402
Ind. Eng. Chem. Prod. Res. Dev. 1084, 23, 492-500
Effect of Evaporation Time on Pore Size and Pore Size Distribution of Aromatic Polyamidohydrazide RO/UF Membranes Kam Chan, Takeshl Matsuura, and S. SourIraJan’ Division of Chemistry, National Research Council of Canada, Ottawa, Ontario K1A 0179,Canada
The average pore size and pore size distribution of reverse osmosis membranes made of aromatic polyamidohydrazMe PPPH 8273 material (abbreviated as PAH membranes) have been determined on the basis of the surface force-pore flow model using separation data for several reference solutes. The results confirm the two-normal distribution of the radius of pores on the membrane surface, and the progressive transition of the pore in the second pore distribution to that in the first pore distribution as the solvent evaporation proceeds. The analysis of the data on pore size distribution also reveals the process of formation of asymmetricity in PAH RO/UF membranes.
Introduction The control of pore size and pore size distribution on membrane surface is one of the most critical factors in RO/UF membrane design, since the selectivity of the membrane made of a given polymer material is determined by the average pore size and the pore size distribution. It is known that the average pore size and the pore size distribution of cellulose acetate membranes depend on many variables involved in the film formation process, each one of which has been identified and discussed in the literature (Sourirajan and Kunst, 1977). Particularly, the pore size and the pore size distribution of the as-cast membrane are modified in the shrinkage process, and the pore structure of the post-treated membranes is controlled primarily by the shrinkage temperature. We have studied in our previous paper (Chan et al., 1984) the shrinkage process, and determined the pore radius of the shrunk membrane as a function of the initial pore radius in the unshrunk membrane and the shrinkage temperature. In contrast to cellulose acetate membranes, there are only a few works in the literature on the study of aromatic polyamide and amidohydrazide membranes in general and on the formation of the above membranes in particular. Some of the pioneering works have been reported by McKinney (1972a,b, 1974) and by Applegate and Antonson (1972), the porous structure has been studied by electron microscopy (Panar et al., 1973), and the separations of many inorganic and organic solutes have been examined in order to establish the physicochemical criteria of such separations (Dickson et al., 1975, 1976; Matsuura et al., 1974b, 1977; Yeager et al., 1981). Most of the above works are summarized in the comprehensive review by Blais (1977). The relationship between the membrane formation and the pore structure has also been discussed (Chan et al., 1980; Fan et al., 1983). In none of the works quoted above, however, has a defmite relationship been established between the average pore size and the pore size distribution on the membrane surface and the corresponding membrane performance. Since the surface force-pore flow model (Matsuura and Sourirajan, 1981; Matsuura et al., 1981) elucidates successfully the process involved in the change of the pore size distribution during the shrinkage of cellulose acetate membranes (Chan et al., 1984), an attempt is made in this paper to apply the same technique to elucidate the change of the average pore size and the pore size distribution on the membrane surface which takes place while the aromatic polyamidohydrazide membranes are being produced. In contrast to cellulose acetate membranes, however, the 0198-4321/84/1223-0492$01.50/0
Table I. Casting Conditions of Aromatic PolvamidohvdrazidePPPH 8273 Membranes Composition of Casting Solution, wt % polymer, PPPH 8273O 14.29 Nfl-dimethylacetamide 80.95 LiCl 4.76 Casting Conditions room temperature of casting solution, OC casting atmosphere air solvent evaporation temperature, O C 95 solvent evaporation time, min variable gelation medium ice cold water membrane thickness before 4 evaporation x io4, m For polymer structure see Matsuura et al. (1977).
shrinkage proce: s is not effective in the case of aromatic polyamidohydrazide membranes and the control of the average pore size and the pore size distribution is usually performed by the change in the solvent evaporation period. The object of this paper is, therefore, to elucidate the change of the pore structure which takes place while the solvent is progressively evaporated during film formation. A major assumption is involved in this study. Since the membrane is gelled in the gelation bath immediately after the solvent evaporation step and the polymer structure in the surface (skin) layer of the membrane can be assumed to be “frozen”in the first moment of the immersion of the cast polymer solution into the gelation bath due to polymer immobilization, and since no post-treatment of the membrane is performed thereafter (except pressure-treatment), the surface pore structure of the membrane finally produced is assumed to preserve the pore structure on the surface of the incipient membrane obtained from the cast polymer solution immediately before gelation. Experimental Section
All the chemicals used for reference solutes are of reagent grade except polyethylene glycol (PEG)solutes (BDHChemicals) in the average molecular weight range of 600 to 6000 (designated as PEG number representing approximate molecular weight). The polymer membrane material used in this study is a wholly aromatic copolyamidohydrazide involving poly 8:2 (mole ratio) m:pphenylene-7:3 (mole ratio) isoterephthalamide hydrazide. The molecular structure, the details of synthesis, and the polymer characterization are given elsewhere (Matsuura et al., 1977). The details of membrane casting are given in Table I. The only variable involved in the membrane casting condition is the solvent evaporation period and all
Published 1984 by
the American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984 493
Table 11. RO Membrane Specificationa NaCl data glycerol data sepn, (PR) x 103, sew, A b x 10", RO membranes A x 107 C % kdh % m PAH-07 1.822 12.72 30.0 18.00 30.7 9.75 2.222 PAH-OS 1.179 91.4 14.16 87.4 5.32 PAH-09 1.104 1.502 94.0 13.34 92.1 4.36 0.742 1.595 PAH-10 92.5 9.32 91.2 4.54 PAH-12 0.868 0.5081 97.7 10.70 94.5 3.87 PAH-15 0.685 0.3721 98.1 8.60 95.6 3.57 "Operating pressure, 1724 kPag (= 250 psig); feed NaCl concentration, 0.06 m;feed glycerol concentration, 100 ppm; k = 22 X lo+ m/s. *Dimension of A, kg-mol HzO/m2 s kPa. (Dm/KG)N,C1,m/s X lo7, m/s.
Table 111. Some Surface Parameters Pertinent to Reference Solutes and Aromatic Polyamidohydrazide (PPPH 8273) Material D~ x 109, r / C A , b x lo'', D X lo'', B x 1030, solute glycerol solute PEG-600 PEG-1500 PEG-2000 PEG-3000 PEG-4000 PEG-6000
mol wta 92.1 600 1500
_- __
3650 6750
mz/s 1.06
m -1.70
m 2.30
Polyethylene Glycerol Solutes 0.389 -4.94 0.245 -3.09 0.214 -1.86 0.174 -0.62 0.159 -1.24 0.098 -1.24
6.27 9.95 11.43 14.06 15.34 25.00
Other Solutes methyl alcohol 32.0 1.69 ethyl alcohol 46.1 1.19 trimethylene oxide 58.1 1.06 1,3-dioxolane 74.1 1.01 p-dioxane 88.1 0.871 12-crown-4-ether 176.2 0.609 a Molecular weight of polyethylene glycol, average molecular weight.
the other casting conditions were set constant. Membranes so produced were subjected to a pure water pressure of 1172 Dag (170 psig) for 3 h and 2068 D a g (300 psig) for 2 h, respectively, prior to the UF or RO runs in order to stabilize the porous structure of the membrane. All experiments were carried out with aqueous feed solutions involving single solute at laboratory temperature (23-25 "C). RO experiments were carried out with the conventional RO cells while UF experiments were carried out with RO cells provided with thin channels for feed flow (Hsieh et al., 1979a). The details of the experimental procedure are the same as reported elsewhere (Kunst and Sourirajan, 1970). The specification of RO membranes in terms of (DM/Ka)NaCI, A, and are given in Table 11. They were determined by the Kimura-Sourirajan analysis of experimental data with sodium chloride solution at a feed concentration of 0.06 m (Sourirajan, 1970). The operating pressure used for the RO experiment with reference alcohol and cyclic ether solutes was 1724 kPag (= 250 psig), while the operating pressure used for the UF experiments with reference to PEG solutes was 345 kPag (= 50 psig). The flow rates of the feed solution were 400 cm3/min and 2200 cm3/min for RO and UF experiments respectively, at which the highest mass transfer coefficients were achieved for NaCl and PEG solutes (Hsieh, et al, 1979a,b). The feed solution concentration was kept at 100 ppm. In each experiment the fraction solute separation, f , defined as f = (ppm of solute in feed) (ppm of solute in product)/(ppm of solute in feed) and product rate (PR) and pure water permeation rate (PWP) in kg/h for given areas of film surface, 13.2 cm2and 14.1 cm2 for RO and UF experiment, respectively, were determined under the specified experimental conditions.
1.89 5.88 16.78 15.32 -1.54 -6.71
1.41 1.94 2.30 2.41 2.80 4.00
m3 -2.987 5894 2194 3313 6012 7536 29575 5.71 21.99 50.30 55.60 5.76 -1.16
The data on (PR) and (PWP) were corrected to 25 "C. Concentrations of NaCl were determined by conductance measurements, while concentrations of organic solutes were determined with a Beckman Total Carbon Analyzer Model 915-A.
Theoretical Calculation of the Pore Size Distributions. The method of calculating the average pore size and the pore size distribution on the membrane surface has already been reported (Chan et al., 1982, 1984). Briefly, the analysis is based on the transport equations derived on the basis of the surface forcepore flow model for RO/UF transport. All the necessary equations and the computation procedures are summarized in the Appendix. Furthermore, the data on the average pore radius, R b , of reverse osmosis membranes and the glycerol separation data on the basis of which R b was calculated are listed in Table 11. The data on r/CA,b, B and D which were calculated in the process of obtaining the pore size distribution are all listed in Table 111. Calculation of the Energy Change Involved in the Decrease of the Pore Size. According to our concept of the membrane pore formation, supermolecular aggregates of spherical shape are formed in the casting solution. These aggregates can be observed by electron microscopy when the polymer solution is quickly frozen (Panar et al., 1973). The interstices of the spherical polymer aggregates as well as the spaces in the polymer network which forms the surface of the spherical polymer aggregate are the two different kinds of membrane pores. The former pores are larger than the latter pores. While the solvent is being depleted from the membrane surface during the evaporation process, the spherical aggregates move toward each other and the interstices of the polymer aggregates become smaller. When the distance between two aggreagtes (the
494
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984
distance between centers of two aggregates) becomes below a threshold value, the two aggregates fuse and the interstitial pore disappears, one or several new network pores being formed in the process. In each of the steps involved an energy input into the membrane system from outside is necessary in order to supply the heat of evaporation for the solvent as well as the increase in the free energy associated with the entropy decrease due to the improved ordering of the polymer segments accompanying the approach of polymer aggregates and their ultimate fusion (Chan et al., 1984). In other words, the change in the pore size distribution can be regarded as the change in the energy level involved in the membrane system as a result of the energy input from outside of the membrane system. An attempt is made in the following to compute such a change in the energy level. Suppose there are nonumber of pores of radius (&,)O on the membrane surface at the start of the solvent evaporation step in the membrane making process and that the radius of such pores is reduced to (&)t at time t. Suppose also that n number of pores retain the initial pore radius (Rb)O at time t. Assuming an equation for the first-order change applies --dn = dt IzPn where k , is the rate constant associated with the change in pore radius. Integrating eq 1with the initial condition n = no (at t = 0) we obtain n In - = -k,t no The rate constant k can be further expressed by a preexponential factor k,8 and the activation energy t in the form (3)
Suppose 99% of the pores of the initial radius (&)O are brought down to the pore radius of (Rb)3 within 3 min from eq 3 In 0.01 = -kpo{exp(-e3/RT)j X 3 (4) where e3 is the activation energy involved in reducing the pore size from (&)O to (Rb)3. Rearranging eq 4, we obtain E~
= -RT In
{-}
+ R T In 3
t 5)
Similarly, 99% of the pores of the initial pore radius (Rp)o are brought down to the pore radius of (&)6 within 6 min, if the activation energy involved in the pore shrinkage is €6 which is given by c6 =
-RT In
{-}
+ RT In 6
(6)
Combining eq 5 and eq 6 e6
6
- e3 = RT In 3
(7)
Though, in the above calculation 99% was used arbitrarily for expressing almost entire transition from one pore size to the other, it is noteworthy that the term related to the 99% transition (In 100) has totally vanished in eq 7 .
Pore radius Evaporation time (Rb)12
Pmin
'Rb) 6
6 min
IRbj3
3 min
0. z
R
0
z
-
Figure 1. Schematic representation of the increase in energy level and the decrease in the pore radius as the evaporation time elapses. Table IV. Pore Size Distribution of Membranes &,I x 10'0, El/ R , , x 10'0, (yz/ membranes m Rb,l m Rb,2 UF Membranes' PAH-03 35.67 0.29 105.9 0.31 PAH-05 29.27 0.32 104.9 0.33 PAH-06 20.87 0.001 115.9 0.23
h2 0.046 0.016 0.004
RO Membranesb PAH-07 6.07 0.001 48.87 0.23 0.011 PAH-08 5.98 0.001 PAH-09 4.62 0.001 PAH-10 4.14 0.001 PAH-12 3.79 0.001 PAH-15 3.62 0.001 a On the basis of PEG reference solutes. On the basis of reference solutes other than PEG.
Therefore, eq 7 implies that the pores of the initial radius (Rb>o were reduced to (R& in the first 3 min and they were further reduced to (Rb)6 in the next 3 min, and the energy level of the latter pore is higher than that of the former pore by R T In (6/3). The shift in the pore radius and the energy level involved in the foregoing process is illustrated in Figure 1. In the foregoing treatment, the activation energy involved in pore size reduction is equated to the change in thermodynamic free energy; this approach is based on linear free energy relationships (Taft, 1956) which have proved successful in physical organic chemistry. Results and Discussion The pore size distributions of the membranes tested were determined following the method described in the Appendix and the results obtained are listed in Table IV. The pore size distributions of PAH-03,05, and 06 membranes were determined by using the PEG solutes as reference solutes and classified as UF membranes, while those of PAH-07, 08, 09, 10, 12 and 15 were determined by using alcohols and cyclic ethers as reference solutes and classified as RO membranes. This classification is arbitrary, and there is no overriding reason for splitting the membranes into RO and UF membranes. Both experimental separation data and the data calculated on the basis of the pore size distributions given in Table IV are listed in Table V. The agreement is reasonable enough to consider that the numerical data obtained on pore size distributions (Table IV) are valid on a relative scale. Figure 2 illustrates the integral pore size distributions of membranes PAH-03 and PAH-06 based on eq A-3 (see Appendix) using the numerical values listed in Table IV. As explained earlier (Chan et al., 1984), the pore of 153 X 10-lom (point A) after 3 min evaporation was reduced to 118 X m (point B) after 6 min evaporation. This conclusion is based on the following two assumptions. (1) The total number of the pores does not change during evaporation. (2) When there are two pores of different sizes, the larger pore remains larger and the smaller pore
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984 405 100
,
E
0-
-
0
x 100 I
I
I
Evaporation time, min
/
/' L
c
2
I
r "-
I
W
,
I
I
8
I
L
&2E
I
I
90 -L 80 50
TRANSITION
I
I
100
150
0
x io'? m Figure 2. Integral pore size distribution of PAH-03 and PAH-06 membranes. PORE RADIUS
I
X
I
A 0
OF PORES
2 to 5 5 10 6 3 to 6 3 lo 7
min
min min
mm
A
CVArUMA
-2
'
I
I
1
1
50
100
150
200
(Rb)x 101O,m
Figure 5. Free energy vs. pore radius at various steps of solvent evaporation period.
V 0
I
50
I
100
I
J
150
200
( Rb)3 x 10I0,m
Figure 3. Pore radius after 5 and 6 min of evaporation time vs. pore radius after 3 min of evaporation time.
remains smaller after the evaporation process, in other words, the solvent evaporation process does not alter the order of the pore size. A note is in order on the first assumption. The preservation of the number of pores may not be necessarily true, particularly when a large pore of the second distribution turns into the smaller pore of the first distribution. When a large pore (aggregate pore) disappears, several small pores (network pores) might arise. However, since the number of pores belonging to the second distribution is so small (at most a few percent), such an increase in the number of pores is relatively insignificant. According to the aforementioned principle, therefore, the pairs of radii, (R& and (&,)e, were obtained and they are plotted in Figure 3. (&)3 vs. (Rb)5 is also shown in Figure 3. Similar results with respect to (Rb) values at the evaporation time of 7 to 15 min vs. (&,)3 are shown in Figure 4. Let us now generate the free energy curve associated with the pore structures of PAH membranes. Figure 3 is used for this purpose. Let us start from (Rb)3of 165 x W0 m (point A in Figure 3) and assume that the energy involved in this pore is the datum energy level, thus setting a scale for the free energy levels corresponding to different pore sizes. According to this scale we plot (0 kJ/mol, 165 X 10-lo m) on the free energy vs. pore radius diagram shown in Figure 5 as an open circle (point A'). At evaporation period of 6 min this pore is reduced to the pore
of 143 X 10-lo m (point B in Figure 3) and the elevation of the free energy level accompanying the pore size reduction is RT In 6/3 = 2.13 kJ/mol. Therefore, we plot (2.13 kJ/mol, 143 X m) in Figure 5 (point B'). Further, 20 X 10-lo m of (&,)e (point D in Figure 3) corresponds to 143 X 10-lo m of (&&,. A further free energy increase of 2.13 kJ/mol is expected in this pore size reduction. Since the pore of 143 X lO-'O m already possesses the free energy of 2.13 kJ/mol, the total of 2 X 2.13 = 4.26 kJ/mol corresponds to the pore of 20 X m. Therefore, m) in Figure 5 (point D'). we plot (4.26 kJ/mol, 20 X Thus we can obtain points A', B', and D' on the free energy vs. pore radius diagram. However, there is a wide blank region between B' and D' and we have to fill this region. Let us choose arbitrarily 103 X lo-'' m of ( & , ) 6 (point F in Figure 3) which corresponds to 150 X 10-l' m of (Rb)3 (point E in Figure 3). Going back to Figure 5, the free energy of 1.6 kJ/mol (point E') is found to correspond to 150 X m by interpolation between points A' and B'. Then, the reduction of the pore radius from 150 X 10-lo m to 103 X m elevates the potential energy to 1.60 + 2.13 = 3.73 kJ/mol. Thus, we can plot (3.73 kJ/mol, 103 X m) in Figure 5 (point F'). (In fact, the procedure of finding positions for points E' and F' is iterative. Their positions are set so that points A', E', B', and F' form a smooth line.) Figure 3 further shows that pore radii are reduced rather moderately until (&)3 reaches 145 X m (point G in Figure 3) and then (&)e is abruptly reduced from 75 X 10-lo m to 20 X 10-lom (from point H to point I) when (Rb)3 = 145 X m, no pore being available as between the points H and I. Again, the free energy of 2 kJ/mol can be assigned to the pore radius of 145 X m by interpolation of Figure 5 (point G') and, m and 20 x therefore, both pores with radii of 75 X m possess the free energy of 2.0 + 2.13 = 4.13 kJ/mol, which are plotted in Figure 5 as the points H' and 1'. It
496
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984
Table V. Comparison of Experimental and Calculated Solute Separation PAH-03
PAH-05
PAH-0 6
pure water permeation rate x
results for U F membranes at feed concn, 100 ppm and operating pressure, 345 kPag (= 50 psig)
169.2
l o 3 ,kg/h
37.4
10.4
solute separation, %
re€ solutes
exptl
calcd
exptl
calcd
exptl
calcd
PEG-600 PEG-1500 PEG-2000 PEG-3000 PEG-4000 PEG76000
1.1 8.5 14.0 20.0 28.0 33.5
1.7 8.6 15.6 23.1 24.3 32.8
11.7 48.0 60.1 71.0 74.0 85.6
5.0 49.8 63.1 70.7 72.5 85.6
65.4 83.5 84.0 84.5 85.8 92.9
65.5 83.2 84.1 85.2 85.9 92.4
PAH-07
PAH-08
PAH-09
pure water permeation rate x
results €or RO membranes at feed concn, 100 ppm and operating pressure, 1724 kPag (= 250 psig)
14.72
l o 3 ,kg/h
16.70
16.49
solute separation, %
re€ solutes
exptl
calcd
exptl
calcd
exptl
calcd
methyl alcohol ethyl alcohol trimethylene oxide 1,3-dioxolane p-dioxane 12-crown-4 ether
4.7 18.6 26.7 29.2 40.0 44.9
3.1 19.3 24.8 26.7 42.4 44.8
9.6 37.9 53.0 56.2 95.2 100.0
6.9 40.6 53.6 58.2 91.6 99.3
16.5 49.9 64.5 68.5 97.2 100.0
21.4 52.4 60.8 67.1 95.9 100.0
PAH-10
5
PAH-12
PAH-1
pure water permeation rate x
results €or R O membranes at feed concn, 100 ppm and operating pressure, 1724 kPag (= 250 psig)
10.87
l o 3 ,kg/h
13.07
10.39
solute separation, %
re€ solutes
exptl
calcd
methyl alcohol ethyl alcohol trimethylene oxide 1,3-dioxolane p-dioxane 12-crown-4 ether
16.5 54.7 69.3 73.9 95.8 100.0
27.4 57.6 66.3 71.0 97.3 100.0
should be noted that there is a discontinuity between the points H and I on Figure 3; therefore, the points H’ and I’ should not be connected in Figure 5. Further, (R& of 75 X 10-lo m (point J in Figure 3) is reduced to (R& of 20 X 10-lom (point K in Figure 3), and therefore the free energy of the pore corresponding to the point K is 4.13 + 2.13 = 6.26 kJ/mol. This is plotted in Figure 5 as the point K’. Note that the points 1’, D’, and K’ form a vertical line. Thus, two discrete free-energy curves were produced on Figure 5: one by connecting points A‘, E’, G’, B’, F’, and H’ and the other connecting the points 1’, D’, and K’. These free-energy curves remind us of the typical freeenergy diagram in which the free energy is plotted vs. the distance of two polymer walls (see Figure 10 of Chan et al. (1984)). The vertical line 1’-D’-K’ of Figure 5 corresponds to the steep potential wall due to the steric repulsion working between two polymer walls, while the line A’-H’ forms the part of the potential barrier caused by the entropy change associated with the change in the distance of polymer walls (Chan et al., 1984). It is further suspected that there is a potential well between the points H’ and I’ which region is unable to be reached in the solvent evaporation process. The method described above is not limited to the shift in pore radii from evaporation period of 3 to 6 min. The same method was applied to the evaporation period of 5 to 6 min and we obtained the data indicated by open triangles in Figure 5. These data are close enough to the ones indicated by open circles, demonstrating that an almost identical free-energy curve can be generated from
exptl
calcd
exptl
calcd
17.2 54.7 70.5 73.9 97.9
30.7 59.2 69.9 75.2 98.1 100.0
21.3 56.1 70.5 76.2 98.2 100.0
33.7 61.3 71.5 76.5 98.6 100.0
100.0
different combinations of evaporation times, provided the evaporation times considered are sufficiently close to each other. The same technique wtu applied further to the shift in the pore size distribution during the evaporation period of 3 to 5 min, 3 to 7 min, and 3 to 15 min, and the results are illustrated in Figure 5 together with the shift involved in the evaporation period of 3 to 6 min. As it is clear, we obtained different lines corresponding to different evaporation times. In order to interpret the free energy function curves such as those illustrated in Figure 5, we go back to the origin of the free-energy curve and decompose it into three elements. As illustrated in Figure 6, the free energy curve is composed of (1) the element representing the steric hindrance, (2) that representing the van der Waals attractive force, and (3) that representing the entropy decrease induced by the decrease in the distance between polymer walls. We are dealing here with the overall A S changes arising not only by the ordering of the relatively small number of supermolecular polymer aggregates but also by the ordering of a far greater number of polymer segments in the aggregates accompanying the closer approach and/or fusion of polymer aggregates. The A S contribution with respect to the former change is relatively small, but that with respect to the latter change is considerable. When a free-energy curve is synthesized by the combination of the first and the third elements we obtain the potential curve a (Figure 6); when the first and the second elements are synthesized we obtain the potential curve e;
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984 497 DECREASE IN T H E SIZE OF AGGREGATE PORE
a
0 STERIC HINDRANCE /, CONTRIBUTON
AG
I F\ '.
DECREASE IN T H E S I Z E OF NETWORK PORE
,f
ENTZPY
' 5 CONTRlbUTlON
l
l
l
--./-I
\
I
\AG
I
@I
Ol
Steric hindrance increase
@
Figure 8. Schematic diagram of the decrease in the size of the aggregate pore and that in the size of the network pore. r
Figure 6. Free energy curves synthesized by the various combinations of steric hindrance, van der Waals force, and entropy effect.
1 1 IF\
J, Steric hindrance
-
/&I
AG
I
I
INCREASE IN EVAPORATION TIME
Figure 7. Transition in the free energy curve as the function of evaporation time.
when all three elements are equally contributing we obtain the potential curve c. In between we can also generate potential curves b and d (Figure 6) depending on the degree of the importance of each element. Suppose all steps indicated by the above free energy curves b to e are passed through as the evaporation time elapses; the transition in free energy functions can be shown as a function of evaporation time as illustrated in Figure 7. The group of free energy curves illustrated in Figure 5 indicates a clear similarity to the ones illustrated in Figure 7. The experimental data, however, show only a part of the functional curves, since the potential well is not susceptible to quantitative analysis by our method. The transition of the free energy curve during the solvent evaporation step is understandable in the following way. Let us consider that there are two kinds of pores on the membrane surface. One is formed as an interstitial space between supermolecular aggregates which are often "seen" as spherical nodules by electronmicrography (Pans et al., 1973; Schultz and Asummaa, 1970). The other is the space formed in the network of the polymer segments themselves in each aggregate. The illustration shown in Figure 8 is the pictorial representation of such pores. The decrease in the size of the former pore, which may be called the aggregate pore, is enabled by the closer approach of the polymer aggregates which surround the pore. When the polymer aggregates approach each other, the bending segments of the polymer molecules have to be realigned so that the order of polymer molecules is increased. As
a result an entropy decrease takes place, and AS < 0. As the pore size decreases progressively entropy decrease is enhanced, leading to the higher contribution of -ThS to the total free energy of the system. When the aggregates approach even closer the steric hindrance increases enormously, particularly when aggregates come in contact with each other. In this process the van der Waals attractive force is overshadowed by the contributions from the entropy effect (3) and the steric hindrance (1)and we obtain the free energy curve such as that represented by Figure 6b. In the case of the pore formed in the network of the polymer segments, which may be called network pores, the polymer segments are, at least locally, well ordered. The approach of the polymer segments, by which the pore size reduction is enabled, does not ephance the order of the polymer segment; therefore the entropy change is practically zero. On the other hand, the approach of polymer segments in the range of D)
(A-9)
Equation for the friction (Matsuura et al., 1981) b= (when Xf 5 0.22) 1/(1 - 2.104Xf + 2.09XB - 0.95Xf5) b = 44.57 - 416.2Xf + 934.9X?
302.4Ap
+ (when Xf > 0.22) (A-10)
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984 499
where
& = D/Rb
(A-11)
Equation for the radial velocity profile (Matsuura and Sourirajan, 1981) d2UB(r)
dr2
1 AP 1 RT + -v -kA3(r) q 6 6
+ -1dUdr) -+-r dr
- cA2)
with the boundary conditions atr=O duB(r)/dr = 0 a t r = R6 - Dwater
UB(r) = 0
(A-13)
All the symbols are defined at the end of the paper. The method of obtaining f for given values of B, D, and the pore size distribution ( & , i ui, t and hi) under the given operating conditions of the operating pressure, the feed concentration, and the flow rate is given in our previous work (Matsuura and Sourirajan, 1981; Matsuura et al., 1981). In order to apply eq A-1 to eq A-13 we need B and D values, which are determined by using the specific surface excess, r/CA,b, obtained from liquid chromatography data on retention volumes (Matsuura et al., 1982). The necessary equations are IVR’1A r/CA,b
=
- [VR’IDZO
(A-14)
A,
The numerical values B and D for glycerol and polyethylene glycol solutes were obtained by setting D equal to the Stokes law radius of the above solutes and using eq A-14 and A-15. The method of determining the average pore radii (average of all the pores on the membrane surface) of reverse osmosis membranes using RO separation data of glycerol on the basis of B and D values obtained above has been described elsewhere (Matsuura et al., 1981). The separation data of glycerol and R b data so obtained are listed in Table 11. Furthermore, the method of calculating B and D values for other reference solutes including methyl alcohol, ethyl alcohol, trimethylene oxide, l,&dioxolane, p-dioxane, and 12-crown-4ether on the basis of the chromatographic retention volume data and the separation data with the RO membrane of the smallest &, is given in our earlier work (Matsuura et al., 1981). The numerical values of r / C A , b , B, and D so obtained for above solutes are all listed in Table 111. Once interaction force constants B and D become available, the pore size distribution of a membrane can be calculated from the separation data of the given set of reference solutes so that the pore size distribution minimizes the quantity sum =
c(fj.cdcd I
- fj,espd2
(A-16)
where fj,dd and fj,exptldenote the calculated and the experimental separation data of j t h reference solute. The procedure involved in the computation is given in our previous work (Chan et al., 1982, 1984). O n the Free-Energy Curve as a Function of Distance between Two Polymer Walls. The free-energy curve and its components, illustrated in Figure 6, are based on the free energy change of the polymer phase only. Since
the membrane formation process involves the process of demixing of solvent from the polymer segments, one might ask whether a similar free energy curve is applicable when both polymer and solvent are considered in the process of polymer-solvent demixing. The answer to this question is affirmative due to the following considerations. The polymer-solvent demixing process during solvent evaporation can be thermodynamicallysplit into two steps; the first one is the separation of a polymer solution into a more concentrated polymer solution phase and a pure solvent phase, and the second one is the evaporation of solvent from the pure solvent phase. The latter step does not contribute to the free energy change of the overall process because, during this step gas-liquid equilibrium is maintained and the free energy change involved in the process is zero. Therefore the overall free-energy change is associated entirely with the first step, where greater extent of polymer-solvent demixing (or polymer concentration) is associated with smaller polymer-polymer distance, and therefore d in Figure 6 decreases. The polymer-solvent demixing process leads to entropy decrease, since improved ordering of polymer segments is necessary when polymer molecules are assembled in a small space (Figure 8). With greater extent of polymer-solvent demixing, the magnitude of entropy decrease may be expected to become higher. Therefore the form of the entropy contribution as a function of d, shown in Figure 6, is also valid when both polymer and solvent are regarded as a single thermodynamic system. Further, as the polyme-olvent demixing proceeds, collisions between polymer aggregates and also between polymer segments in the ag, gregates may be expected to occur when the quantity d becomes smaller than some critical value characteristic of the system. Consequently, the form of the steric hindrance contribution shown in Figure 6 is also valid for the polymer-solution system as a whole. It has to be noted that the contribution of solvent phase to the entropy and steric hindrance factors is relatively small compared to that from the polymer phase due to the high rigidity and significantly smaller size of the solvent molecule. The question is therefore focused on the enthalpy contribution (AH) which is indicated in Figure 6 as the van der Waals contribution. In the presence of the solvent, AH can be either positive (endothermic demixing) or negative (exothermicdemixing) depending on the sign and relative magnitude of the polymer-polymer, polymer-solvent, and solvent-solvent interaction forces. However, the enthalpy of polymer demixing is usually negative, which is indicated by expressing the enthalpy of mixing (the reverse of demixing) as the square of the difference in solubility parameters of polymer and solvent (van Krevelen, 1976). Therefore, the enthalpy of mixing is positive and consequently that of demixing is negative. Further, the enthalpy decrease is enhanced at a higher degree of polymer-solvent demixing as can be shown by the Scatchard-Hildebrand equation (Hildebrand and Scott, 1950). From the foregoing considerations, one can conclude that the form of the free energy change shown in Figure 6 for the polymer phase only is also valid for the polymer-solution system as a whole in the process of polymer-solvent demixing. Nomenclature A = pure water permeability constant, kg-mol/(m2 s kPa) A,, = the surface area of polymer powder in the chromatography column, m2 B = constant characterizing the van der Waals attraction force, m3 b = frictional function cA2 = molar concentration of the solute at the pore inlet, mol/m3
500
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984
cA3(r)= molar concentration of the solute at the pore outlet, (a function of radial distance r ) , mol/m3 CA,b = bulk solute concentration, mol/m3 D = constant characterizing the steric repulsion at the in-
terface, m Dwater. = molecular radius of water (0.87 X m) d = distance between polymer material surface and the center of the solute molecule, m d = distance between two polymer walls, m D m = solute diffusivity, m2/s (DAM/Fb)NaCI = transport parameter of reference sodium chloride in water (treated as a single quantity), m/s f = fraction solute separation based on the feed concentration f’ = fraction solute separation based on the solute concentration in the boundary phase g = density of solution, kg/m3 hi= ratio defined by eq A-2 k = mass transfer coefficient for solute on the high-pressure side of the membrane, m/s k , = rate constant associated with the transition of the pore radius before evaporation to that after evaporation, l / s nj = number of pores belonging to the ith normal distribution nT = total number of pores n = number of pores with radius (R& at the evaporation time 1
L
n, = number of pores with radius (&,)O at the evaporationtime -0 hp = operating pressure, kPag (PR) = product rate through given area of membrane surface,
kg/h (PWP) = pure water permeation rate through given area of membrane surface, kg/h i?b = pore radius, m R b = average pore radius, m Rb,i = average pore radius of the ith distribution R = gas constant r = radial distance in the cylindrical coordinate of the pore, m A S = change in entropy, J/mol K sum = quantity defined by eq A-16 T = absolute temperature, K t = evaporation time, s uB(r)= velocity of water in the pore as a function of r, m/s [ VR’IA= chromatography retention volume of solute A, m3 v, = permeation velocity of product solution, m/s Yi(&,) = normd pore size distribution function, l / m Greek Letters
r = surface excess of the solute, mol/m2 6 = length of cylindrical pore, m tt = change in the energy level involved in the reduction of the pore radius (&)O to ( & , ) t ,J/mol 7 = solution viscosity, Pa s A, = quantity defined by eq A-10 ui = standard deviation of ith normal pore size distribution, m
potential function of interaction force exerted on the solute from the pore wall, J/mol xm = proportionality constant between solute velocity relative to water and friction force working on solute, (J s)/(m2mol) Registry No. (Isophthaloyl chloride).(terephthaloyl chloride).(m-aminobenzhydrazide).(p-aminobenzhydrazide) (copolymer), 41706-75-6. $ =
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Received for review September 19, 1983 Revised manuscript received March 2 , 1984 Accepted March 12, 1984
Issued as N.R.C. No. 23346.
