J. Phys. Chem. 1993, 97, 9817-9824
9817
Diffusion and Interaction in Gels and Solutions. 6. Charged Systems Lennart Johansson' Department of Physical Chemistry, Chalmers University of Technology. S-412 96 Gbteborg, Sweden
Urban Skantze and Jan-Erik Liifrotb Department of Drug Delivery Research, Pharmaceutical R&D, Astra Hassle AB, S-431 83 Mblndal, Sweden Received: March 30, 1993; In Final Form: June 20, 1993'
Results are presented from Brownian dynamics simulations and from experimental measurements of selfdiffusion coefficients of monovalent ions in systems of oppositely charged polymers. The simulations were carried out in ordered and in random polyion systems. The ordered systems consisted of parallel straight chains on a square lattice, while in the other systems the straight polymer chains were randomly oriented. At low polymer concentrations there was good agreement between the results obtained in both types of systems. However, when the polyion concentration was increased, the ion diffusion was slower in the random than in the ordered systems. It is suggested that the observed results emanate from the distribution of interpolyion distances and the relative orientation of the chains present in the random systems. The interpretation is based on the transport in the bulk between the polyions as the rate-limiting process. The simulation data are in qualitative agreement with the experimental results. However, the experimental results also emphasize the importance of steric obstruction, which can be the dominant effect for large ions and in concentrated polyelectrolyte systems.
Introduction The diffusion of solute molecules in polymer gels and solutions is of both fundamental and applied interest. In our continuing work we have so far focused on the obstruction effect on diffusion due to the excluded volume caused by the polymer chains, both in the solution and in the gel state. Our studies have considered the effect of solute size, polymer concentration, and the structure in such systems. As examples, we have studied the diffusion of small monodisperse poly(ethy1ene glyco1)s of different sizes1 and of nonionic micelle-forming surfactants* in polysaccharide solutions and gels. It has been shown that when there are no attractive interactions between the diffusing solutes and the polymers, a hard-spheretheory3 could describe the experimentally measured self-diffusion coefficients1.* as well as Brownian dynamics simulation data.4 The theory utilizes theoretical results for the distribution of nearest-neighbor distances in polymer systemsSand the so-called cylindrical cell m ~ d e l . ~This , ~ model consists of an infinite cylindrical cell, containing polymer and solvent, with the polymer represented as a rod, centered in the cell. Our theory contains parameters that are possible to determine from independent experiments and has provided fundamental knowledge about the diffusion in polymer systems with repulsive interactions. In this paper we address the question of how attractive interactions, between the diffusing species and the polymers, influence the diffusion. Attractive interactions are found for example in polyelectrolyte systems, between counterions and charged polymers. Characteristic for such interactions is the accumulation of counterions close to the polymer. This might be described by Manning's counterion condensation theory>.9 which differentiatesbetween counterionsin the immediatevicinity of the polyion as being 'bound" or condensed and the "free" counterions. Alternatively,the ion distribution in charged polymer systems might be described by the Poisson-Boltzmann (PB) equation in the cylindrical cell model.1° Results from such calculations show that there is a continuously varying concentration of ions in the radial distance from the polyion. Moreover, the PB theory has been found to give a good description of the To whom correspondence should be addressed. Abstract published in Advance ACS Abstracts, August 15, 1993.
distributionfor monovalent ions, when compared to more accurate Monte Carlo' 143 or Brownian dynamics simulations.14 Several theoretical descriptions of counterion self-diffusion in systems of charged polymershave been presented. Thus, Manning derived a theory utilizing his condensation concept.15 Also, a theory based on the PB equation in the cylindrical cell model was derived by Yoshida,16 Belloni et a1.,6 and Nilsson et al.7 In this theory, called the Smoluchowski-Poisson-Boltzmann (SPB) theory, the diffusion coefficient is given as the weighted average of the diffusion parallel to the polymer (assumed to be unperturbed) and of the radial diffusion. This is of course a correct result considering the geometry of the cell. Calculations with the SPB theory are in qualitative agreement with experimentally measured self-diffusion coefficient^.^*^^-*^ However, the theory in general predicts too high diffusion coefficients compared to experimental data. This has been interpreted as effects from the assumed free diffusion parallel to the chains in the theory or from assumptions inherent in the PB theory. A more general version of this theory was derived by JBnsson et al.,zo who showed that it is possible to calculate the effective ion diffusion coefficient, given only the ion concentration distributionand the local diffusion coefficient in such systems. The theory by JBnsson et al. is a very important improvement, since using their approach it is possible to examine how ion diffusion is influenced by, for example, ion size and ion correlation, effects neglected on the PB level. Thus, ion distribution profiles obtained from Monte Carlo simulations or more refined electrostatic theories21can be utilized together with the theory by JBnsson et al. to calculate the effectivediffusion coefficient. However, almost all theoretical work in the area of counterion diffusion in polyelectrolyte systems has been carried out with simplified geometrical models, such as the cylindrical cell model. Such treatment neglects, for example, effects from the distributions of the relative orientation and in the separation of neighboring polyions, certainly present in real polymer systems. In this paper we therefore compare results obtained from Brownian dynamics simulationsof the diffusion of a pointlike ion in systems with randomly arranged polyions to the diffusion in more regular systems. Moreover, a limited amount of reported experimental work has considered the combined effect of obstruction and electrostatics on solute diffusion in real systems.22 Thus, we
0022-3654/93/2097-98 17$04.00/0 0 1993 American Chemical Society
Johansson et al.
9818 The Journal of Physical Chemistry, Vol. 97, No. 38, 1993
present results from measurements of the diffusion coefficient in polyelectrolytesolutions and gels of an ion and of an uncharged molecule of comparable sizes.
Computational Methods SimulationSystems. The simulationof ion diffusion (see below) was carriedout in both ordered and random systems. The ordered systems were composed of parallel charged straight chains arranged on a square lattice. The random systems, hereafter denoted polymer networks, contained randomly oriented charged wormlike chains with a persistence length of lo6 A. The chains in both systems consisted of spherical subunits-beads-with a radius a* = 1.51/2a, where a is the radius of the corresponding volume-adjusted cylinder. The construction of the polymer network has been described in detail elsewhere? Briefly, a network was obtained by first putting a bead at one of the sides of a cubic box. One bead at a time was then added at touching distance, by sampling bond angles from a distribution characteristic for the given persistence length and bead radius.23 When a bead was added outside the box, a new side was chosen and the procedure was repeated until the desired polymer volume fraction was reached. The last chain was always allowed to reach the side of the cube, modeling infinite chain lengths. Also, the polymer chains were not allowed to cross each other, and the overlappingchains were then discarded. Each simulationdescribedbelow was camed out in 600 different polymer networks, each consisting of approximately 250 beads, which also was the number of beads used in the ordered systems. For comparison,systemscontaining randomly arranged parallel charged straight chains were also studied to some extent. These systems then differed from the ordered parallel systems described above by their random positions of the chain centers and from the polymer networks by their parallelism of the chains. Dynamic Simulations. Brownian dynamics simulations were carried out of the diffusion of a pointlike ion (valence z = +1) in an electrostatic background created by the polyions together with the ions present in the systems. The algorithm by Ermak and McCammon was used.24 Thus, neglecting the hydrodynamic interactions, the displacement Ar from a position r during a time step At was assumed to be described by
Ar = S ( A t )
+ DoAt -F(r) kT
Here, S(Ar) is a random displacementobtained by sampling from a Gaussian distribution with (S) = 0 and ( S s p ) = 6,,92DoAt (Greek indices denote Cartesian coordinates and 6 is the Kroenecker delta), DOis the diffusion coefficient of the ion in a polymer-free system, k is the Boltzmann constant, and T i s the temperature. In the simulations DOwas taken to be 2.45 X m2/s and T = 298.15 K. The interaction between the ion and the charged polymer chains was described by an electrostatic forceF(r) and a hard-core interaction. The hard-core interaction was accounted for by rejecting displacements that positioned the ion within polymer beads. The calculation of the electrostatic force was based on the fact that F(r) can be written as
F(r) = -zeV\k(r) where z is the valence of the diffusing ion, e is the electron charge, and 9 ( r ) is the electrostatic potential. For 9(r), the D e b y e Hiickel potential ~ D can H be used, i.e.
with 9DH(r) given by
In this expression ri is the distance between the ion and the ith
TABLE I: Values of the Apparent Charge Separation P Obtained from Fitting Eq 6 to Solutions of the Poisson-Boltzmann Equation in the Cylindrical Cell Model for the r-Carrageenan Coil and Helix Conformation' system coil ( a = 3.5 A, I = 5.0 A)
helix ( a = 5.3 A, 1 = 2.2A)
4 0.00108 0.00524 0.0155 0.0256 0.00108 0.00524 0.0154 0.0256
I* (A) 6.78 6.44 6.16 6.04 4.59 4.18 3.75 3.55
a The temperature was 298.15 K, and the concentrationof added 1:l electrolyte was 10 mM.
polymer bead, q = e2a*/l is the charge of a polymer bead where I is the projected charge separation, and crtO is the dielectric constant of the solvent. Moreover, K is the reciprocal Debye screening length calculated as
(
1 03NAe22Z trtOkT
)
where NA is the Avogadro number and I is the ionic strength (in molar units) of the system including all mobile ions, i.e., both polymer counterions and added salt. This approach then represents a linear superpositionof screened Coulomb potentials. However, it has been shown that if effective charge values of polyelectrolytes areused, it is also possible to account for nonlinear effects in an approximate way with the use of screened Coulomb potentials.2s29 This corresponds in our case to the useof somewhat larger distances between the charges on the polymer chain than those obtained from structural information. In order to estimate the apparent chargeseparation,denoted I*, weused the nonlinear Poisson-Boltzmann equation in the cylindrical cell model (PBCM).IO In this model, the rod radius a and the cell radius b are related to the polymer volume fraction, 4, by 4 = a2/b2. A comparable, although not equivalent, model in our approach is to consider a (straight) polymer chain placed at the origin and oriented along the z axis. To account for the effect of polymer concentration, we placed another chain, parallel to the first one, at a radial distance p = 26. This assured the electroneutrality condition at the cell border 6, i.e., d9/dp = 0, used in the PBCM. Thus, for the resulting potential at p we have
where Az = 2a* is the bead diameter and ~ D isH the DebyeHiickel potential with an effective charge q = -e2a*/l*, eq 4. Furthermore, in eq 6 we also set 9 ( b ) = 0. An estimate of I* was then obtained by fitting eq 6 to a numerical solution of the PBCM. It should be noted that the effect of an apparent charge also alters the Debye screening length, due to a change in the concentration of the polymer counterions. This is in analogy with the concept of counterion conden~ation.s.~ As described above, Brownian dynamics simulations were carried out in both ordered and random systems. Furthermore, we also considered two different polymer conformations, characterized by radii a and charge separations 1. The values of a and 1 were taken to represent warrageenan in the coil and helix conformation, respectively.3'J Thus, applying the fitting procedure, we obtained the effective charge separation I* for these two polymer conformations. The results obtained with 10mM added 1:1 electrolyte at different polymer volume fractions are given in Table I. Thesevalueswere then used in thesimulationstocompute
The Journal of Physical Chemistry, Vol. 97, NO. 38, 1993 9819
Diffusion and Interaction in Gels and Solutions the force in the equation of motion (see eqs 1-4). In the simulations we used periodic boundary conditions to simulate an infinitesystem. Also, the minimum image convention was used when the distances between the ion and the polymer beads were determined and, thus, when the force in eq 2 was calculated. Furthermore, to account for the equilibrium distribution of ions in our systems, the following procedure was applied to each trajectory. First, the starting point was chosen at random in the simulation cube. Then we checked whether this point resided within the polymer beads. If it did, a new starting point was chosen at random; if not, the probability of finding the ion at this particular point was evaluated as (7) using the sum of screened Coulomb potentials given by eqs 3 and 4. From each trajectory a (time-dependent) diffusion coefficient of the ion was calculated as
D(t)
-61 dw dt
where w(t) is the mean-square displacement. The diffusion coefficient was then finally multiplied by the starting point probability p ( r ) / z p ( r ) . The way of calculating the diffusion coefficient,i.e. eq 8, is analogous to D(t) = w(t)/6r but approaches the long-time limit faster, as shown by Cichocki and Hinsen.3' According to these authors the time-dependent diffusion coefficients in eq 8 is given by
(9) where D is the long-time self-diffusion coefficient and T the characteristic time for the relaxation from short-time to longtime behavior. Thus, each DIDO, which was the primary interest in this work, was estimated from simulations of 600 trajectories with 2 X 104, 4 X 104, 8 X lo4, and 1.6 X lo5 time steps. The time steps used were in the range 0.075-0.6 ps. From each trajectory, 2 X lo4 configurations evenly spread in time were collected and used in the averaging procedure described above. Equation 9 was then fitted to the resulting (time-dependent) diffusion coefficients, giving estimates of DIDO and T for each time step. Within the experimental accuracy, the long-time selfdiffusion coefficients showed no significant influence from the length of the time steps used, in contrast to previous hard-sphere simulations.4 Thus, we finally calculated DIDO as the mean over the time steps, giving standard deviations of approximately 5% or less. It should be mentioned that several simulations were carried out with a larger number ( 1000) of polymer beads than reported above (-250). However, for these larger systems we obtained the same diffusion results. This is because that, already for the smaller systems, the chains are appreciably longer than the Debye screeninglength.32 Thus, end effectson the electrostatic interactions and on the diffusion are negligible for such systems.
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Experimental Section
Materiala A sample of choline chloride, HOCHzCHzN(CH3)3Cl, was obtained from Aldrich and used without further purification. Radioactivity labeled choline+ (I4C, CFA 424, 54 mCi/mmol) as the chloride salt and glycerol (3H, TRA 244,2.8 Ci/mmol) were purchased from Amersham. Commercialsamples of K-carrageenan (C- 1263) and i-carrageenan (C-4014) were obtained from Sigma and converted to their pure ionic forms, Na+ or K+,as described elsewhere.' The presence of i-carrageenan impurities in the purified K-carrageenan samples was determined to be 10% (mol/mol) by 'HNMR measurements at 90 OC,1,33 Approximately the same amount of K-carrageenan impurities was found in the purified i-carrageenan sample. The amount of counterions in the purified samples was measured
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with atomic absorption spectroscopy. The K+-K-carrageenan contained 2.5 mmol of K+/g, the Na+-K-carrageenan 2.7 mmol of Na+/g, and the Na+-r-carrageenan 4.0mmol of Na+/g. These values should be compared to the values 2.4, 2.5, and 3.9 mmol of ionlg, respectively,calculated from the ideal molecular weights per charge of the polysaccharides. The polymer volume fraction, 4, in gels or solutions of carrageenan (using water as the solvent) was calculated from the partial specific volumes, u = 0.50 mL/g for K-carrageenan and u = 0.47 mL/g for i-carrageenan34 DiffusionMeasurements. The measurements were carried out as described elsewhere.35 Briefly, a 1-mL plastic syringe with the top cut off was used. The syringe was filled with a gel or a solution (including salt at a chosen concentration) and allowed to equilibrate in a temperature box for 24 h. When solutions were examined, small glass beads were also added to prevent disturbances from for example environmental shaking. An experiment was started by applying a 0.2-ccL drop of a solution containing radioactively labeled choline+ and glycerol on the top of the syringe. At the end of the experiment (after 15-24 h at the chosen temperature), the contents of the syringe was carefully pressed out and simultaneouslycut into pieca, which were sampled in scintillator vials and analyzed in a @-counter. All experiments were carried out in at least triplicate, giving standard deviations less than 5%. Rheological Measurements. The elastic modulus G' of K+K-carrageenan and Na+-warrageenan gels was measured with a CarriMed CSL rheometer. A parallel plate geometry was used (gap 4 mm, plate diameter 2 cm, stainless steel) with grooves in the upper plate to prevent gel slippage. The bottom plate had an edge to hold solutions in place. A hot solution (-8 mL) was transferred to the bottom plate, which was pretemperated to 55 OC. The solution was covered with a viscosity standard oil (ASTM, 10 cP) to prevent evaporation. The temperature was then lowered to 5 OC at a cooling rate of 1 OC/min and the sample allowed to equilibrate overnight. The elastic modulus was finally measured at 1 Hz in intervals of 5 OC. The desired temperature was attained with a heating rate of 0.3 OC/min, and an equilibration time period of 30 min at each temperature was used.
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R@ults
Brownian Dynamics Simulation Data. The results from our Brownian dynamics simulations are presented as the ratio of the ion self-diffusion Coefficientin a polymer system, D, to thediffusion coefficient in a polymer-free system, DO. The simulations were carried out for the diffusion of one pointlike monovalent ion in an electrostatic background created by the polyions together with the ions present in the systems. The electrostatics was treated on the Poisson-Boltzmann (PB) level, thus neglecting for example ion-ion correlations. Moreover, the simulations were carried out for two differently charged polymer conformations representing the i-carrageenan in the coil and the helix state. The charge separation 1 used for the coil conformation was 5.0 A, while 2.2 A was used for the more densely charged helix conformation.30 In Figure 1 results are presented from simulations of ion diffusion in systems of parallel stiff chains placed on a square lattice, as functions of the polymer volume fraction 4. The figure shows the diffusion coefficientsfor diffusion parallel to thechains, Dp,and for the diffusion in the radial direction to the chains, 4. For such systems, the isotropic diffusion coefficient is given by D -p-D- -- -l + DO
3 DO
2Dr 3 DO
These systems then resembled the cylindrical cell model, except for the square lattice arrangement and the discreteness of the charges used. The simulation data can be compared to the theoretical results presented for ion diffusion in the cylindrical
9820
The Journal of Physical Chemistry, Vol. 97,No. 38, 1993
Johansson et ai.
1.o
0.8
0.6
0.44 0
1
0.02
0.01
0.03
0.70-
Q Figure 1. D/Dovs the polymer volume fraction Q from simulationsof ion diffusion in systemsof parallel straight chains on a squarelattice with polymer parameters representingthe carrageenan coil (filled symbols) and helix (open symbols). The circles show the results obtained for diffusion parallel to the chains and the squares the radial diffusion. The concentration of added 1:l electrolyte was 10 mM, and the temperature was 298.15 K. The two lower solid lines show the theoretical results for the radial diffusion coefficient using the Smoluchowski-Poisson-Boltzmann theory in the cylindrical cell model.'
2
1.o'L 0
C T
.r
I
10
20
40
30
Figure 3. Elastic modulus G' (upper part) and experimental diffusion quotients for choline+ DIDO(lower part) vs temperature in 3% (w/w) Na+-c-carrageenan with 1 mM choline chloride.
6000
C 4000h
0.41 0
0.01
0.02
0.03
m
5
0 Figure 2. D/& vs the polymer volume fraction Q from simulationsof ion diffusion in systemsof randomly orientedstraightchainswith polymer parameters representingthe L-carrageenancoil (filled symbols) and helix (opensymbols). Theasterisksshowresultsobtainedfortheradialdiffusion coefficient in systems with randomly arranged parallel straight chains. The concentration of added 1:l electrolyte was 10 mM, and the temperature was 298.15 K.
cell model, called the Smoluchowski-Poisson-Boltzmann (SPB) equation.6.7.16 The lower solid lines in Figure 1 show the prediction with this theory for the radial diffusion coefficient. It is seen that the agreement between this theory and the simulation data was good, which supported the validity of the computational procedures. Furthermore, in the SPB theory the diffusion in the direction parallel to the chains is assumed to equal the free diffusion, DO. This is inherent in the model where the polymer charge is smeared out on the cylinder surface. Thus, the variation of the electric field along the chain is neglected in the model. However, in our simulations we used a discrete charge model. As seen in Figure 1, the discreteness of the polymer charge results in a reduction of D,/Do to at most -0.85 for our systems. In Figure 2 results are presented from simulations of the isotropic diffusion coefficient in systems with randomly oriented chains, using the same polymer parameters as for the ordered systems (see Table I). It can be seen that a t low polymer concentrations (4 0.001) we obtained the same results in the random systems as for the isotropic diffusion coefficient in the ordered systems (obtained with eq 10 applied to data of Figure 1). However, when the polymer concentration was increased, the diffusion in randomly oriented systems was even lower than the radial diffusion obtained in the ordered systems. We also carried out simulations in systems with randomly arranged parallelchains, a t one polymer concentration (4 = 0.0256). The
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2000-
0 0
10
20
30
40
50
10
20
30
40
50
1
0.65
0
c
T(") Figure 4. Elastic modulus G' (upper part) and experimental diffusion quotients for choline+ D/& (lower part) vs temperature in 2% (w/w)
K+-K-carrageenan with 1 mM choline chloride.
radial diffusion coefficients obtained in such systems are shown as the asterisks in Figure 2. Diffusion Data The experimental work concerned the diffusion of choline+ in polymer gels and solutions of carrageenans. In Figures 3-6 the diffusion results are presented as the ratio of the diffusion coefficient of choline+ in a polymer system, D, to the diffusion coefficient in a polymer-free system, DO.In Figure 3, DIDOis presented as a function of temperature for 3% (w/w) Na+-&-carrageenan with 1 mM choline chloride. Also, mea-
Diffusion and Interaction in Gels and Solutions
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The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 9821
II \y-\* -
L \., t
I
0.01
0
0.02
@
0.61 0
I
0.03
?gum 5. D/& vs the polymer volume fraction 4 for choline+ diffusion Na+-K-carrageenan at 25 OC (triangles), K+-K-carrageenan at 5 OC (circles), and Na+-&-carrageenan at 5 O C (squares). The arrows indicate
in
the gel points for the K+-K-Carrage" and for the Na+-r-carrageenan. (Na+-K-carrageenan did not gel within the investigated concentration range). The concentration of added choline chloride was 1 mM.
0.01
@
0.02
0.03
Figure 7. Ratio of the diffusion coefficient of a counterion, D+, to the diffusion coefficient of an uncharged molecule of comparable size, D,,, vs the polymer volume fraction 4. The results were obtained for choline+ andglycerol inNa+-K-carrageenanat 25 O C (triangles), K+-rc-carrageenan at 5 OC (circles),and Na+-r-carragccnan at 5 O C (squares). The arrows indicate the gel points for the K+-K-carragc" and for the Na+-r carrageenan. (Na+-K-carrageenan did not gel within the investigated concentrationrange). The concentrationof added choline chloride was 1 mM.
0.8-
1.0,
0
0:1
0.2
0.3
cs (M)
-6. D/&vs the concentration of added salt C, for choline+diffusion in 3% (w/w) Na+-rcarrageenanat 5 OC, using NaCl (circles) and choline (squares) as the added salt.
surements of the elastic modulus G'are presented, which showed that the polymer underwent a gelation in the temperature range investigated. It is seen that at low temperature (-5 "C), where the polymer was in the gelled state, the choline+ diffusion was lower (DID0 0.6) than at high temperature (DIDO 0.7). At high temperature the polymer was in the solution state (G'- 0). The results in Figure 4 show the choline+diffusion together with the elastic modulus for 2% (w/w) K+-K-carrageenan. Again, a somewhat lower diffusion was found in the gelled state than in the solution state. However, DIDOwas somewhat higher in this latter polymer system for both the gelled and the solution state. In Figure 5 results from choline+ diffusion are presented for three different polymer systems as a function of the polymer volume fraction 4. However, it should be noted that the data are presented for different temperatures. Thus, the Na+-K-carrageenan was in the coil conformation at all concentrations at 25 OC. On the other hand, the K+-K-carrageenin and the Na+&-carrageenanwere gels above 4 0.0035 and 4 0.005, respectively, at 5 OC. It is seen in Figure 5 that the choline+ diffusion was faster in the Na+-K-carrageenan coil system (I 10 A)30 than in the K+-K-carrageenan system. Above 4 0.01 the K+-K-carrageenan system seemed to be completely in the helix conformation (I 4 A).30 This was concluded from the temperature-independent choline+ diffusion at 4 0.01 at temperatures below - 5 OC, as seen in Figure 4. However, the differences in DID0 were not pronounced, especially not in the high concentration range. Furthermore, the slowest diffusion was found in the Na+-&-carrageenansystem, which above 4
-
-
-
-
-
--
-
-
0.64
0
c
0.
0.1
C,(M) Oa2 Figure 8. Ratio of the diffusion coefficient of a counterion, D+, to the diffusion coefficient of an uncharged molecule of comparable size, D,,, vs the concentration of added salt C,. The results were obtained for choline+and glycerol in 3% (w/w) Na+-c-carrageenan at 5 OC, using NaCl (circlcs) and choline chloride (squares) as the added salt.
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0.015 was completely in the helix conformation ( I 2 A);N see Figure 3. The effect of the concentrationof added salt on the ion diffusion in polyelectrolyte systems was also studied. Results obtained fromsuch studiesare shown in Figure 6 for the diffusion of choline+ in 3% Na+-&-carrageenanat 5 'C, with two different added salts (NaCl and choline chloride). At the chosen temperature and polymer concentration, &-carrageenanwas completely in the helix conformation. It is seen that increasedsalt concentration resulted in faster ion diffusion, as expected. However, this effect seemed to be somewhat more pronounced when using NaCl than with choline chloride. All results presented so far in this section have considered the diffusion of choline+in different polymer systems. The diffusion of an ion of finite size in such systems is influenced not only by electrostatic interactions but also by obstruction effects. To estimate the obstruction effects, we also measured the diffusion of an uncharged molecule, glycerol, in these systems. This molecule is of comparable size to choline+,as determined from the diffusion coefficients obtained in aqueous solutions (differed by less than 5%). In Figures 7 and 8 we show the ratio of the choline+ diffusion coefficient, D+, to the glycerol diffusion coefficient, D,. The D+/D, quotient obtained in this way was correctedfor the small differences in aqueous diffusion coefficients (
