Ind. Eng. Chem. Prod. Res. Dev. 1985, 2 4 , 655-605
655
Viscoelastic and Statistical Thermodynamic Approach to the Study of the Structure of Polymer Film Casting Solutions for Making RO/UF Membraned T. D. Nguyen, Kam Chan, Takeshl Matsuura, and S. SourIraJan' Division of Chemistry, National Research Council of Canada, Ottawa, Canada K1A OR9
Using viscometric data, the size of the supermolecular polymer aggregate of poly(m-phenylene-iso(70)-co-tere(30)-phthabmMe)(abbreviated as PA) polymer in the solution of filmcastlng composition was determined at various temperatures with respect to polymers of different molecular weights. On the basis of the size of the aggregate so determined and the dlstrlbutlon of aggregates at the solution surface obtained from a statistical thermodynamic analysis, the size of pores, called aggregate pores, that originate from the space devoid of aggregates was estimated; further, on !he basis of a similar analysis the size of another type of pores, called network pores, existing in each aggregate was also estimated. The sizes of the above two kinds of pores were found comparable to those obtained by analysis of experimental RO data on the basis of the surface force-pore flow model.
Introduction A transport model, called the surface force-pore flow model, was developed in our previous work (Matsuura and Sourirajan, 1981; Matsuura et al., 1981; Sourirajan, 1983a) as a quantitative expression of the preferential sorptioncapillary flow mechanism of reverse osmosis (Sourirajan, 1970). In the above model, the interaction force working between membrane polymer material and solute in water environment and the average pore size and the pore size distribution on the membrane surface are incorporated in transport equations in order to obtain data on membrane performance such as solute separation and membranepermeated product rate under a given set of experimental conditions. When the transport equations arising from the above model are applied to experimental ultrafiltration (UF) and reverse osmosis (RO) data, it has been shown that two distinct Gaussian normal distributions of pore sizes on the membrane surface are required to predict the experimental separation data of the reference solutes used in the UF and RO experiments (Chan et al., 1982; 1984a,b; Liu et al., 1984; Nguyen et al., 1984). Further, it has also been shown that the reduction in the average pore size, either by the hot-water shrinkage procedure in the case of cellulose acetate membranes or by the solvent evaporation procedure in the case of the aromatic polyamide membranes, is actually the process of transforming some of the pores in the second distribution to those in the first distribution; and, in this transformation procedure, the potential barrier due to the entropy reduction (of the membrane polymer system) accompanying the pore size reduction has to be overcome (Chan et al., 1984a). That pore size distributions have significant effects on UF membrane fouling and on solute fractionations in RO has also been shown (Liu et al., 1984; Matsuura and Sourirajan, 1983). The object of this paper is to discuss the origin of these two distinct pore systems on the membrane surface. In an earlier work (Sourirajan, 1983b), it has been suggested that there are indeed two kinds of membrane pores, each having a distinct origin. The first kind, called the aggregate pore (represented by the second distribution), origiIssued as NRC No. 24893. 0196-4321/85/1224-0655$01.50/0
nates from the interstitial spaces surrounded by polymer aggregates; the other kind, called the network pore (represented by the first distribution), originates from spaces between polymer segments within each polymer aggregate. This paper initiates a process of inquiry on the physicochemical basis of the above concept of membrane pores. In this work, the size of the polymer aggregate in the polymer solution of the film casting solution composition was determined by viscosity measurements and the results compared with the average size of the aggregate pores indicated by the analysis of RO/UF experimental data. A similar comparison of the average size of the network pore with that of the space surrounded by polymer segmenta within the aggregate is not possible because of the difficulty in determining the distance between such adjacent polymer segments. Therefore, a statistical thermodynamic approach was undertaken to study the dispersion of water in polymer networks in order to inquire into the possibility of the existence of such network pores within the polymer aggregate, particularly when they are in equilibrium with aqueous environment. The analysis shows that the existence of two distinct origins of pores on the membrane surface is indeed possible; further, the analysis also revealed how the variables involved in the membrane formation process, such as molecular weight of the polymer and the casting solution composition, could affect the pore size and pore size distribution on the surface of the resulting membrane. Thus this paper presents a new approach to the study of the structure of the polymer film casting solutions for making RO/UF membranes. Experimental Section Polymer Synthesis, Viscosity Measurements, and Liquid Chromatography Experiments. The polymer (poly(m-phenylene-iso(70)-co-tere(30)-phthalamide), abbreviated as PA hereafter; see Figure 1) samples of different molecular weights were laboratory-synthesized according to the method reported elsewhere (Gan et al., 1975). The viscosity measurement of the polymer was conducted by means of a Cannon-Ubbelohde viscometer; the capillary size was chosen so that kinetic energy corrections were not necessary. The temperature was controlled to f0.2 OC, and flux times were reproduced to &0.1%. The polymer concentration ranged from 0.1 to 0.5 g/dL of solvent. Intrinsic viscosities were determined by Published 1985 American Chemical Society
656
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985 Table 11. Intrinsic Viscosity of Polymer Solutions and Molecular Weight of Polymer Aggregates
Figure 1: Structure of the repeat unit of poly(m-phenylene-iso(70)-co-tere(30)-phthalamidepolymer). Table I. Characterization of Reverse Osmosis Membranes Studieda film PA-1 PA-2 PA-3 PA-4 pure water permeability constant A X 0.511 0.294 0.570 0.533 IO7, kmol of HzO/(m2wkPa) solute transport parameter DAM/K6X 2.555 2.113 1.807 1.284 io7, m/s sodium chloride data 60.2 76.3 86.7 92.5 soIute separation, % product rate X lo3, kg/h 5.40 3.95 6.95 6.80 Operating pressure, 1724 kPag (=250 psig); feed molality, 0.06.
extrapolation of qsp/c to zero concentration. The plots were linear, with correlation coefficients of more than 0.98 in all cases. The liquid chromatography experiments in which the above polymer and water were used as the column packing material and solvent, respectively, were conducted with respect to all reference solutes involved in this study. The details of LC experiments have been reported elsewhere (Matsuura et al., 1982). Membrane Formation and RO Experiments. The aromatic polyamide membranes were produced from polymers of different molecular weights according to the method reported in our previous work (Matsuura et al., 1974). The casting solution composition was polymer 12.5 w t % , dimethyl acetamide 83.7 wt % , and calcium chloride 3.8 w t %. Solvent evaporation temperature was 95 "C, and solvent evaporation period was 9 min. After solvent evaporation the membrane was immersed in ice-cold water for gelation for more than 1 h. In RO experiments the separations of reference solutes were tested by all the membranes produced above. The apparatus and experimental procedure used were the same as those reported earlier (Kunst and Sourirajan, 1970). The specifications of membranes used are given in Table I in terms of pure-water permeability constant A (in kmol of H20/ (m2.s.kPa)) and solute transport parameter (Dm/KS) for sodium chloride (m/s) a t 1724 kPag (=250 psig). Table I also includes solute separation and product rate data for the membranes used at the operating pressure of 1724 kPag (=250 psig) with 3500 ppm NaCl-H20 feed solution at feed flow rates corresponding to a mass-transfer coefficient of 22 x 10" m/s on the high-pressure side of the membrane. All experiments with respect to organic reference solutes were performed at the feed concentration of 100 ppm. All experiments were of the short-run type, and they were carried out a t the laboratory temperature (23-25 "C). The (PR) and (PWP) data used in calculation were then corrected to 25 "C by using the relative viscosity and density data for pure water. The terms "product" and "product rate" refer to membrane-permeated solutions. The fraction solute separation f obtained in each experiment was calculated from the equation solute ppm in feed - solute ppm in product f= solute ppm in feed In each experiment, (PWP) and (PR) in grams per hour per given area of film surface (13.2 cm2) and f were determined a t the operating conditions employed. The concentration of organic solutes in feed and product solutions was determined by carbon analyzer (Beckman Model 915-A), and sodium chloride concentration was
polymer PA-1 PA-2 PA-3 PA-4 PA-5
96 % H2S04 0.6928 1.1303 1.2534 1.7063 1.7972
+
DMA 0.6161 0.8552 0.9672 1.3855 1.4722
DMA CaClz 1.0302 1.8800 2.0452 3.1052 4.1896
M, 10500 18400 20700 29500 31300
(M,,)C~C~~ n 22500 2.14 47850 2.60 53030 2.56 90330 3.06 129090 4.12
"Data at 20 "C. *CaC12to polymer weight ratio = 1:3.3.
determined by conductivity bridge.
Theory Determination of Molecular Weight of Polymers. The intrinsic viscosities, [77] of five polymer materials (PA-1, PA-2, PA-3, PA-4, and P A 4 were determined in 96% sulfuric acid solution a t 20 "C. Since [77] vs. number-averaged molecular weight, M,, relationship is reported by Leibnitz et al. (1979) with respect to the same structure of the polymer repeat unit, the above relationship was used to determine the molecular weight, M,, of the laboratory-synthesized polymers by assuming the molecular weight distributions of polymer samples used in this work are the same as those used in Leibnitz et al. Both [77] in sulfuric acid solution and the resulting M, values are given in Table 11. Determination of the Number of Polymer Molecules in a Polymer Aggregate. The intrinsic viscosity of all the polymer materials (from PA-1 to PA-5) was measured in dimethyl acetamide solution (DMA) at 20 "C and was correlated to the molecular weight according to the Mark-Houwink equation
[VI
= KMn"
(1)
Therefore, K and a values are known for polyamide polymers in DMA solvent. Then, the intrinsic viscosity of the polymer solution in DMA solvent was measured in the presence of calcium chloride. While making the solution of different polymer concentrations, the ratio of the weight of calcium chloride to that of the polymer was maintained constant as k3.3, which is the same as that in the casting solution. Assuming that K and a values determined for polymer-DMA solutions in the absence of CaC12are valid also in the presence of very low concentration of CaC12, the apparent molecular weight of the polymer aggregate designated as (Mn)CaClz was obtained. Since we know the molecular weight, M,, for each polymer sample, we can calculate the number of polymer molecules in one polymer aggregate as The same procedure was repeated at temperatures 30,40, 55, 70, and 90 "C. The quantity, n, as well as (M,)c,cl, obtained above are assumed to remain constant as the polymer concentration in the solution changes. Size of Polymer Aggregate in the Membrane Casting Solution. The determination of the aggregate size of the polymer solution with casting solution composition was made according to the method developed by Rudin and Johnston (1971) with a modification with respect to the effective volume factor a t the limiting concentration t, based on Johnston and Sourirajan (1973). According to Rudin and Johnston the radius of the spherical particle in solution (equivalent to the radius of the spherical polymer aggregate) can be calculated by
Ind. Eng. Cham. Prod. Res. Dev., Vol. 24, No. 4. 1985
(S) = (
3 ’ 3
657
ASSEMBLY 0 ~ 1 3 ISOLATED FILLED I\CI\NT CUBES VACANT CUBE CUBE
(3)
where u is the volume of an unsolvated polymer molecule and t is the effective volume factor. Further u can be written as u =MdpN0
(4)
where M A is the molecular weight of polymer which is approximated in this work by M.. The quantities p and No are the amorphous polymer density and Avogadro’s number, respectively. The quantity 6 in eq 3 can be calculated by
for a given polymer concentration g, where to and ex are the limiting values of t when g approaches zero and a limiting concentration, g,, respectively. The numerical value for to can be calculated by to =
KMnap
2.5
(6)
while 6% is given by (Johnston and Sourirajan, 1973) t,
= 2.6
+ (1.7 X 10%
(7)
where z is the number of main-chain atoms in a polymer molecule. The correspondingvalue of g, is given by (Rudin and Johnston, 1971) g, = 0.507p/tx
W S U R F A C E TENSION.6
TOTAL NUMBER OF UNIT CUBES. “73
....I
Figure 2. Distribution of vacant unit cubes.
space. Therefore, in the following discussion, a spherical polymer aggregate is approximated by a cube, the length of whose edge is the same as the diameter of the spherical aggregate of the polymer. Distribution of Polymer Aggregates. Now let us assume a cube that consists of m3number of unit cubes. Such a large cube can be regarded as a solution space. Each unit cube is either filled with a polymer aggregate or unfilled. Let us call hereafter the unit cubes filled with a polymer aggregate “filled cubes” and those filled by solvent only “vacant cubes”. Let us then count the number of methods by which B number of vacant unit cubes can be distributed, separately, in the aforementioned large cube of m3 unit cubes (see Figure 2). Let us also assume m3 >> F >> 1. The number of such methods, wl,is the number of combinations of m3things P a t a time and can be written as
(8)
The calculations of (S) was performed for each polymer sample by using the intrinsic viscosity values of the polymer solution in the presence of calcium chloride at two temperature levels of 20 and 95 O C . Equations 3-8 are those of Rudin and Johnston (1971) and Johnston and Sourirajan (1973). They involve several assumptions arising from colloid and polymer chemistry considerations. It is not the object of this paper to justify those assumptions. The above equations are used here merely for purposes of obtaining a few of the numerical data neceasmy to illustrate the basic approach to the study of fh-casting polymer solution structure presented in this paper. Calculation of Pore Size Distribution on Reverse Osmosis Membranes. 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). The analysis is based on the transport equations arising from the surface forcepore flow model for RO/UF separations; this analysis stands on firm physiochemical grounds (Sourirajan, 1983a). All the necessary quantities and computation procedures are summarized in the Appendix.
Statistical Thermodynamic Approach t o the S t u d y of t h e S t r u c t u r e s of Polymer Aggregates in the Film Casting Solution In this theoretical approach, an attempt is made to examine the distribution of polymer aggregates in a given solution space with the aid of statistical thermodynamics. For the statistical calculations, we need to specify a unit cell, either occupied or unoccupied by a polymer aggregate. It may be convenient to consider a number of unit cubic cells whose size is equivalent to that of polymer aggregates and that are distributed three-dimensionally in the solution
w1 = ,E@=
(m3)! (9)
(m3- P)!(P)!
This combination includes both the case where vacant unit cubes are distributed without touching each other and the case where a vacant cube touches one or more other vacant unit cubes. The total number of the latter case, w;, would be the number of combinations of (m3- 1)things (P - 1) at a time and can be written as
w ; = mhlC&l = The ratio wl/w; =
(m3- I)!
W ~ J W is ’ ~ written
(m3)!
(10)
(m3- 13)!(13 - I)! therefore
/
(m3 - I)!
(m3- 13)!(P)! (m3- 13)!(P - I)!
m3
=->>1
P
(11)
Equation 11 indicates that vacant cubes are distributed separately from each other in most cases when m3 >> B. Therefore, as a first approximation, the total w1 number is regarded as the distribution where the vacant cubes are isolated from each other. Let us now put l3 vacant cubes together into a single cube that consists of P unit cubes (see Figure 2, left side). Let us count the number of ways of positioning such a cube into the large cube (of m3 unit cubes). Such a number, w2, is equal to the number of methods of positioning a unit cube which occupies a corner of the vacant cube of P units and is expressed as
w 2 = ( m- 1
+ 1)3
(12)
The entropy increase resulting from splitting an assembly
658
Ind. Eng. Chem. Prod. Res. Dev.. VoI. 24, No. 4, 1985 TOTAL NUMBER OF THESE "*,CANT CELLS
TOTAL NUMBER OF THESE "FC" CELLS
UMI SQUARE
ISOLATED
2
"*CANT
TOT*L NUMBER OF "NIT SQUARES
Y I C A N I SQUARE
"NlT
SQUARES
+p"RF*CE
VACANT
FILLED SQUARE
TE"SION.6
I2PP FLLED
Figure 3. Distribution of vacant unit squares.
of l3 vacant unit cubes into isolated vacant unit cubes (see is then calculated as Figure 2), designated as AS,,
-
w1 = k In -
AS..,
W9
(m3)!
= k In
(m3- 13)!(13)!(m- 1 +
]
u3
Figure 4. Distributionof vacant squares of four units (total number of vacant squares of four units = h2,). (2A)
(13) CORNER W E R E FOCR W A R E S MEET
Using Stirling's formula
ASeubeN k[{(m3)In (m3)- (m3)l- l(m3 - 13) In (m3P) - (m3 - 13)) - (0)In (13) - (a))]- k In (m- 1 + Figure 5. Vacant square positioned at the corner where four unit squares meet.
Since m3 >>
one occupied by solvent only as a "vacant square". Let us then distribute a total X2 number of vacant squares in such a way that XZ1 of them are distributed in a completely isolated manner, while the rest are grouped into a numher of squares, each one of which is composed of four unit squares. Supposing the number of latter squares to he Xz4
P >> 1
AScU,
N
=
k In
3 (m7 ) - 3k In m
(.7,)
3kL31n - - 3 k l n m
= 3k In
x2
m(@-l)
113
3kP In
(y)
Splitting a large vacant cube into a number of small vacant unit cubes is also accompanied by an increase in the surface area of contact between fded and vacant cubes, resulting in an increase of surface energy that can he regarded as an increase in enthalpy AHcubs.The latter can he calculated by
AHH,,,
613L2u- 6(1L)'u
= AHH,,
- TAS,.,
= (613L2- 6(1L)2Jo- 3kT13 In ( m / l )
+ 4x2,
(19)
(see Figure 3). First we count the number of methods of distributing X2, vacant squares the length of whose edge is 2A in an isolated manker. In order 6 count this number, the large square that consists of ( 2 ~ number ) ~ of unit squares is regarded BS the array of squares consisting of 4 unit squares (see Figure 4). Obviously, there are pz number of such squares. The method of distributing X2, vacant squares (with the length of the edge of 2A), w3, is then
(16)
where L and u denote the length of the edge of the unit cube and the interfacial tension between polymer aggregate and solvent, respectively. When eq 15 and 16 are combined, the free energy change involved in splitting a large vacant cube is written as AG,.,
=
(17)
(18)
Distribution of Polymer Aggregates in the Solution Surface and the Size of Aggregate Pores. The situation a t the surface of the solution is different from that in the bulk. Because of the surface tension, the polymer a g p g a t e at the airaolution interface is flattened (Kesting et al., 1965); therefore, a two-dimensional distribution of polymer aggregates has to he considered rather than a three-dimensional one. Let us now assume that there is a large square that consists of ( 2 ~ ) 'numbers of unit squares. This large square corresponds to a two-dimensional solution space. The length of the side of a unit square is A. Again, we define the unit square occupied hy a flattened polymer aggregate as a "filled square" and the
Obviously, this is an overestimation of the number of isolated distribution, since the above calculation also includes the case where the vacant squares touch each other. However, it is also an underestimation, since the above calculation does not include the case where the center of a vacant square is positioned a t the corner where four squares with length 2A meet (see Figure 5). However, if we assume that both effects cancel each other, eq 20 becomes valid a t least as a first approximation. Then, there are ( 2 ~-) (2XJ2 ~ numher of unit squares left to he occupied, and our next attempt is to distribute PI number of vacant unit squares among them in an isolated manner. Such a number is calculated by
The total number of distribution w g is, therefore w 5 = w3 x W I =
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985 659
The number of methods by which an assembly of X2 vacant unit squares is positioned in the solution space (see Figure 3, left side) is calculated to be
wg = (2p -
+
0 -0-WATER MOLECULE
(23)
The entropy increase accompmying the two-dimensional distribution of vacant unit squares, AS,,, is given by
AS,, = k In
NUMBER OF WATER M0LECULES;N
(2)
\
sm
WATER MOLECULE NUMBER OF WATER MOLECULES IN A SMALL PORE,
--
HIGH DENSITY REGION
k In (2p - X
+
I LOW DENSITY REGION
+ 1)2
Figure 6. Dispersion of water molecules into polymer network.
(24)
One might ask why only the above distribution modes were considered in the statistical thermodynamic calculation. In fact, the probability of the formation of larger vacant cells than those with four unit squares was considered in the preliminary calculation and it was found that eq 29 was not satisfied by such distribution modes. On the Network Pore. While the aggregate pores are associated inherently with the polymer structure in the casting solution, the network pores are associated with the structure of the polymer aggregate itself, which is in equilibrium in water environment. For calculating the size of such network pores, let us now consider a polymer aggregate that contains N number of water molecules. Let us also consider that most of the polymer chains are in high-density (more crystalline) form, and there are also low-density (more amorphous) regions (see Figure 6) that connect two high-density (more crystalline) regions. Let us assume v number of water molecules intrude into one low-density (more amorphous) region and form a pore; thereby, the length of the polymer segment is slightly stretched. Since the polymer aggregate is flat on the surface due to surface tension consideration (Kesting et al., 1965), we regard this problem again as that of twodimensional distributions of water molecules. Then, the number of methods by which total N number of water molecules are grouped into v number of water molecules are
The interfacial energy increase due to the distribution of vacant unit squares is caused only by the generation of boundaries between filled squares and vacant squares. Hence, the enthalpy change accompanying the splitting of X2 vacant squares should be AHB, = ( 8 h ~ X ~ 44hTX21)a- 4 h ~ X a
v
(26)
where T is the thickness of the flattened polymer aggregate. The free energy change involved in the splitting of a large vacant square consisting of X2 unit can be calculated under the condition p2 >> X2 >> 1 as AG,q = AH,,- TAS,, (27)
Therefore the entropy increase accompanying the dispersion of water molecules in a polymer aggregate in the aforementioned manner can be written as N! ASdiep = k In (w7/1) = k In (31) (V!)NIY
Introducing Stirling’s formula hSdisp k[(N In N - N) - ( ( N / v ) ( vIn u - u)]] = kN(1n N - In Y) =kNln (N -\ \ V I
As it will be shown later, the most thermodynamically stable state in the bulk solution is where the vacant spaces are all assembled in one place, under which conditions AG,h is equal to 0. Consequently, the equilibrium between the bulk- and surface-phase solutions requires the condition AGSq = 0
(29)
(32) The surface energy increase involved in the generation of the interface a t the water-polymer boundary can be considered as the enthalpy change and can be written as N AHinterface = - - 2 ~ u ~ / ~ r , , ~ ~ ~ a ’ (33) V
where v112rwaterrepresents the radius of a circle that is formed when v number of water molecules are fused twodimensionally and a’ is the interfacial tension between polymer and water. In addition, we have to consider the
680
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985
Table 111. Apparent Molecular Weight of Polymer Aggregates at Different Temperatures in the Presence of CaCl," temp, "C Dolvmer 20 30 40 55 70 90 22200 23000 22200 PA-1 22000 23500 20600 PA-2 46200 48600 43800 46400 47600 49200 PA-3 51300 53900 49400 54000 51500 54500 PA-4 85900 91500 83100 88400 90000 92400 PA-5 124300 128400 121100 127400 128800 128800 a CaC1,
to polymer weight ratio = 1:3.3.
work stored in the polymer segment when the latter is stretched by the intrusion of water molecules in the lowdensity (more amorphous) region. When a polymer segment of cross-sectional area S and of initial length Lo is stretched by hL, the work required for the stretching, which may be regarded as further addition to the enthalpy increase in the system, can be written as (34)
polymer samples in DMA solvent are given in Table 11. From the available data, K and CY values in eq 1 were obtained as 3.13 X dL/g and 0.81, respectively, at 20 "C. The intrinsic viscosity data in the presence of calcium chloride additive are also shown in Table 11. When K and cy values obtained above are applied in eq 1, the apparent molecular weight of the polymer aggregate in the presence of calcium chloride (Mn)CaC1was computed with respect to each polymer sample. T i e results are also shown in Table 11. (Mn)CaClz values at different temperatures are shown in Table 111. Apparently the temperature has only very little effect on (Mn)CaClz. The intrinsic viscosity data and the Mark-Houwink constants of different polymer samples were obtained at various temperatures, and results are shown in Table IV (in the absence of CaC12)and Table V (in the presence of CaC1,). By the use of the intrinsic viscosity data in Tables IV and V the radius of the spherical agglomerate, was obtained by eq 3-8. The numerical values necessary in the equations are MA N Mn
(s),
p = 1.3 X lo3 kg/m3
where E is Young's modulus of the polymer. Since there are N/v number of such noncrystalline regions that are intruded by water molecules, the enthalpy increase due to the stretching is (35) Therefore, the free energy change involved in the process of two-dimensional dispersion of water molecules can be written as AGdisp
=
minterface
+ matretch
- TASdisp
N 1NES = - 2 7 r ~ ~ / ~ r ~+~ -~ ~ -AL2 ~ 7 u ' - kTN In v 2 u L" When the system in which water molecules are dispersed in the polymer is in equilibrium with that where water and polymer exist separately, AGdisp should be equal to 0. Furthermore, AL can be approximated by the area occupied by intruding water molecules divided by the distance between two polymer segments, d,.
AL = u7rrZwater/ds
(37)
Setting AGdisp= 0 in eq 36 and inserting eq 37, we obtain
Equation 38 points out the factors governing the size of network pores. Results and Discussion Molecular Weight and Size of Polymer Aggregate. The intrinsic viscosity data obtained with respect to five
No = 6.023
X
z =
no. of main-chain atoms in a polymer repeat unit molecular weight of polymer repeat unit
X
The polymer concentrations g is based on (polymer kg)/(DMA solvent m3) and 0.14 X lo3 kg/m3 (=12.5/ (83.7/0.94 X lo3)) for the solutions of casting solution composition a t 20 OC. A t higher temperatures solvent volume was modified by considering solvent density. Inclusion of the volume of CaC1, in the solvent volume when CaC12is present in the solution had only very little effect on the final ( S ) value. The results of the calculation of ( S ) are listed with respect to different polymer samples in dilute solution at different temperatures (Tables VI and VII) and in the solution of casting composition a t 20 O C and a t 95 OC (Table VIII). The data indicate that while the spherical agglomerate size decreases with the increase in polymer concentration, it does not change significantly with the increase of solution temperature. Moreover, the spherical agglomerate size increases from 31.5 to 57.2 X m with the increase in the molecular weight of the polymer sample from 10500 to 31 300 at 20 O C . The polymer aggregate of a comparable size was observed by electron microscopy with respect to aromatic polyamidohydrazide polymer (Panar et al., 1973). Pore Size and Pore Size Distribution on the Membrane Surface. The pore size distribution on the membrane surface obtained for the membrane produced from polymer samples PA-1-PA-4 is listed in Table IX together
Table JV. Intrinsic Viscosity and Mark-Houwink Constant of Polyamide Polymers at Different Temperatures in the Absence of CaCl, Mark-Houwink [TIP dL/g constants temp, O C PA-1 PA-2 PA-3 PA-4 PA-5 K X lo4, dL/g cy 20 0.6161 0.8552 1.3855 0.9672 1.4722 3.13 0.81 30 0.6015 0.8497 0.9549 1.3629 1.4320 3.29 0.80 0.9523 1.3040 1.3918 40 0.5893 0.8343 4.25 0.78 1.2586 1.3515 4.46 0.76 0.8266 0.9450 50 0.5835 1.2711 5.12 0.75 0.8829 1.1803 70 0.5533 0.7855 0.8746 1.1005 1.1906 90 0.5216 0.7665 5.78 0.73
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985 661
Table V. Intrinsic Viscosity of Polyamide Polymers at Different Temperatures in the Presence of CaClza Inl, dL/a temp, "C PA-1 PA-2 PA-3 PA-4 PA-5 20 1.0302 1.8800 2.0452 3.1052 4.1896 30 1.0335 1.8475 2.0078 3.0633 4.0194 3.9188 40 0.9832 1.7738 1.9470 2.9220 55 0.9589 1.6921 1.9054 2.7819 3.6844 3.4799 1.6499 1.7512 2.6613 70 0.9304 90 0.8843 1.5378 1.6583 2.4370 3.1071 CaC12 to polymer weight ratio = 1:3.3.
Table VI. Radius of the Aggregate of Polyamide Polymers in Dilute Solutions at Different Temperatures in the Absence of CaCl, radius of Dolvmer anareaate X lolo, m polymer M,, 20 OC 30 OC 4OoC 5OoC 70 OC 90 OC 46.5 46.0 45.2 44.4 PA-1 10500 46.9 46.2 PA-2 62.5 62.3 61.3 60.8 18400 63.0 62.9 PA-3 20700 68.3 68.0 68.0 67.8 66.3 66.1 PA-4 29500 86.7 86.2 84.9 83.9 82.2 80.3 PA-5 31300 90.1 89.2 88.4 87.5 85.8 83.9
(s)
Table VII. Radius of the Aggregate of Polyamide Polymers in Dilute Sohtions at Different Temperatures in the Presence of CaC12" radius of polymer aggregate x lolo, m Dolvmer (A4,)c.m. 20 OC 30 OC 40 OC 55 OC 70 OC 90 OC PA-1 22500 71.7 71.8 70.6 70.0 69.4 68.2 47850 112.7 112.1 110.6 108.8 107.9 105.4 PA-2 53030 120.0 119.2 118.0 117.2 113.9 111.9 PA-3 90330 164.7 163.9 161.4 158.7 156.4 151.9 PA-4 129090 205.0 202.1 200.4 196.4 192.7 185.5 PA-5
(s)
"CaC1, to polymer weight ratio = k3.3.
Table VIII. Radius of the Aggregate of Polyamide Polymers in Solutions of Casting Solution Composition' at Different Temperatures radius of polymer aggregate (S) x IO'O, m polymer 20 OC 95 "C PA-1 31.6 31.5 PA-2 40.9 40.8 PA-3 42.3 42.3 PA-4 50.7 50.7 PA-5 57.2 57.2 (I
Polymer:DMA:CaCl, = 12.5:83.7:3.8.
with the experimental and calculated separation data of reference solutes used for the determination of the pore size distribution. In the table the pore size distribution is given in a form of bimodal distributions where &,I, RbZ, ul,u2, and h2 denote the average pore radii of the fiit and second distributions, the standard deviation of the first and second distributions, and the ratio of the number of pores belonging to the second distribution to that of pores belonging to the first distribution. Several features of the average pore size and the pore size distributions are revealed in this table, namely: (1)The pore size distribution on the membrane is characterized by biomodal distribution, as in the case of membranes produced from cellulose acetate (Chan et al., 1984a) and aromatic polyamide hydrazide (Chan et al., 1984b). The small pores are regarded as the network pores, while the larger pores are regarded as the aggregate pores (Sourirajan, 198313). (2) The average pore radii of the network pores range from 6 to 6.7 X m, while those of the aggregate pores range from 48 to 50 X 10-lom. Therefore, the average radii of both pores do
Table IX. Data on Pore Size Distribution and Solute Separation of Some Polyamide Membranes pore size PA-1 PA-2 PA-3 PA-4 distribution mol wt 10500 18400 20700 29500 Rb,J-x 10", m 6.65 6.00 6.40 6.20 al/Rb,l 0.01 0.10 0.10 0.01 Rb,2-X 10", m 50.0 50.0 50.0 48.0 0.45 0.40 0.45 a2/Rb,2 0.48 hz 0.004 0.003 0.002 0.001 Solute reference solute exptl (calcd) ethanol 25.48 (23.47) trimethylene 35.86 oxide (33.23) 1,3-dioxolane 39.50 (37.77) p-dioxane 51.84 (57.00) 12-crown-4 70.00 (71.31) 15-crown-5 70.00 (71.79) 18-crown-6 78.22 (71.98)
Separation, % exptl exptl (calcd) (calcd) 32.00 30.10 (33.08) (29.93) 40.72 40.50 (39.17) (39.88) 46.08 44.79 (43.92) (45.48) 64.50 68.90 (64.40) (68.66) 72.00 83.00 (78.23) (85.52) 75.00 83.30 (78.34) (85.85) 85.73 91.00 (78.13) (85.80)
exptl (calcd) 37.27 (37.59) 47.37 (45.74) 50.92 (50.58) 74.32 (75.51) 90.00 (91.74) 91.00 (92.18) 95.00 (92.26)
not change significantly with the change in the polymer molecular weight. (3) The ratio of the number of aggregate pores to that of network pores, h2,decreases as the molecular weight of the polymer increases. The separation data of the reference solutes increase accordingly. Comparison of the Polymer Aggregate Size and the Pore Size on the Membrane Surface. Our next attempt is to compare the polymer solution structures characterized by the polymer aggregate size with the pore size on the membrane surface obtained above. Two steps in the membrane formation process, namely solvent evaporation and membrane gelation, affect the structure of the polymer during film formation before the final pore structure on the membrane surface emerges. Furthermore, the measurement of the aggregate size does not provide us with the information on the size of the space between the polymer segments within the aggregate (network pore) when polymers are in the casting solution. Therefore, any immediate relationship between the aggregate size and the pore size on the membrane surface is not expected. In fact, a t first glance there seems to be no correlation between data on given in Table VI11 and the average pore radii given in Table IX. Our attempt is therefore to connect the above two experimental results by applying the statistical thermodynamics approach that was developed in the earlier section. The calculation according to the statistical thermodynamic approach was conducted with respect to polymer samples PA-1 and PA-5, representing the lowest and the highest molecular weight, respectively. Thus, the effect of the molecular weight can be quantitatively evaluated by this calculation. Numerical Values Required for the Calculation. In order to apply eq 18, 19, 28, 29, and 38 derived by the statistical thermodynamics to polymer samples PA-1 and PA-5, several numerical values are required. These values are supplied for 20 "C as follows: m3, 13. One kg of casting solution consists of 0.125 kg of polymers and 0.837 kg of DMA solvent. The balance is calcium chloride. Since the densities of polyamide polymer and DMA solvent are 1.3 X lo3 kg/m3 and 0.9366 X lo3 kg/m3 at 20 "C, respectively, the volumes occupied by the polymer and the solvent in the solution space are
(s)
882
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985
96.15 X lo+ m3 (0.125/1.3 X lo3) and 893.7 X lo+ m3 (0.837/0.9366 X lo3),respectively. With respect to PA-1 polymers the length of the edge of one unit cell is 63.14 X 10-lom (=2 X 31.57 X 10-lo). Therefore, the volume of a unit cell is equal to 2.517 X m3. Accordingly, there are in total 3.93 X 1021(=(96.15 X lo4 893.7 X lo4)/ 2.517 X lo-%)numbers of unit cells in the polymer solution. The volume occupied by CaC12is ignored in this calculation as a first approximation even though the weight fraction of CaC1, in the casting solution (3.5%) is not insignificant. The molecular weight of a polymer aggregate was calculated to be 22 500 kg/kmol with respect to PA-1 polymer. Then, a unit cell, which is equivalent to one aggregate of PA-1 polymer, occupies the volume of 2.873 X m3 (=22500/6.023 X 1023/1.3X lo3). Note that this is the volume of one aggregate occupied by polymer only. Accordingly, there are 3.35 X lG1 (96.15 X 104/2.873 X lo-%) numbers of unit cells filled by polymer aggregates, which means 0.58 X loz1(=3.93 X 10,' - 3.35 X 1021)numbers of unit cells are vacant, filled by DMA solvent only. As a result of the above calculation, m3 = 3.93 X loz1and P = 0.58 X loz1 were obtained with respect to the PA-1 polymer sample. Similarly, m3 = 0.662 X lo2' and l3 = 0.079 X loz1with respect to the PA-5 polymer sample. L. The length of the edge of a unit cube is equated to the diameter of the polymer aggregate. Therefore, L = m (=2 X 31.57 X and 114.38 X 10-lo 63.14 X m (2 X 57.19 X 10-lo)for PA-1 and PA-5 polymer samples, respectively. u, d. The interfacial tension between the polymer material and solvent is calculated by (Omenyi et al., 1981)
A
B
co
+
or u' = [(surface tension of polymer)1/2(surface tension of s o l ~ e n t ) ~ / ~ ] ~ / { 0-. 1.5[(surface 001 tension of polymer)(surface tension of solvent)]lj2) (40)
u
When surface tension of aromatic polyamide = 36.85 X N/m (approximated by the average of those for Nylon 6 and Nylon 6,12), surface tension of DMA solvent = 33.16 X N/m, and surface tension of water = 72.75 X N/m a t 20 "C are used, interfacial tensions are calculated as u =
0.202
X
N/m
u' =
27.08 X
N/m
A. The spherical polymer aggregates are assumed to be flattened to circles consisting of two polymer layers, and their diameters are set equal to A. Setting the surface area of the spherical polymer aggregate equal to that of the circle
rL2 = r A 2 / 2 Therefore, A = 2lI2L. Accordingly, A = 89.3 X m and 161.76 X 10-lo m for PA-1 and PA-5 polymer samples, respectively. ( 2 ~ )X2. ~ , Assuming the ratio of vacant unit squares to is equal to that of vacant the total unit squares, (X2/(2~)2), unit cubes to the total unit cubes (Z3/m3),X2/(2p)2 = 0.147 (=0.58/3.93) for PA-1 polymer. Consider 1 m2 of the surface. Since a unit square of PA-1 polymer aggregate occupies the area of (89.3 X 10-10)2m2, there are 12.54 X 1015 (=1/(89.3 X 10-10)2)unit squares a t the surface. Therefore ( 2 ~ and ) ~ X2 are 12.54 X l O I 5 and 1.843 X 1015 respectively, with respect to PA-1 (=0.147 X 12.54 X polymer. Similarly, ( 2 ~and ) ~ X2 are 3.822 X 1015and 0.459 X 1015, respectively, with respect to PA-5 polymer. 7, S, Lo, d,. These quantities are properties of the polymer studied. They are estimated from the structure given for poly m-phenylene isophthalamide by Herlinger et al. (1973). The distance between stacked polymer layers
\
HN
c?
0 Hb
(f4 NH
~
~
NH
TN
5 Hb
4 3 . 1 8 ; 1 i 6.7iA
Figure 7. Structure of poly(m-phenylene-iso-phthalamide). Reprinted with permission from Herlinger et al. (1973). Copyright 1973, Deutsches Kunststoff-Institut,
(see Figure 7A) is known to be 4.7 X m when the distance is measured between main backbones, considered as chains of flat benzene rings. Considering contributions from C=O and N-H groups, which are extruding from the main-chain backbones, 6.8 X m (=4.7 X X 1.44) is adopted as the thickness of a flattened polymer aggregate, 7. (This factor (1.44) was determined by the three-dimensional projection of molecular model shown in Figure 7.) The area occupied by the cross section of one polymer chain was calculated to be 22.1 X (=(4.7 X 10-10)2),which is considered to be S. Furthermore, mphenylene isophthalamide structure is considered to form the low-density (more amorphous) part of the polymer, while m-phenylene terephthalamide structure is considered to form the high-density (more crystalline) structure. This consideration is justified by the relative higher crystallinity of Kevlar @-phenylene terephthalate) compared with crystallinity of Nomex (m-phenylene isophthalate) (Tadokoro et d., 1979). Since one repeat unit of m-phenylene isophthalamide is connected to the m-phenylene diamino structure of the adjacent terephthamide (see Figure l), the length including three benzene rings is considered to be Lo (see Figure 7A). Therefore, Lo = 16.5 X m (=11.0 X (3/2)). The quantity, d,, is equated to the distance between two polymer chains lying on the same polymer layer (see Figure 7B); therefore d, = 6.7 X m. E. Young's modulus of polyhexamethylene terephthalamide (=4.84 X lo9N/m2) reported in the literature (Sprague and Singleton, 1965) is used as E. F,,,,. The Stoke's law radius of the water molecule was calculated on the basis of the self-diffusivity data of the water molecule (Robinson and Stokes, 1959) to be 0.87 X m. N. Since the molecular weight of one polymer aggregate is 22 500 kg/kmol with respect to PA-1 polymer and the saturated water content of polyamide polymer is 0.391 (Matsuura et al., 1983),the number of water molecules in one polymer aggregate is N = 488 (=22500 X 0.391/18.02). Similarly, with respect to PA-5 polymer, N = 1960. Distribution of Polymer Aggregate and Solvent in the Bulk Solution. AGmb value was calculated by using eq 18 and the numerical values given above. The results are 23.42 J/kg solution and 11.84 J/kg solution for PA-1 and PA-5 polymers, respectively. The positive free energy values indicate that the splitting of vacant cubes occupied by solvent only into small unit cubes is thermodynamically unfavorable, which means that vacant cubes (filled by solvent only) tend to gather in the bulk solution phase.
Ind. Eng. Chem. Prod. Res. Dev.. VoI. 24. No, 4. 1985
Distribution of Polymer Aggregates and Solvent in the Solution Surface and the Size of Aggregate Pores. The ratios X12/X2 and (2XJ2/X2 were calculated by using eq 19,28, and 29 and the necessary numerical values obtained above. The results are hZl/h2 = 0.029 and 0.980 ((2X4)2/X2 = 0.971 and 0.020) for PA-1 and PA-5 polymers, respectively, which means that 97.1% vacant unit squares (fdedby solvent only) exist as isolated squares comprising four units (the length of the edge = 2A) with respect to PA-1 polymer, while 98% vacant unit squares exist as isolated single squares (the length of the edge = A) with respect to PA-5 polymer. The above results indicate that the space unfilled by the polymer aggregate tends to be split into small cells at the polymer solution surface. This may offer a thermodynamic reason for the asymmetric structure of RO/UF membranes, though it is usually attributed to kinetic effects (Strathmann et al., 1971). The vacant cells may be considered as incipient aggregate pores. The above results therefore indicate that the equivalent radius of the vacant cell (therefore, of the incipient aggregate pore) is 2 X (89.3/2) X = 89.3 X m for PA-1 polymer (radius of polymer aggregate = 31.6 X m) and (161.76/2) X = 89.0 X m for PA-5 polymer (radius of polymer aggregate = 57.2 X m). From these data, one may conclude that the size of the molecular weight does not affect very much that of vacant cells and, consequently,that of incipient aggregate pores. Furthermore, the aggregate pore size may be expected to decrease as solvent evaporation proceeds, as indicated in our earlier paper on aromatic polyamide hydrazide membranes (Chan et al., 1984b). Size of the Network Pore. When eq 38 and numerical values obtained above are used, v is calculated to be 24.1 and 38.5 for PA-1 and PA-5 polymers, respectively. When water molecules are fused two-dimensionally, they form pores of radii 4.3 X m and 5.4 X lo-" m, respectively. The above calculation is very enlightening in its scope and direction. It indicates there are indeed two distinct kinds of membrane pores. The larger one, called aggregate pores, originates from the spaces in the polymer solution surface, which are devoid of polymer aggregates. The radius of such pores is less than about 90 X 10." m, depending on the evaporation period. Obviously, the sizes and numbers of these pores are strongly related to the polymer structure in the casting solution, which sets the beginning of the membrane formation process, and these pores are of transient nature. The smaller ones, called network pores, are related to the structure of polymer segments in the polymer aggregate that is in thermodynamic equilibrium in water environment. Such an environment sets the end of the membrane formation process, which in turn seta a limit to the size of obtainable network pores. These pores are the ones that the polymer tends to acquire ultimately in water environment. The transition from aggregate pores to network pores is possible, as discussed in our earlier papers (Chan et al., 1984a). It is also interesting to note that the network pore radius depends on the intrinsic property of the polymer as well as its disposition in the aggregate. In the process of the numerical calculation according to eq 38, it was found that the first term of the equation (AHhbdae) can be ignored in comparison to the rest of the terms involved in the equation. Then, looking into eq 38, the number of water molecules in the network pore, v, increases when N increases. Obviously, increase in water content in the polymer increases N as well as Y. The value of v decreases with increase in Young's modulus E and with decrease in Le Therefore, the material of less elmticity decreases the
663
NETWORK PORE 4 - 5 i 20-50 IN ONE AGGREGATE rAGGREGATE PORE D
(A-3)
where d is the distance between the polymer surface and the center of the solute molecule, D is a constant associated with the steric hindrance (distance of steric repulsion), and B expresses the nature and the magnitude of the van der Waals force. The quantity D is always positive, and when the solute is assumed spherical, it can be approximated by the molecular radius such as the Stoke's law radius, while B may be either positive (corresponding to an attractive force) or negative (corresponding to a repulsive force). The para_meters associated with the pore size distribution, i.e., Rb,i,ui,and hi, and the interfacial interaction force parameters B and D are related to the specific
Table X. Some Physicochemical Parameters Pertinent to Reference Solutes reference D m X r A / C A , b X D X lolo, B X IOM, solute MW los. m2/s 1O'O. m m ms ethyl alcohol 46.1 trimethylene 58.1 oxide 1,3-di74.1 oxolane p-dioxane 88.1 12-crown-4 176.2 15-crown-5 220.3 18-crown-6 264.3
1.19 1.06
-0.538 16.70
1.94 2.30
3.60 50.41
1.01
14.40
2.41
54.53
0.871 0.609 0.524 0.460
1.86 -3.53 -3.17 -2.33
2.80 4.00 4.65 5.30
38.63 -13.26 25.18 102.0
surface excess, r A / C A , b , obtainable from the chromatographic retention volume data, and the solute separation f , obtainable from RO experiments by (rA/cA,b)j = k(B, D)]J fJ
(A-4)
= (h(Rb,,,u,, h,, B, D; under given operating conditions)], 64-5)
where subscript j indicates the j t h solute and g( ....) and h(....) are some functional forms established in the surface forcepore flow model (Matsuura et al., 1981; Chan et al., 1984a; Sourirajan, 1983a). By the use of the above equations, numerical parameters involved in the interfacial interaction forces and the pore size distribution can be determined as follows. Let us f i t choose seven reference solutes such as those give in Table X. Setting eq A-4 and A-5 for each reference solute, we have seven eq A-4 and seven eq A-5 corresponding to j = 1, 2, 3, ... 7. Then, if we set (D) equal to the Stoke's law radius of jth reference solute, (BI, can be calculated from eq A-4 set for the j t h solute, so that experimental (rA/CA,b), can be satisfied. Further, five pore size distribution parameters q,R b g , u2, and h,) can be calculated by nonlinear regression analysis of seven eq A-5 by using B and D values obtained above for each reference solute. The B and D values so obtained are listed in Table X.
Nomenclature A = pure water permeability constant, kmol H20/(m2.s.kPa) B = constant characterizing the van der Waals attraction force, m3 CA,b = bulk solute concentration, moi/m3 D = constant characterizing the steric repulsion at the interface, m DAM/K6 = solute transport parameter, m/s d, = distance between two polymer segments, m d = distance from polymer material surface to the center of solute molecule, m E = Young's modulus, N/m2 f = fraction solute separation based on the feed solution AG = change in Gibbs free energy, energylmass (or area) AGC,b =-A-G associated with the splitting of a large vacant cube, J / k g AG,, = AG associated with the splitting of a large vacant square, J/m2 AG&p = AG associated with the dispersion of water molecules in a polymer aggregate, J/weight of a polymer aggregate g = polymer concentration, kg/m3 of solvent g, = limiting polymer concentration, kg/m3 of solvent AH = change in enthalpy, energylmass (or area) AH,,b = AH associated with the splitting of a large vacant cube, J/kg AHw = AH associated with the splitting of a large vacant square, J/mZ AHinterface = A H associated with the generation of interface at the water-polymer boundary, J/weigbt of a polymer aggregate
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985 665
AHBtretch = AH associated with the stretching of polymer
segments, J/weight of a polymer aggregate hi = ratio defined by eq A-2 K = Mark-Houwink constant k = Boltzmann constant L = length of the edge of a unit cube, m Lo = initial length of a polymer segment involved in the low-density (more amorphous) region, m AL = amount of the stretching of a polymer segment, m P = number of vacant unit cubes MA = molecular weight of polymer = number-average molecular weight Mf m = number of total unit cubes No = Avogadro's number n = quantity defined by eq 2 ni= number of pores belonging to the ith normal distribution (PR) = product permeation rate per given area of the membrane surface, kg/ h (PWP) = pure water permeation rate per given area of the membrane surface, kg/h R b = pore radius, m R b = average pore radius, m Rb,, = average pore radius of the ith distribution, m R = gas constant r,,,, = Stoke's law radius of water molecule (0.87 X 10-lom) S-= cross-sectional area of the polymer segments, m2 (S) = radius of the spherical polymer aggregate, m A S = entropy change, [energy/mass (or area)]/temperature AScube = A S associated with the splitting of a large vacant cube, J/(kg.K) ASw = A S associated with the splitting of a large vacant square, J/(m2-K) = A S associated with the dispersion of water molecule m a polymer aggregate,J/(weight of a polymer aggregate-K) T = absolute temperature, K u = volume of an unsolvated polymer molecule, m3 wl,wtl, w2, w3,w4,w5,w6,w7 = quantities defiied by eq 9, 10, 12, 20, 21, 22, 23, and 30, respectively Yi(Rb) = normal pore size distribution function, l / m z = number of main-chain atoms in a polymer molecule
qdbp
Greek Letters = Mark-Houwink constant = surface excess of solute A, mol/m2 c = the effective volume factor eo, ex = e at g = 0 and g = g,, respectfully [7]= intrinsic viscosity, dL/g A = length of the side of a unit square, m X2 = number of vacant unit squares ( 2 ~ =) number ~ of total unit squares N = number of water molecules in a polymer aggregate v = number of water molecules in a network pore p = amorphous polymer density, kg/m3 u, d = interfacial tension between polymer and DMA solvent and between polymer and water, respectively, N/m a!
ai = standard deviation of the ith normal pore size distribution,
m = thickness of a flat polymer aggregate, m 4 = potential function of interaction force exerted on the solute from the pore walls, J/mol Registry NO.PA (SRU), 53414-70-3; (isophthalic acid).(mphenylenediamine)-(terephthalicacid) (copolymer),26876-90-4; ethanol, 64-17-5; trimethylene oxide, 503-30-0; 1,3-dioxolane, 646-06-0;p-dioxane, 123-91-1;12-crown-4,294-93-9;15-crown-5, 33100-27-5;18-crown-6,17455-13-6; calcium chloride, 10043-52-4. 7
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Receiued for review January 28, 1985 Revised manuscript received May 30, 1985 Accepted June 29, 1985
