concentration- and timedependences of D, but ignored the boundary movement and changes in surface concentration. The latter calculation is similar to the one presented here, but it is postulated that the weight decrease is exponential, In our experiments, this was correct only until l/2-6/7 of the solvent had evaporated. Discussion In the case of vapor sorption to, or desorption from, polymer films, it is well known that gravimetric measurements are the powerful, and even almost exclusive, method with which to follow the sorption process. As long as Fickian diffusion occurs, the weight change proceeds in proportion to square root of time, and the evaporation rate varies as (film thickness)-2. It is often assumed, to a first approximation, that the vapor surface concentration takes a constant equilibrium value immediately after an experiment starts. While a maximum of only 0.2 g acetone vapor can be sorbed to 1 g CA film [ 91, the membrane casting solutions usually contain more than 3 g solvent/g

21 CA. The solvent concentration at the surface is sometimes considered to decrease continuously with time [ 3,7 ,11,20] . It has also been shown that the thickness decreases with time [7]. However, the solvent evaporation process reported so far in the literature did not permit straightforward analysis, because of the complexity caused by the presence of two or more evaporable components in the casting solution. We have presented the shape of the evaporation curves of only one component, acetone, and their changes with thickness, temperature, and solvent:polymer ratio. When the data obtained were plotted on w - t”* scale, no better correlation than Figs. 1 and 2 was obtained between w and t. At the same time, the observed thickness dependence of the desolvation rate was difficult to explain by simple analogy with sorption studies, even if attempts to account for non-Fickian diffusion [ 8-101 were considered. Therefore, in order to interpret the characteristics of desolvation, or membrane casting processes, we have carried out model calculations by taking into account the time dependence of both the solvent surface concentration and the position of the air/solution boundary. One of the simplest, and yet physically meaningful, expressions which permit the surface concentration to change gradually with time is eqn. (11). Although the shape of the experimental desolvation curves was complex, they all contained a region in which log w vs. t relation was almost linear. Such relation was predictable from the solutions of the diffusion equation with a fixed or moving boundary. The upward bending observed below 19°C is similar to the solution with a moving boundary. The bending at around w = 0.2, observed at all of the temperatures and casting thicknesses, is ascribable to gelation of CA. As normal, at a given temperature, the desolvation rate was determined by an initial solvent weight per unit area, W(O)/A, irrespective of solvent:polymer ratio in the prepared solution. The reason is that W(O)/Aisproportional to c,,, since, by definition, c,, = W(O)/As(O). Themodel used in the above calculation predicts that the desolvation rate is determined only by co for fixed b, D, and s(6). The rapid decrease in solvent weight just after casting was also simulated in Figs. 5 and 7. An alternative explanation of this phenomenon may be that the surface concentration takes an equilibrium value, somewhat lower than co, immediately after casting. However, experimental evidence seems insufficient at present for further reasoning of this rapid evaporation. The observed casting thickness dependence of T (power law with y = 1.4-1.6) was explicable from Fig. 6(b). It is expected, from this figure, that a decrease in b or s(0) tends to lower the value of 7. In fact, 7 decreased from 1.6 to 1.4 on decreasing temperature (and hence b). However, a change in 7 with s(0) was not clear within the observed range of s(0). It can also be seen from Fig. 6(b) that 1.4 Q 7 < 1.6 corresponds to 1.8 Q P G 4 (fixed boundary) or to 2 Q P < 6 (moving boundary), and that the temperature increase from 12” to 28°C is supposed to cause two- to threefold in-

22

crease in /I. The corresponding increase in the evaporation rate (Fig. 6(a)) is 50-80%. In order to explain the observed threefold increase in the evaporation rate, a change in the diffusion coefficient must also be taken into account. However, detailed analysis of Fig. 4 is difficult, because the temperature of the casting solution is not strictly constant. In the particular thermal surroundings of Ref. [ 111, and by using a solution containing CA, acetone, and formamide (25:46:30 by weight), a maximum decrease of 5.6 degrees has been reported at t = 160 set by casting 9.35 mg acetone/cm2. We can calculate D from the above values of p, Fig. 6(a), and the observed 7. D for 600 pm casting at 28°C is 2 X 10S5 (fixed boundary) or 4 X lo-‘ cm2/ set (moving boundary), which is lower than the reported values of Anderson and Ullman [16]. The reason for this discrepancy is not clear at present. Thus, although some simplification was made in the calculation, most of the characteristics observed in CA membrane casting could be explained. In order to achieve better approximation, we consider it necessary to take into account: (1) The dependence of D on concentration and temperature. (2) The dependence of D and b on the state of the polymersolvent system. (3) The non-additivity of the volumes of polymer and solvent. The advantages of gravimetric approach to membrane casting will be as follows: (1) The desolvation can, at least in principle, be expressed rigorously by the diffusion equation. (2) While there are a number of methods to investigate casting solutions and cast membranes, the process which connects these two is more difficult to study experimentally. Weight change measurement is a simple, as well as direct, method by which to follow this process. (3) The shape of the desolvation curves is primarily determined, as shown in Results, by the solvent weight content (w). This means that even in the case in which precise and complete control of every casting condition is not easy, the casting process can be described and controlled fairly accurately by gravimetry. (4) On solving the diffusion equation, information on surface concentration, concentration distribution within the solution, and the position of the air/solution interface is available. Conclusions The process of CA membrane casting has been followed gravimetrically, and the results analysed on the weight basis. The main experimental observations were as follows: (1) The desolvation curves showed linear log w vs. t relation with a time constant of 7, at least within a limited range of w, where w is the solvent: polymer weight ratio. (2) The extrapolation of this part to t = 0 sometimes gave a lower solvent :CA value than expected from the composition of the casting solution.

23

(3) At 28”C, the linear region lasted until the solvent decreased to l/4-1/7 of the initial content, whereas at 19’) 16”) and 12”C, upward bending was observed after about a half the solvent had evaporated. (4) 7 increased in proportion to the 1.4-1.6th power of the initial casting thickness. (5) A temperature decrease of as small as 16 degrees (28” to 12°C) increased 7 by about three times. (6) T was determined by the initial solvent amount per unit area, irrespective of the solvent : CA ratio. It was difficult to interpret these observations only by analogy with vapor desorption from polymer films, or with drying engineering. Therefore, the results were compared with the solution of the diffusion equation on the assumption that the surface concentration gradient is proportional to the surface concentration, and taking into account the effect of the movement of the air/casting solution boundary. All of the above experimental results were explicable by the calculations. It is considered that the gravimetric approach is useful for the characterization of membrane casting processes and for the control of the fabrication procedures of cast membranes. Acknowledgements The authors wish to thank Mr. Tadae Yamanaka and Dr. Yasuo Ressho for discussion on solutions of the diffusion equation, and Dr. Kensaku Mizoguchi for reading the manuscript. List of Symbols

A b

wm =c(mlco CW) CO D d

fw) = s(ww s(t) T = Dt/s(O)' (= t UP

us WI

wg

a)

Area of the casting plate (cm2). Proportional constant defined by eqn. (ll), and a measure of the surface evaporation rate (cm-‘). Reduced solvent concentration. Solvent concentration (g/cm’). Solvent concentration in the prepared casting solu. tion (g/cm3). Diffusion coefficient of the solvent (cm2/sec). Initial solvent weight per unit area (g/cm2). Reduced position of the air/solution interface. Position of the air/solution interface (cm). Reduced time. Time after casting (set). Partial specific volume of polymer (cm3/g). Partial specific volume of solvent (cm3/g). Solvent weight at time t (g). Weight of the casting plate (g).

24

WP wt

WV’3

Km

Weight of polymer (g). Experimentally measured weight at time t (g). Experimentally measured weight after vacuum drying (g). Experimentally measured weight after complete solvent evaporation (g) . Solvent:polymer weight ratio, defined by eqn. (1). Reduced distance. Distance measured perpendicularly from the casting plate (cm). = m/s(o)% = bs(O).

The power expressing the thickness dependence of desolvation rate (eqn. (3)). The power obtained when the time constant of desolvation in a reduced tie scale is considered to be proportional to p6 . ROOtsofCOt~i=~i/P(O<~,<~,<~,...) Time constant of desolvation, defined by eqn. (2) (set). 7-l is the evaporation rate.

s @i 7

References R.E. Kesting, Synthetic Polymeric Membranes, McGraw-Hill, New York, 1971, pp. 116-121. B. Kunst and 5. Sourirajan, Evaporation rate and equilibrium phase separation data in relation to casting conditions and performance of porous cellulose acetate reverse osmosis membranes, J. Appl. Polym. Sci., 14 (1970) 1983. G.J. Gittens, P.A. Hitchcock, D.C. Sammon and G.E. Wakley, The structureof cellulose acetate membranes for reverse osmosis, Part I, Membranes prepared from a dioxan based dope, Desalination, 8 (1970) 369. H. Strathmann, P. Scheible and R.W. Baker, A rationale for the preparation of LoebSourirajan-type cellulose acetate membranes. J. Appl. Polym. Sci., 16 (1971) 811. M.A. Frommer and D. Lance& The mechanism of membrane formation: Membrane structures and their relation to preparation conditions, in: H.K. Lonsdale and H.E. Podail (Eds.), Reverse Osmosis Membrane Research, Plenum Press, New York, 1972, p. 85. G.J. Gittens, P.A. Hitchcock and G.E. Wakiey, The structure of cellulose acetate membranes for reverse osmosis. Part 2. Transmission and scanning electron microscopy of membranes cast from an acetone-formamide dope, Desalination, 12 (1973) 315. C. Lemoyne, C. Friedrich, J.L. Haiary, C. Noel and L. Monnerie, Phyricochemicai processes occurring during the formation of ceilulose diacetate membranes. Research of criteria for optimizing membrane performance, V, Cellulose diacetate-acetonewater--inorganic salt casting solutions, J. Appl. Polym. Sci., 26 (1980) 1883. J. Crank and G.S. Park, Diffusion in high polymers: Some anomalies and their significance, Trans. Faraday Sot., 47 (1961) 1072. E. Bagiey and F.A. Long, ‘Iwo-stage sorption and desorption of organic vapors in cellulose acetate, J. Amer. Chem. Sot., 77 (1966) 2172.

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F.A. Long and D. Richman, Concentration gradients for diffueion of vaportr in glassy polymers and their relation to time dependent diffurion phenomena, J. Amer. Chem. Sot., 82 (1960) 613. 11 H. Ohya and 5. Sourirajan, Determination of concentration changes on membrane surface a~ function of time during evaporation of reverse-osmosis film casting solutions, J. Appl. Polym. Sci., 15 (1971) 706. 12 R. PiIon, B. Ku& and S. Sourirajan, Studies on the development of improved reverse osmosis membranes from cellulose acetate~cetone-formamide casting solutions, J. Appl. Polym. Sci., 16 (1971) 1317. 13 J. Vinit, J.L. HaIary, C. No&I and L. Monnerie, Physicochemicai processes occurring during the formation of cellulose diacetate membranes - Research of criteria for optimizing membrane performance - III, Evaporation rate and equilibrium phase separation, Eur. Polym. J., 11 (1975) 71. 14 W.R. Marshall, Jr. and S.J. Friedman, Drying, in J.H. Perry (Ed.), Chemical Engineers’ Handbook, 3rd edn., McGraw-Hi& New York, 1960, Section 13. 16 H. Strathmann and K. Kock, The formation mechanism of phase inversion membranes, Desalination, 21 (1977) 241. 16 J.E. Anderson and R. Ullman, Mathematical analyeis of factors influencing the skin thickness of asymmetric reverse osmosis membranes, J. Appl. Phye., 44 (1973) 4303. 17 M. Jakob, Heat Transfer, Vol. 1, John Wiley, New York, 1949, pp. 270-276. 18 M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1973, p. 225. 19 C. Castellari and S. Ottani, Preparation of reverse osmosis membranes. A numerical analysis of asymmetric membrane formation by solvent evaporation from cellulose acetate casting solutions, J. Membrane Sci., 9 (1981) 29. 20 H. Strathmann, K. Kock, P. Amar and R.W. Baker, The formation mechanism of asymmetric membranes, Desalination, 16 (1975) 179.

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