The role of swelling, amorphous vs. crystalline content, charge density, and applied stress on the transport properties of polymeric films

Journal of Membrane Science, 3 (1878) 131-148 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

131

THE ROLE OF SWELLING, AMORPHOUS VS. CRYSTALLINE CONTENT, CHARGE DENSITY, AND APPLIED STRESS ON THE TRANSPORT PROPERTIES OF POLYMERIC FILMS

A. GLIOZZI Laboratory of Cybernetics and Biophysics, Camogli, Genoa (Italy) A CIFERRI Istituto di Chimica Industriale, University of Genoa, Genoa (Italy)

Summary Previously reported results on the transport behavior of crosalinked collagen membranes are reviewed and discussed . Under isoelectric conditions, alteration of the degree of swelling and of the state of the membrane are induced by changing salt type and concentration . The filtration coefficient L y increases when swelling is increased in the amorphous state, decreases when swelling is increased in the crystalline state, and increases during the crystal-amorphous transition . Under non-isoelectric conditions (low pH, low ionic strength), L . and swelling have the same trend in the amorphous state while they have opposite trends in the crystalline state . The trend of the reflection coefficient a as a function of pH (a maximum is exhibited at pH " . 3) is quantitatively explained on the basis of the competition between swelling and fixed charge density (the former includes a contribution due to the charge-induced phase transition at pH 's . 2, in addition to the usual Donnan contribution) . The permeability to THO, co T , increases with increasing strain on the amorphous membrane and then goes through a maximum and a minimum when a stress-induced phase transition occurs . The relationship between frictional coefficients derived from irreversible thermodynamics and polymer-salt-water interactions deduced from equilibrium thermodynamics is pointed out .

Introduction The role of both biological and artificial membranes in a variety of processes has been extensively investigated in recent years . The wide variety of processes in which membranes are relevant span from the selective control of transport through cellular walls, to biomedical and technological applications. The model according to which the membrane is simply a passive boundary between two physico-chemically different media has been definitely abandoned. For both biological and artificial membranes, the membrane "phase" has to be regarded as a thermodynamic system in which the occurrence of interactions and reactions between the various components modifies the fluxes of water and solutes, and thus the permeability of the membrane itself .

1 32

Our work [1-4] has been based on the transport properties of collagen films, due to the fact that for this particular system a variety of interactions between the protein substfate and the solution components can occur . For instance, by alteration of variables such as pH, ionic strength, and applied stress, it is possible to induce variations of net charge, swelling, degree of crystallinity and orientation, and even induce a phase transition in the membrane . It is therefore possible to systematically analyze the role of these physical and structural parameters on the transport properties, avoiding the additional complexity associated with the use of chemically different substrates. This article represents a comprehensive review of the work we have performed in recent years [1-41 . The membranes we have used [1-4] are collagen films, 25 pin thick, slightly oriented during the fabrication process . They were produced by the Ethicon Company (Somerville, N .J.) by deposition, under no stress, of a dispersion of beef tendons . Scanning micrographs of the membrane surface at two different magnifications (2200 X and 6300 X) are shown in Fig .1 . Electron micrographs of these films at higher magnification reveal the typical tropocollagen pattern with the long period of 700 A [1] . Membranes were chemically cross-linked by exposure to p-benzoquinone solutions [1] . Cross-linking was performed in order to avoid dissolution of the membrane when the transport behavior was investigated in the amorphous state . Moreover, the cross-linking reaction allows an alteration of the degree of swelling in a given solvent medium. The degree of swelling can also be altered by variation of pH and salt type and concentration . The membrane has a net fixed charge equal to zero at the isoelectric point which occurs at pH ti 5. At pH ti 2 the membrane carries about 1 positive charge/10 monomers [2] . A phase change from the crystalline to the amorphous state can be induced, for instance, by increasing CaCl2 or KSCN concentration [1] . Our membranes are highly permeable to water and exhibit a large degree of swelling (30-90%) . While we study them as a model system for the physicochemical description of transport processes, from a practical standpoint their properties and behavior may be related to some processes of biological and technological relevance . In particular, in some biomedical applications of hydrogels, such as artificial corneas, burn dressings, and cartilage replacement, the relevant transport features should be analogous to the ones investigated in our work . Moreover, our system may be also a valid model for several natural membranes, for instance those which serve shape-retaining functions and offer osmotic protection to the cell . Our approach [1-4J to the physico-chemical description of transport processes through collagen films is based on the determination of transport parameters defined by the thermodynamics of irreversible processes [5] . These parameters are then compared with molecular parameters determined applying equilibrium thermodynamics to the open polymer system . We shall see that under appropriate experimental conditions it is possible to measure

133

Fig.1 . Electron micrographs of collagen films at two different magnifications (2200 x and 6300 X). (From reference 1(b)).

134 the filtration coefficient, LP, the reflection coefficient, a, and the solute permeability, wi, described by Katchalsky and Currant [ 6] . The phenomenological coefficients, being gross parameters functionally dependent in turn upon a variety of more fundamental parameters, are not well-suited to separate and characterize the various physical changes induced in the membrane by variations of electrolyte concentration in the bathing solutions . A first attempt at a more detailed description can be made from a determination of the frictional coefficients [6,6] ftk between the various components : solvent, solute, and membrane . These are determined from the phenomenological coefficients once the membrane thickness, Ax, and the degree of swelling, V/Vo are known . However, the interaction between the membrane and the components which are able to diffuse through can be better described using equilibrium theories . We shall discuss in detail these theories in connection with the analysis of experimental data . Now we simply want to show the meaning of the tools which we need for describing the relevant interactions . When a collagen film is swollen by a salt solution we can distinguish [7] between specific polymer-water and polymer-salt interactions from non-specific interactions between the polymer and the remaining solution . The strong, specific interactions can be described by binding constants (Kw and Ks) involving amorphous collagen and water, or collagen and salt, as described for instance by Orofino et al . [7] . Occurrence of these strong interactions can drastically affect the stability of the crystalline vs . the amorphous phase, and quantitative relationships between binding constants and melting temperature exist [7] . The weaker interaction can be described by a partial molar free energy of dilution parameter, x , which is used in polymer solution theories [8] . The x s parameter describes the degree of swelling of the network, as described quantitatively, for instance, by Flory [ 8] . The latter author [8] has also considered the increase of the degree of swelling of the network due to a Donnan effect when the polymer carries a net fixed charge . An alteration of the degree of swelling also results from the application of a tensile stress of the membrane as discussed in detail by Treloar [9] . The tensile stress also affects the crystalline-amorphous equilibrium [10] . All these effects must be quantitatively considered for a proper interpretation of the transport behavior . We now turn to the presentation of the experimental results beginning with the role of swelling and of crystalline vs . amorphous content on the transport properties under no stress and under isoelectric conditions .

(A) Effect of swelling and of amorphous us, crystalline content [1] The net flux of solution, Jv , in cm sec', when a hydrostatic pressure gradient was applied across the membrane, was determined with the apparatus shown in Fig .2 . The two half cells were filled with identical salt solutions . Volume displacements were determined following the movement of the meniscus in the horizontal capillary as a function of time . Reported in Fig.2 is a typical experimental run showing the volume displacement per unit area



1 35

AP I

capillary

rotating magnet X

magnetic stirrer

7 a) ', membrane

b)

2

1

3

A t, hrs

Fig. 2 . (a) A schematic diagram of the apparatus for the hydraulic permeability measurements. (b) A typical experimental run showing volume displacement vs . time. (From reference 1(b)) .

(cm'/cm') vs. time. The volume flow determined by a hydrostatic pressure difference, A P, can be coupled with an electrical driving force, A 1P , arising from the streaming potential through the pores of the membrane . It can also be coupled with an osmotic pressure difference, air, which might be induced by a different rate of permeation of water and salt in the course of the experiment . However, streaming potential and reflection coefficient measurements have shown that under the adopted experimental conditions (isoelectric pH and large volume of the half-cells with respect to volume displacements), the coupling phenomena can be neglected . The filtration coefficient (or hydrualic permeability), L p , can thus be obtained by the simple relationship [5]

L p = J' /AP

(1)

Fig.3 shows the trend of L p with KCI concentration, when the membrane is in the crystalline state. Also reported in Fig .3 is the corresponding degree of swelling (i.e. swollen volume divided by dry volume) . We may observe that the trend of L p is opposite to that of V/Ve , i .e . an increase of swelling is accompanied by a decrease in filtration coefficient, while a decrease of swelling is accompanied by an increase in the filtration coefficient . Fig .4 shows the trend of L p and of V/V e with salt concentration when the membrane is in the amorphous state (obtained by using salt concentrations greater



136

KSCN

17 21

6 w

16

5 C

a

15

E

4= 0 n 1

N

2

J

3 4 C5, Mal/ I

5

18

3 I

7

18

3 N

4

cT

E 6

2 E E

0

5

10 n J

1

2 3 Cs, Mal/I

4

3

4 5 C s , Mal/I

Fig .3 . (a) The filtration coefficient, L P ,and (b) the swelling volume, V/V, for crystalline cross-linked collagen membranes plotted as a function of KCI concentration ; T = 26° C, pH = 5 .5. (Data from ref. l(a)). Fig .4 . The filtration coefficient, Lp , and the swelling volume V/Vo plotted as a function of salt concentration for amorphous, cross-linked collagen films in (a) KSCN and (b) CaCl, solutions . T= 26'C, pH = 5 .5 . (Data from reference 1(a)) . than the critical salt concentration at which the transition occurs, cf . seq .) for two different set of experiments, namely KSCN and CaCI, solutions . It is of interest to observe that, at variance with the case prevailing in the crystalline state, in the amorphous state L p and V/Vp both vary with salt concentration in the same way . We have already mentioned that binding of particular salts to amorphous collagen may induce a conformational transition accompanied by dimensional changes even at room temperature . As an example, Fig .5 shows length changes as a function of CaC12 concentration . The abrupt change observed on increasing salt concentration corresponds to the phase transition . The lower curve shows length changes for a subsequent decrease of salt content . The latter changes are completely reversible . Thus, by increasing salt concentration of salts such as KSCN or CaC1 2 , which are able to induce a phase transition, we may analyze the correlation between swelling and filtration coefficient during the transition . Results are shown in Fig .6 . The data show that during the transition, the degree of swelling and the filtration coefficient have the same trend as in the case of the amorphous state .



1 37

15 CaC12

05

1

2

3

4

Cs, mot/ 1 Fig. 5 . Length relative to dry length, L, . The path of increasing salt concentration through the "first" transition is denoted by o • . The path of decreasing salt concentration is denoted by ~-k T = 26 ° C, pH = 5 .5 . (Data from ref. 4) . (Open and filled circles in the graphs refer to the first and subsequent transitions, respectively, throughout) .

21

KSCN 18

17

16 18 15 I I I

6

5 T b 6 4 O J3 I

2

3

4

1

5

2

3

4

5

C5,MoI/I Fig.6 . The filtration coefficient, L P , and the degree of swelling, V/Vo plotted as a function of salt concentration for two different salts . The dotted line refers to the phase transition and to the amorphous state . Data refer to the "first" induced transition (cf. Fig.5) . T = 26° C, pH = 5 .5 . (Data from refs . 1(a) and 1(b)) .



1 38

It thus appears that the permeability of the membrane differs depending on whether the crystalline phase, the amorphous phase, or the phase transition are considered . Obviously, we cannot explain the observed behavior in the simplest way, that is by an increase in the thickness of the membrane . We are led therefore to consider the effect of the interaction between salt, water, and the polymeric substrate as a function of the state of the latter . It appears that in the crystalline phase the absorption of water and salt reduces the permeability by an effect equivalent to a closing up of the porous structure of the membrane . Since deswelling in the crystalline state caused by KC1 brings about an increase of permeability, one has to conclude that the porous structure of the crystalline state does not collapse by simply desorbing the solution . In the amorphous state, however, and during the transition, since an absorption of solvent leads to an increase of permeability, one concludes that the swollen portion of the amorphous membrane is in a relatively looser state than in the swollen crystalline state . In other words the membrane behaves as a sponge-like structure in the amorphous state and during the transition, but not in the crystalline state . We shall show later that the frictional coefficients between polymer and salt solution do indeed decrease during the transition . (B) Effect of fixed charges [2] An alteration in the degree of ionization of the polymer may be induced by changing the pH at low ionic strength (for instance 10 -2 M) . In this case we must consider the following phenomena : (1) An osmotic swelling (note that this is different from the lyotropic swelling observed under conditions of zero net charge) ; (2) A destabilization of the crystalline structure, which cannot be ascribed to selective ion binding, but rather to electrostatic repulsion between charged side-chains ; and (3) Non-vanishing values of the streaming potential and of the reflection coefficient resulting from the discrimination between cations and anions under conditions of zero current flow . 1 .7 shows the degree of swelling on lowering the pH for a membrane at 52"C. The dotted line, which gives the degree of swelling of a polymeric network on increasing the fixed charge density, has been calculated according to Flory's relationship [8] . V

V°,

sis =[(X°/2S'12)2+(i-Xi)/V1]se/Vo

(2)

where X° is the concentration of fixed charges referred to the unswollen network, S is the ionic strength, V, is the molar volume of the solvent, and v,/V° is a parameter characterizing the degree of cross-linking of the network . The curve has been calculated taking X, = 0 .5 and fitting the network structure parameter v e /Vo to the experimentally observed value of V/V° at pH 3 .5 . The experimentally observed swelling at a pH of 2-3 exceeds the theoretical swelling curve, which describes only the osmotic swelling. This additional contribution to swelling is attributed to the phase transition .



139

9 0 7 5 3 1 2 pH Fig .7 . Experimentally observed pH-variation of the degree of swelling (solid line) for unrestricted collagen networks at 52° C (10 - ' M KCI) compared with the corresponding variation (dotted line) calculated using Flory's theory for the swelling of ionic networks . (Data from ref.2). 4

3

From irreversible thermodynamics equations, when Air = 0 we have

OP -

Lp

- LE U-)

(3)

The term LE(Ay/AP) 2 may be evaluated from measurements of conductance and of streaming potential [2] . Therefore, values of L p may be calculated from the Jv/AP ratio with the proper corrections due to the streaming potential effect (which accounts for ti 0.1% of the L p value at pH . 5, and for ti 6% at pH n 3) . Fig .8b shows the trend of the filtration coefficient as a function of pH . The Lp - pH variation may be compared with the corresponding pH variation of the degree of swelling (Fig .8a). The behavior of the filtration coefficient exhibits features similar to those previously observed when the salt concentration was varied at constant pH . In fact, we may observe that L P and V/Vo have the same trend in the amorphous state and during the conformational transition, while they have opposite trends in the crystalline state . The reflection coefficient a (a = (OP/Air) j . 0 ) was determined using the same cell employed for the determination of the volume flow . Two different salt solutions were used in the two half-cells . The hydrostatic pressure (applied to the most concentrated solution side) necessary to reduce the volume flow to zero was determined by extrapolating J v vs . AP plots at any given pH . Results are shown in Fig .8c . A maximum in a at pH - 3 may be observed. The determination of the reflection coefficient allows a quantitative comparison between the experimental results and the theoretical expectations



1 40

8

0 7

6 4 2

02 U 01 1

I

I

6

5

4

1

3

2

1

pH

°

Fig . 8. (a) pH variation of the degree of swelling, V/ V , (b) of the filtration coefficient, Lp, and (c) of the reflection coefficient, o for unrestricted collagen networks at several temperatures. Bathing solution 10 - 'M(10 - 'M and 5 x 10" M, and T a 52 ° C for reflection coefficient data) . Data from reference 2 .

from irreversible thermodynamics . We shall make use of the expression of a derived by Kedem and Katchalsky for the permeation of a uni-univalent electrolyte through a charged membrane {5] a = 1-

WT' p

a

-Xow

(4)

1

o

a

where w is the permeability coefficient of the salt when J = 0, u is the partial molar volume of the salt, is the arithmetic mean of the external salt concentration in both compartments, t° is the transport number of the

ca



141

counter-ion in free solution, 0, is the volume fraction of diluent inside the membrane, and the other symbols are as previously defined . Substituting plausible values of the filtration coefficient and of the salt permeability, the second term in eqn . (4) is found to be negligible with respect to the others . Therefore, the expression for a reduces simply to (5)

a =

to, X

1-camw

where cs and Of may be regarded as constants on lowering the pH (at least down to pH' 2 .5) . Eqn . (5) shows that the trend of the reflection coefficient should be determined mainly by the competition between swelling (mw) and fixed charge-density (X) . The expectation is fully supported by the data collected in Table 1 . The values of 0. reported in the second column were derived from the experimentally observed variation in degree of swelling with pH . The number n o of milliequivalents of HCl fixed per gram of collagen, and the corresponding charge density, reported in the third and fourth columns, were determined using typical titration data for native collagen, as reported by Gustavson [11] . When the ratio ¢ w /X (reported in the fifth column of Table 1) is used in conjunction with eqn . (5), it appears that in the crystalline state (pH > 3) X increases more than Ow does, thus leading to the experimentally observed increase of a, while at pH < 3 the decrease of a with pH is accounted for by the prevailing increase of 0w . In the last column of Table 1 we report values of X calculated using eqn. (5) and the experimental a and Ow values. The agreement with the data obtained from independent titration measurements up to pH^. 3 is satisfactory . Thus, we are able to predict the reflection coefficient behavior from titration data or vice versa . It should be observed that when pH < 2 .5, even the approximation under which eqn . (4) has been derived is no longer valid . However, an evaluation of the corrective terms [2] gives the values at the bottom right

TABLE 1 Charge density X . Comparison between transport and titration data . (Data from ref . 2) pH

5 4 3.5 3 2.5 2 1 .5

Qw

0.56 0.67 0.72 0.76 0.86 0.89 0.87

na (mEquiv/g)

Xt x 10' (moth)

QW /Xt x 10 - '

(limo!)

XU x 10 2 (mol/1)

0.09 0.3 0.5 0.7 0.83 0.9 0.9

0.5 1 .3 1 .57 2.1 1 .48 1 .3

1 .1 0.5 0.46 0.36 0.58 0.68 0.67

0 .38 1 .24 1 .38 1 .67 2 .2-1 .5 3 .3-1 .3 -

142 of the sixth column and reveals that, even at low pH, the above interpretation is qualitatively correct . Finally, we may observe that when values of the degree of swelling obtained from the dotted line of Fig .7 are used in conjunction with the expression for a given by eqn . (5), it is confirmed that the decrease of a with decreasing pH, when pH < 3, can be attributed only to the contribution to swelling given by the phase transition . (C) Effect of stress [3] The characterization of the stress-permeability behavior for a relatively simple model system is of interest in connection with a number of biological phenomena . For instance, in the case of frog gastrocnemius muscle, the filtration coefficient of water (due to an osmotic pressure gradient) increases with increasing strain up to elongations of about 20%, and decreases upon further increase of strain. Similar behavior is exhibited by the human iliac artery . In the case of red blood cells, it is known that hemolysis may occur even without a permanent alteration of the membrane, when the latter is subjected to deformation caused by osmotic swelling in hypotonic solutions . It appears that a satisfactory description of these phenomena is a rather complex one, and simple mechanical models, i.e. pore deformation, are certainly inadequate. The complex physico-chemical interaction between the membrane and the solution components, as affected by the application of stress, as well as the occurrence of stress-induced conformational transitions, must all be taken into account for a proper interpretation of the phenomena observed. Our experiments were performed by stepwise stretching a strip of membrane (2 cm X 1 cm) held in a frame which allowed unidimensional elongation . The membrane was immersed in 6 M KSCN and stretched up to elongation ratios of about 1 .5. The determination of the filtration coefficient, LP, was particularly difficult under these conditions . Therefore, the permeability to tritiated water under free diffusion conditions was measured . The permeability coefficient to tritiated water, defined as w f -

JT RTAC,.

(6)

was determined with the experimental apparatus shown in Fig .9. Small amounts of tritiated water were put into one of the two half-cells and the diffusion of THO followed at constant intervals of time . Determination of water and salt contents of the stretched membrane in equilibrium with a deuterated solution of KSCN was determined by means of an IR spectrophotometer by measuring the area under the peak at 2500 cm'' . Results are shown in Fig. 10 which includes the swelling and the permeability behavior. Since the permeability appears to be essentially correlated to the swelling behavior, the peculiar variation of the latter with the applied

1 43

Fig.9 . A schematic diagram of the permeability cell and of the frame holding the membrane in the strain-dependent permeability and swelling measurements . (Data from ref. 3).

32

30 a 9

28

26

24 09 0w

v E 08

0 3 11

13

15

a Fig.10. (a) Variation of the overall degree of swelling, for collagen membranes in 6 M KSCN, with strain a . T = 25° C . The dotted line is drawn from eqn . (7) with Mc = 1160 and x, = 0 .63 . (b) strain variation of the permeability of THO, WT, under free diffusion. 6 M KSCN, T = 26° C. (Data from ref. 3) .

1 44

stress deserves special consideration . An increase in the degree of swelling with increasing stress is generally expected in unidimensional elongation for both amorphous and crystalline material . The dotted line is a theoretical curve calculated from the rubber elasticity theory [9] t, = RT [ln (1-V) + V°+ x, (Vo ) 2 + V E E V0 V, J V• V V M, V (V°)2isa2, V W

(7)

where t, is the true stress (force per unit area measured in the swollen, strained state), (Vo /V)H, is the volume fraction of polymer in the swollen unstrained state, p is the polymer density, M c is the molecular weight of a network chain, a is the extention ratio, and the other symbols are as previously defined. This equation is valid for a cross-linked polymer not undergoing the conformational transition . The ascending portions of the V/Vo curve are therefore not unexpected in terms of the rubber elasticity theory and of general phenomenological considerations [9] . The occurrence of the descending portions of the V/V o vs . a plot may be attributed to the stress-induced amorphous to crystalline transformation . Thus, in the present system, depending upon the strain level, either swelling (induced by stress on the homogeneous material) or de-swelling (induced by a partial crystal-amorphous transition) might occur . This combination of effects determines the permeability behavior. The increase of W T with strain up to a " 1 .25 is in line with the filtration behavior observed for amorphous collagen under no strain, when an increase of swelling is accompanied by an increase in hydraulic permeability . The decrease of WT when 1 .25 < a < 1 .35 may then be attributed to the increase of the degree of crystallinity of the membrane during the phase transition, in line with the decrease of hydraulic permeability observed during the amorphous to crystalline transition under no stress. The crystallizing membrane must be viewed as a non-homogeneous, granular type structure where, however, a large degree of crystallinity never develops [3] . Thus, the increase in permeability when 1 .35 < a < 1.45 may be attributed to the fact that the increase in the degree of swelling due to increased strain prevails over an additional increase of crystallinity . (D) Determination of the frictional coefficients [4] We have shown that, in the case of charged membranes, it is possible to evaluate the fixed charge density by means of reflection coefficient measurements, making use of irreversible thermodynamics . We shall now show, as an example, a system in which, from phenomenological coefficient measurements, it is possible to deduce the frictional interactions between the various components of the system . To this end we need to consider a four component system : membrane, sucrose, salt, and water . The water-salt solution will be considered here as a one-component diluent, in other words we attribute to firm binding any possible "enrichment" effects, i .e. disproportions between water and salt 17] . The membrane is viewed as a flexible struc-



145

ture, whose physical state is determined, among other factors, by the concentration of electrolytes in the bathing solution . Thus, the permeation of uncharged solutes and solvent is strongly dependent upon the type and concentration of salt in the bathing solution . We have already seen that induced changes of the structure of the membrane are reflected in variations of several hundred percent of the values of the phenomenological coefficients L p and a . The sucrose molecule may therefore be viewed as a probe of a given structure. Fig.11 shows the trend of the phenomenological coefficients, together with the degree of swelling, as a function of CaC1 2 concentration, when, at variance with the cases previously considered, sucrose (I or which we use the subscript z) is also present . In this figure the values of a and w a (the latter determined using sucrose labelled with C'") refer to the reflection and permeability coefficients of sucrose . In order to explain the occurrence of the maximum (minimum in the case of a) in these curves, the frictional interaction between the various compo-

m T a d 0 E

25

>° 2 1

0 N

4

3 2

3

17

t I 1 2 3 4 5

1

2

3

4

05 18

1D

03

01 1

2

3

4

1 2 3 4 C 5 , Mo1/I Fig. 11 . (a) Degree of swelling as a function of CaCI, concentration in a solution containing 0 .2 M sucrose . (b) Filtration coefficient, L P , as a function of salt concentration in a solution containing 0 .2 M sucrose . (c) Permeability coefficient of sucrose, w i , as a function of salt concentration (sucrose 0 .2 M). (d) Sucrose reflection coefficient, a, as a function of salt concentration . Sucrose (0.2 M) present only in one side of the cell . T = 29'C . (Data from ref. 4) .



146

nents (sucrose-diluent : zw ; sucrose-membrane : zm; diluent-membrane : wm) was evaluated by means of the following relationships [5,6] fzw = 9tiw(1-a)/wzOx fzm

= $w

fwm

=

(8)

[1- (1 - a)(1+ozaz)]/w z Ox

mw ow [' -Ax

(9)

(1-a) (a+coin/Lp)ez]

Lp-

wz

(16)

where the symbols are as previously defined . An equivalent description would be to give, instead of fwm , the "equivalent pore radius" r, determined from the eqn . [5] 8nLpAx) i/z

r= l ') Ow

(11)

22

14

m

p13

(a)

6 1

1

1

I

1

E 14 0 E LI ~m

6

18

1!7 cb)

T O

I

60 16

20

J

~

10 fCl

'

I

I

1

I

16 14 aG

12 10 8

Id)

6

I

1

3

5

CCaC12' Mol/1 Fig.12. The frictional coefficients, (deduced from data in Fig . 11), (a) fwm, (b) fzm and (c) fzm+ and the equivalent pore radius, r, as a function of salt concentration . (Data from ref.4 ) .



1 47

where n is the viscosity of the solvent. The results of the calculations are shown in Fig .12. For all the parameters considered, the point of minimal interaction - or the point of maximum pore radius - corresponds to the salt concentration at which the phase transition is at about its midpoint . The friction between solvent and membrane, fw,,,, (Fig. 12a) is much smaller than the friction involving sucrose . At this point it is appropriate to remember that the degree of swelling of the membrane in CaCI2 solutions reflects, through the X, parameter [12] the thermodynamic affinity of the salt solution (regarded as a one-component diluent) for the collagen matrix (possibly including bound salt and water). Thus it appears that a large thermodynamic affinity between a membrane and the bathing solution corresponds to a reduced frictional resistance for the flow of that solution through that membrane . This finding more adequately accounts for the previously given qualitative description of swelling of the amorphous membrane by a solvent which is in a relatively "looser state" (note that these considerations refer to a membrane analyzed through its second transition when a high degree of crystallinity was never present). The change of the frictional interaction between sucrose and membrane with salt concentration appears to be correlated with the variation of the pore radius (Fig. 12b and d) . It is also interesting to observe that the frictional interaction between sucrose and solution (Fig .12c), which does not directly involve the collagen component, shows a minimum as a function of salt content. List of symbols • • • • • • • • • • • • • • • •

mean concentration of species i frictional coefficient between components i and k volume flow per unit area and time flow of THO binding constant of species i filtration coefficient molecular weight of polymer network equivalent pore radius gas constant ionic strength transport number of counter-ion in free solution true stress absolute temperature partial molar volume of species i volume of swollen membrane volume of dry membrane molar volume of solvent concentration of fixed charges in the swollen membrane concentration of fixed charges in the unswollen network

1 48 a OP Ax A

>V

77

ve P a X1

Wi

extension ratio hydrostatic pressure difference membrane thickness osmotic pressure difference electric potential difference viscosity of solvent cross-linking parameter polymer density reflection coefficient interaction parameter permeability coefficient of species i

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