J. Chem. Thermodynamics 68 (2014) 293–302
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Transport, and thermodynamic investigations on sodium polystyrenesulfonate in ethylene glycol–water media Chanchal Das a, Bijan Das b,⇑ a b
Department of Chemistry, Sikkim Government College, Tadong, Gangtok 737 102, India Department of Chemistry & Biochemistry, Presidency University, Kolkata 700 073, India
a r t i c l e
i n f o
Article history: Received 1 July 2013 Received in revised form 14 September 2013 Accepted 16 September 2013 Available online 27 September 2013 Keywords: Ethylene glycol–water mixed solvent media Sodium polystyrenesulfonate Conductivity Fractions of uncondensed counterions Colby model
a b s t r a c t Polyion–counterion interactions in sodium polystyrenesulfonate dissolved in (ethylene glycol + water) mixed solvent media have been investigated conductometrically with special reference to their variations as functions of polyelectrolyte concentration, relative permittivity and temperature. Manning counterion condensation theory for polyelectrolyte solutions failed to describe the present experimental results. The data have, therefore, been analyzed using a new model for semidilute polyelectrolyte conductivity which takes into account the scaling arguments proposed by Dobrynin et al. The fractions of uncondensed counterions were found to depend on the polyelectrolyte concentration varying from 0.27 to 0.37, within the concentration range investigated here, indicating a strong interaction between counterions and polyion. A considerable fraction of the counterions is shown to migrate in the same direction as the polyions. The results further demonstrate that the monomer units experience more frictional resistance in solutions as the ethylene glycol content of the mixture increases or as the temperature decreases. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Polyelectrolytes are polymers with many ionizable groups, which in polar solvent media dissociate into a polyion and counterions of opposite charge [1,2]. The thermodynamic and transport properties of polyelectrolytes in solutions are mainly controlled by the interactions between the polyion and counterions. For example, the transport properties, studied by electrical conductivity, are of central importance in accounting for the solution behavior of polyelectrolytes because the electrical conductivity takes into account the movement of any charged entity present in the system under the influence of an externally applied electric field. Current interest in charged polymer solutions has also been supported by the needs of biophysics since biopolymers are charged under physiological conditions and many of their biological functions are governed by the polyelectrolyte behavior [3]. Although the electrical conductivity has so far been measured for a great variety of polyelectrolytes [4–17], only a few studies reported the influence of medium and temperature on the interaction between a polyion and its counterions derived from conductivity measurements [18–21]. We have, therefore, initiated a program to investigate the behavior of different polyelectrolytes in various mixed solvent media as a function of temperature [22– 25]. Here we present a study on the electrical conductivity of the
⇑ Corresponding author. Tel.: +91 94752 49401. E-mail address:
[email protected] (B. Das). 0021-9614/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jct.2013.09.022
semidilute solutions of sodium polystyrenesulfonate in (ethylene glycol + water) mixed solvent media at different temperatures and the data have been analyzed on the basis of the Manning counterion condensation theory as well as the model derived from the scaling concept. The objective of this contribution is to examine the influences of the polymer concentration, the temperature and the medium on (i) the fractions of uncondensed counterions, (ii) the polyion equivalent conductivities, (iii) the standard state free energies of counterion association, (iv) the polyion transference numbers, and (v) the solvodynamic friction coefficients of the polyion in the solution to provide a comprehensive understanding of the counterion condensation phenomena in polyelectrolyte solutions. The results are discussed from the viewpoint of the general solution behavior of polyelectrolytes. 2. Theory Generally, the equivalent conductivity of polyelectrolyte solutions is given by [26]
K ¼ f ðk0c þ kp Þ; k0c
ð1Þ
where and kp are the limiting ionic equivalent conductivity of the counterion and the equivalent conductivity of the polyion at a finite concentration, respectively, and f is the fraction of uncondensed counterions.The description of different electrical properties of polyelectrolytes in solutions and of their interactions with counterions is generally based on the Manning counterion condensation
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C. Das, B. Das / J. Chem. Thermodynamics 68 (2014) 293–302
theory [27–29]. In this model, the polyion is represented by an infinitely long charged line. The small counterions are assumed to form an ionic atmosphere whose density depends on the frame of the polyion and their interactions with the charged polyions are purely Coulombic in nature, so that the screening effect extends over the Debye length. The uncondensed mobile ions are treated in the Debye–Hückel approximation. The solvent is assumed to be a dielectric continuum characterized by a spatially uniform relative permittivity e. Interactions among the polyions are neglected, the theory being addressed to highly diluted solutions.According to the Manning counterion condensation theory, polyelectrolytes have been characterized by a linear charge density parameter defined by [26,28]
n¼
e2 ; ekB Tb
ð2Þ
where e is the protonic charge, e the relative permitivity of the medium, kB the Boltzmann constant and T the absolute temperature and b the contour distance per unit charge. This theory states that if n > 1, enough counterions condense onto the polyion chain to yield the critical value n = 1. If, on the other hand, n < 1, ionization takes place to reach this critical value. Considering the electrophoretic and relaxation contributions to the equivalent conductivity, Manning [26] theoretically derived the equivalent conductivity of a polyion, kp, with counterions each bearing a charge of zc.
kp ¼
279Ajzc j1 j ln jaj 1
1 þ 43:2Aðjzc jk0c Þ j ln jaj
;
ð3Þ
where the parameter a is the radius of the polymer chain, while
A¼
ekB T ; 3pg0 e
ð4Þ
with g0 being the coefficient of viscosity of the solvent. In equation (3), j is the Debye screening constant, which is defined by
j2 ¼
4pe2 n1 cjzc j ; ekB T
ð5Þ
where c is the stoichiometric equivalent polyion concentration. In accordance with this model, the equivalent conductivity (K) is given by equation (2) in conjunction with equation (3) with f being defined as
f ¼
0:866 : n
ð6Þ
Since the Manning theory applies, as stated above, to highly diluted systems where polyion–polyion interactions are assumed to be absent and to polyions modeled as a linear array of point charges, its validity is limited to very low concentration regimes of polyelectrolyte solutions. In most of the earlier studies on the electrical conductivity of polyelectrolyte solutions [4–25], the concentrations are far from being very dilute and are primarily the semidilute solutions (c > overlap concentration, c⁄) have been studied. The application of the Manning model to these systems is, however, less straightforward because these semidilute polyions do not assume a fully stretched conformation in solution. Careful measurements [7,9,14–16,18,22] of the electrical conductivity of aqueous salt-free polyelectrolyte solutions, however, demonstrated a major deviation from the Manning theory. In the case of semidilute polyelectrolytes, the polyion concentration modifies the flexibility of the chain, giving rise to different conformational aspects and hence the Manning model is not applicable to these systems. An alternative theory for the electrical conductivity of semidilute solutions of polyelectrolytes without added salt has been
proposed by Colby et al. [30] using the scaling description put forward by Dobrynin et al. [31] for the configuration of a polyelectrolyte chain. In semidilute solutions, the polyion chain is modeled as a random walk of Nn correlation blobs of size n0, each of them containing g monomers. Each blob bears an electric charge qn = zcefg (zc being the counterion valence and e is the electronic charge) and the complete chain, of contour length L = Nn n0, bears a charge Qp = Nn qn = zcefg Nn. Due to the strong electrostatic interactions within each correlation blob, the chain is a fully extended conformation of ge electrostatic blobs of size ne. This means that for length scales less than n0, the electrostatic interactions dominate (and the chain is a fully extended conformation of electrostatic blobs of size ne), and for length scales greater than n0, these interactions are screened and the chain is a random walk of correlation blobs of size n0. Following this model, in absence of an added salt, the equivalent conductivity of a polyion in a semidilute solution is given by
kp ¼
Fzc efcn20 n ln 0 ; 3pg0 ne
ð7Þ
where F is the Faraday number and the other symbols have their usual significance. Thus according to this model the equivalent conductivity of a polyelectrolyte solution is given by equation (1) with the kp value obtained from equation (7). Within this model, the parameter f – that defines the fraction of uncondensed counterions in the Manning sense – has been treated as an adjustable quantity. Whereas the Manning theory applies to polyelectrolyte solutions in the highly diluted regime and predicts a fraction of condensed counterions independent of the polymer concentration, the Colby model can be applied to more concentrated systems, up to the concentration cD, where the electrostatic blobs begin to overlap and the electrostatic length equals the electrostatic blob size. This new model has been applied, so far, to a limited number of aqueous polyelectrolytes [30,32] and there has, so far, been only one report [25] by our group on the application of this model to a polyelectrolyte dissolved in methanol–water mixed solvent media, and the good agreement with the experiment is very encouraging. Moreover, this model has been successfully employed to identify concentration regimes differing in the fractions of uncondensed counterions [33].
3. Experimental 3.1. Materials Ethylene glycol (E. Merck, India, 99.9% pure) was dried over molecular sieves and fractionally distilled. The middle fraction was collected and redistilled. Triply distilled water with a specific conductance of less than 106 S cm1 at 308.15 K was used for the preparation of the solvent mixtures. The physical properties of (ethylene glycol + water) mixtures used in this study at 298.15, 308.15, and 318.15 K, namely the coefficients of viscosity (g0), and the relative permittivities (e), are reported in table 1. Also included in this table are the limiting equivalent conductivities of the counterion (Na+), k0c in (ethylene glycol + water) mixtures containing 10, 20, and 30 mass percent of ethylene glycol taken from the literature [34]. Sodium polystyrenesulfonate employed in this investigation was purchased from Aldrich Chemical Company, Inc. The sample had an average molecular weight 70,000 and a degree of sulfonation of 1.0.
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298.15 308.15 318.15
1.9025 1.4622 1.1638
0.3 mass fraction ethylene glycol 69.77 3.19 66.30 3.25 63.00 3.31
29.19 36.75 45.81
3.2. Procedure Conductance measurements were carried out on a Pye Unicam PW 9509 conductivity meter at a frequency of 2000 Hz with negligible polarization effects using a dip-type cell with a cell constant of 1.14 cm1 and having an uncertainty of 0.01%. The measurements were made in a water bath maintained within ±0.01 K of the desired temperature. The details of the experimental procedure have been described earlier [35]. Due correction was made for the specific conductance of the solvent by subtracting the specific conductance of the relevant solvent medium (j0) from those of the polyelectrolyte solutions (j). In order to avoid moisture pickup, the experimental solutions were prepared in a dehumidified room with utmost care. In all cases, the experiments were performed at least in three replicates and the results were averaged. 4. Results and discussion
100
-1
monomol )
34.65 42.87 53.01
2
298.15 308.15 318.15
0.2 mass fraction ethylene glycol 1.4693 72.68 3.06 1.1494 69.47 3.10 0.9243 66.27 3.15
k0c /S cm2 eqv1
Λ / (S cm
42.93 52.59 64.12
n
c
80 60 40 20 0
0
0.02
0.04
0.06
0.08 1/2
√c / (monomol
.L
-1/2
0.10
0.12
)
FIGURE 1. Concentration dependence of the equivalent conductivities (K) of sodium polystyrenesulfonate in ethylene glycol–water mixed solvent media containing (a) 0.1 mass fraction ethylene glycol, (b) 0.2 mass fraction ethylene glycol, and (c) 0.3 mass fraction ethylene glycol. Experimental values: 298.15 K (circles), 308.15 K (squares), and 318.15 K (triangles). Predicted values according to Manning counterion condensation theory: solid lines. Predicted values according to scaling theory using poor solvent correlation: dashed lines.
120 100
-1
298.15 308.15 318.15
0.1 mass fraction ethylene glycol 1.1520 75.65 2.94 0.9078 72.07 2.98 0.7381 68.68 3.06
e
monomol )
g0/mPa s
2
T/K
120
Λ / (S cm
TABLE 1 Coefficients of viscosities (g0) and the relative permittivities (e) of ethylene glycol– water mixtures at different temperatures, and the charge density parameters (n) of sodium polystyrenesulfonate and the corresponding limiting equivalent conductance (k0c ) values of sodium ion in these media.
80 60 40 20
4.1. Experimental equivalent conductivity 0
Figure 1(a)–(c) displays the variation of the equivalent conductivity of sodium polystyrenesulfonate solutions as a function of the square root of the polyelectrolyte concentration in three different ethylene glycol–water mixtures (containing 10, 20, and 30 mass percent of ethylene glycol) at 298.15, 308.15, and 318.15 K over the entire concentration range investigated. The representative figure 2, on the hand, shows the polyelectrolyte concentration dependence of the equivalent conductivity at a given temperature (308.15 K) in three different ethylene glycol–water mixtures namely those containing 0.1, 0.2, and 0.3 mass fraction of ethylene glycol. The equivalent conductivities exhibit a slight increase with decreasing polymer concentration. Figure 1(a)–(c) demonstrated an increase the K values with temperature over the entire concentration range in a given solvent media. The effect of medium on the equivalent conductivity values, on the other hand, is directly evident from figure 2, where at a given temperature, the K values are found to decrease as the medium gets richer in ethylene glycol in going from 0.1 mass fraction to 0.3 mass fraction of ethylene glycol in ethylene glycol–water mixtures over the entire polyelectrolyte concentration range. 4.2. Comparison with the Manning counterion condensation theory Now we will compare the experimental K values with those calculated using the Manning theory. The charge density parameters n were calculated from equation (2) using the literature value
0
0.02
0.04
0.06
0.08 1/2
√c / (monomol
L
-1/2
0.10
0.12
)
FIGURE 2. Concentration dependence of the equivalent conductivities (K) of sodium polystyrenesulfonate in ethylene glycol–water mixed solvent media at 308.15 K. Experimental values: 0.1 mass fraction ethylene glycol (circles), 0.2 mass fraction ethylene glycol (squares), and 0.3 mass fraction ethylene glycol (triangles). Predicted values according to Manning counterion condensation theory: solid lines. Predicted values according to scaling theory using poor solvent correlation: dashed lines.
for the length of the monomer unit having one charged group [36] and these are listed in table 1. The theoretical values of kp and hence of K, are dependent on a, the radius of the polyion cylinder. For the radius of the assumedly rod-like polymer cylinder, we used a value of 0.8 nm for the present analysis [8]. Theoretical predictions (lines) based on this model along with the experimental K values (points) are shown in figures 1(a)–(c) and 2. The experimentally obtained equivalent conductivities have always been found to be considerably higher than the theoretical values calculated following the Manning model. Deviations [9,14,15,18] of the theoretical values from the experimental values were also noticed earlier for other polyelectrolyte solutions. The discrepancy probably arises from the fact that the polyelectrolyte solutions investigated here are different from the model that underlies equation (3).
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The reason for the failure of the Manning model can be understood if one estimates the overlap concentration (c⁄) for the polymer chain investigated using the following equation [17]
c ¼
1 NA L2 a
ð8Þ
;
T/K
where L is the countour length and the other symbols have their usual significance. It is observed that the polymer solutions in the present study are essentially in the semidilute regime where the Manning limiting law does not apply (c⁄ 0.0001 eqv L1). 4.3. Scaling theory and the fractions of uncondensed counterions Since the Manning theory fails to describe the conductivity behavior, we have employed the scaling theory approach of the polyelectrolyte conductivity proposed by Colby et al. [30] for semidilute polyelectrolyte solutions for the analysis of the present conductivity data. The electrostatic blob size (ne) and the correlation blob size (n0) appearing in equation (7) depend upon the quality of the solvent and are, for poor solvents, given by [30]
A2 ne ¼ b n
n0 ¼ ðcbÞ
!1=3 ð9Þ
;
1=2
A2 n
!1=3
n0 ¼ ðcbÞ
ð11Þ
;
A2 n
r
298.15 308.15 318.15
3.28 3.91 5.54
298.15 308.15 318.15
0.2 mass fraction ethylene glycol 0.33 0.33 0.32
2.57 3.43 4.23
298.15 308.15 318.15
0.3 mass fraction ethylene glycol 0.34 0.33 0.32
2.01 2.83 3.94
Considering f as the only adjustable parameter.
Figures 3(a)–(c) and 4 demonstrate that the fractions of the uncondensed counterions do not remain fixed rather they vary over the concentration range investigated in the present study. It is also observed that a preponderant proportion (63% to 73%) of the counterions get condensed onto the polyion chain.
ð10Þ
!3=7
1=2
a
f 0.1 mass fraction ethylene glycol 0.33 0.32 0.31
4.4. Effect of polyelectrolyte concentration on counterion condensation
:
For good solvent cases, on the other hand, these are given by [30]
A2 ne ¼ b n
TABLE 2 The fractions of uncondensed counterions (f)a for the best-fit of the experimental equivalent conductances for sodium polystyrenesulfonate in ethylene glycol–water mixed solvent media as a function of temperature following the scaling theory approach using poor solvent correlation along with the respective standard deviations (r).
!1=7 :
ð12Þ
For the present system, poor solvent correlations are always found to provide a better description of the experimental results. In figures 1(a)–(c) and 2, the predictions in accordance with the Colby model (dashed lines) for the semidilute regime have been compared with the experimental equivalent conductivity data treating the mixed solvent media as a poor solvent for sodium polystyrenesulfonate. Figures 1(a)–(c) and 2, however, reveal that although the scaling theory approach, with only one adjustable parameter f (listed in table 2) over the entire concentration range provides a significant improvement over the Manning model, a quantitative description of the experimental results is yet to be achieved. It is thus apparent that the assumption of the independence of the effective charge on a polyion chain of the polymer concentration is no longer valid for the system under investigation. The deviations of the calculated values from the experimental results clearly demonstrate that the fraction of uncondensed counterions varies with the polyelectrolyte concentration for the system under consideration. We have, therefore, calculated the fractions of uncondensed counterions from our conductivity data using equation (1) in conjunction with equations (7), (9), and (10). The concentration dependence of f thus obtained is shown in figures 3(a)–(c) and 4. The polyion equivalent conductivities have also been computed on the basis of these f values and are depicted in figures 5(a)–(c) and 6.
From figures 3(a)–(c) and 4 one observes that the fractions of the uncondensed counterions increase as polyelectrolyte concentration increases at a given temperature in a given solvent medium. As polyelectrolyte concentration increases, the relative permittivity of the medium is known to increase, owing to the polarizability of polyelectrolytes [1]. While the increase in the effective relative permittivity of the medium is not completely understood yet [37], the experimental evidence is well established [1,38]. For sodium polystyrenesulfonate, in particular, the frequency-dependent relative permittivity is known to increase with the polyelectrolyte concentration [39]. The increase in the relative permittivity of the medium (e) would result in a decrease in the Bjerrum length (lB = e2/ekT). Since the Bjerrum length sets the scale for the distance between the dissociated counterions on the polyion chain [1,27,40], there will be fewer condensed counterions as the relative permittivity of the medium increases. 4.5. Effect of temperature on counterion condensation The fraction of uncondensed counterions is found to decrease with increasing temperature over the entire polyelectrolyte concentration region in a given mixed solvent medium (cf. figure 3(a)–(c)). This can be ascribed to a change in solvation and condensation behavior of counterions upon changing the temperature. Raising the temperature has the effect of gradual desolvation for the counterions and the polyions which results in an increase of counterion condensation on the polyion chain. This is reflected in the decreasing fraction of uncondensed counterions at higher temperatures. Desolvation of the sodium counterions with increasing temperature is directly evident from our earlier investigation where we noted a significant increase in their mobility with temperature [34]. This might be ascribed to the decreasing size of their solvodynamic entity and hence an increasing surface charge densities resulting in a greater mobility under the action of the applied electric field. A similar behavior was also observed for aqueous solutions of sodium and potassium dextran sulfates [41] where an increase in the charge density parameter (and hence, according
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a
b
0.40
0.40
0.35
f
f
0.35
0.30
0.30
0.25
0.25
0
0.02
0.04
0.06
0.08
√c / (monomol
1/2
c
0.10
0.12
0
0.02
-1/2
L
)
0.04
0.06
√c / (monomol
0.08 1/2
0.10
0.12
-1/2
L
)
0.40
f
0.35
0.30
0.25
0
0.02
0.04
0.06
√c / (monomol
0.08 1/2
0.10
0.12
-1/2
L
)
FIGURE 3. Concentration dependence of the fractions of uncondensed counterions (f) for sodium polystyrenesulfonate in ethylene glycol–water mixed solvent media containing (a) 0.1 mass fraction ethylene glycol, (b) 0.2 mass fraction ethylene glycol, and (c) 0.3 mass fraction ethylene glycol. Experimental temperatures are: 298.15 K (circles), 308.15 K (squares), and 318.15 K (triangles).
4.6. Effect of medium on counterion condensation
0.40
f
0.35
0.30
0.25
0
0.02
0.04
0.06
√c / (monomol
0.08 1/2
-1/2
L
0.10
0.12
)
FIGURE 4. Concentration dependence of the fractions of uncondensed counterions (f) for sodium polystyrenesulfonate in ethylene glycol–water mixed solvent media at 308.15 K in 0.1 mass fraction ethylene glycol (circles), 0.2 mass fraction ethylene glycol (squares), and 0.3 mass fraction ethylene glycol (triangles).
to equation (6), a decrease in the fraction of uncondensed counterions) was reported with the rise of temperature. Similar results were observed with sodium carboxymethylcellulose [22] and sodium polystyrenesulfonate [25] in mixed-solvent media earlier.
The measured fraction of uncondensed counterions is found to increase with increasing ethylene glycol content of the mixed solvent media at each temperature over the entire polyelectrolyte concentration range investigated (please see the representative figure 4). This might be ascribed to the preferential solvation of the polyionic sites and the counterions by the ethylene glycol molecules as the medium becomes richer in ethylene glycol. That is, with increasing amount of ethylene glycol in the mixed solvent media, there will be a gradual replacement of the coordinated water molecules around the polyionic sites and the counterions by ethylene glycol molecules. Since the ethylene glycol molecules are bigger in size compared to the water molecules [42], the electrostatic forces exerted either by the polyionic sites and the counterions would be reduced as the medium becomes richer in ethylene glycol which will obviously result in a gradually lower level of counterion binding. 4.7. The association constant (KA) and the standard state free energies of counterion condensation (DG0A ) and their variation with polyelectrolyte concentration, solvent medium, and temperature In order to obtain an insight into the spontaneity of the counterion condensation process, an information on the standard state free energies of counterion association (DG0A ) is essential. For this
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b
300
-1
monomol )
250
200
300
250
200
2
λp / (S cm
λp / (S cm
2
-1
monomol )
a
150
100
50
0
0.02
0.04
0.06
√c / (monomol
0.08 1/2
-1/2
100
50
0.12
0
0.02
0.04
0.06
√c / (monomol
)
0.08 1/2
L
-1/2
0.10
0.12
)
300
250
200
λp / (S cm
2
-1
monomol )
c
L
0.10
150
150
100
50
0
0.02
0.04
0.06
√c / (monomol
0.08 1/2
L
-1/2
0.10
0.12
)
300
Free site þ Naþ () Combined site;
250
have been calculated as a function of concentration from the fractions of uncondensed counterions using the following equation:
ln K A ¼ ln
200
λp / (S cm
1f lnðfcÞ: f
150
DG0A ¼ RT ln K A ; 100
50
ð13Þ
ð14Þ
The standard state free energies of counterion association (DG0A ) can then be easily obtained from:
2
-1
monomol )
FIGURE 5. Concentration dependence of the polyion equivalent conductivities (kp ) in ethylene glycol–water mixed solvent media containing (a) 0.1 mass fraction ethylene glycol, (b) 0.2 mass fraction ethylene glycol, (c) 0.3 mass fraction ethylene glycol. Experimental temperatures are: 298.15 K (circles), 308.15 K (squares), and 318.15 K (triangles).
0
0.02
0.04
0.06
√c / (monomol
0.08 1/2
L
-1/2
0.10
0.12
)
FIGURE 6. Concentration dependence of the polyion equivalent conductivities (kp ) in ethylene glycol–water mixed solvent medium at 308.15 K in: 0.1 mass fraction ethylene glycol (circles), 0.2 mass fraction ethylene glycol (squares), and 0.3 mass fraction ethylene glycol (triangles).
purpose, the values of the association constants (KA) for the binding of the counterions onto to polyionic sites defined as the equilibrium constant for the reaction
ð15Þ
where R is the universal gas constant. Figure 7(a)–(c) displays the variation of DG0A values of sodium polystyrenesulfonate solutions as a function of the square root of the polyelectrolyte concentration in three different ethylene glycol–water mixtures containing respectively 0.1, 0.2, and 0.3 mass fraction of ethylene glycol each at 298.15, 308.15 and 318.15 K over the entire concentration range investigated. Figure 8, on the hand, shows the polyelectrolyte concentration dependence of the DG0A values at a representative temperature (namely 308.15 K) in three different ethylene glycol–water mixtures mentioned. The negative DG0A values indicate that the counterion condensation process is spontaneous for the present polyelectrolyte system over the entire concentration range although the process becomes less spontaneous as the concentration increases. Addition of increasing amount of ethylene glycol to the medium makes the counterion
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C. Das, B. Das / J. Chem. Thermodynamics 68 (2014) 293–302
b
-15.0
-1
ΔG / (kJ momomol )
-17.5
-20.0
-22.5
-15.0
-17.5
-20.0
-22.5
0
0
-1
ΔG / (kJ momomol )
a
-25.0
-27.5
0
0.02
0.04
0.06
0.08 1/2
-1/2
√ c / (monomol
0.12
-27.5
0
0.02
0.04
0.06
0.08 1/2
√ c / (monomol
)
-1/2
L
0.10
0.12
)
-15.0
-17.5
-20.0
-22.5
0
-1
ΔG / (kJ momomol )
c
L
0.10
-25.0
-25.0
-27.5
0
0.02
0.04
0.06
0.08 1/2
√ c / (monomol
-1/2
L
0.10
0.12
)
(DG0A )
FIGURE 7. Concentration dependence of the standard state free energy change in ethylene glycol–water mixed solvent medium containing (a) 0.1 mass fraction ethylene glycol, (b) 0.2 mass fraction ethylene glycol, and (c) 0.3 mass fraction ethylene glycol. Experimental temperatures are: 298.15 K (circles), 308.15 K (squares), and 318.15 K (triangles).
4.8. Transference number (Tp) of the polystyrenesulfonate anion in ethylene glycol–water mixed solvent media and its variation with polyelectrolyte concentration, solvent medium and temperature
-17.5
-1
ΔG / (kJ.momomol )
-15.0
The transference number of the polyion (Tp) has been obtained from the equivalent conductivity of the polyion obtained earlier (kp ) and the experimentally determined equivalent conductivity of the polyelectrolyte as a whole (K) using the following relationship
-20.0
0
-22.5
-25.0
-27.5
Tp ¼
0
0.02
0.04
0.06
√ c / (monomol
0.08 1/2
-1/2
L
0.10
0.12
)
FIGURE 8. Concentration dependence of the standard state free energy change (DG0A ) in ethylene glycol–water mixed solvent medium at 308.15 K in 0.1 mass fraction ethylene glycol (circles), 0.2 mass fraction ethylene glycol (squares), and 0.3 mass fraction ethylene glycol (triangles).
condensation process less favorable. The spontaneity of the counterion condensation process is, however, found to increase as the temperature increases in a given mixed solvent medium although the effect of temperature is rather small.
kp : K
ð16Þ
In figure 9(a) we present the transference number data of the polyion as a function of the square root of the polyelectrolyte concentration in 0.1 mass fraction ethylene glycol–water mixed solvent medium at three different temperatures. Similar behavior was observed in the other two media (figure 9(b) and (c)). Figure 10, on the other hand, shows the influence of medium on the pffiffiffi Tp vs. c profiles at a given temperature, namely 308.15 K. We also noticed similar behavior at the other two temperatures investigated. The polyion transport number values are always found to be greater than unity over the entire concentration range investigated in this work, and decrease monotonically with increasing polyelectrolyte concentration. Accordingly, the transference numbers of the counterions are negative. This negative sign of the
300
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a
b
3.0
3.0
2.5
Tp
Tp
2.5
2.0
1.5
2.0
0
0.02
0.04
0.06
0.08 1/2
√c / (monomol
c
-1/2
L
0.10
1.5
0.12
0
0.02
0.04
0.06
0.08 1/2
√c / (monomol
)
-1/2
L
0.10
0.12
)
3.0
Tp
2.5
2.0
1.5
0
0.02
0.04
0.06
0.08 1/2
√c / (monomol
-1/2
L
0.10
0.12
)
FIGURE 9. Concentration dependence of the transference number (Tp) of the polystyrenesulfonate ion in ethylene glycol–water mixed solvent medium containing (a) 0.1 mass fraction ethylene glycol, (b) 0.2 mass fractionethylene glycol, and (c) 0.3 mass fraction ethylene glycol. Experimental temperatures are: 298.15 K (circles), 308.15 K (squares), and 318.15 K (triangles).
unity for several aqueous polyions are also available in the literature [43–47]. It is observed that an increase in temperature facilitates counterion binding (cf. figure 9(a)–(c)), while an increase in the amount of ethylene glycol in the mixed solvent media facilitates counterion dissociation (cf. figure 10). The values of Tp confirm our earlier conclusions derived from the measured fraction of the uncondensed counterions.
3.0
Tp
2.5
2.0
4.9. The coefficient of friction between the polyion and the solvent (fps) and its variation with polyelectrolyte concentration, solvent medium and temperature
1.5
The friction coefficient provides a measure of the friction between a monomer unit of the polyion and the solvent and can be estimated from the expression [47]
0
0.02
0.04
0.06
0.08 1/2
√c / (monomol
-1/2
L
0.10
0.12
)
FIGURE 10. Concentration dependence of the transference number (Tp) of the polystyrenesulfonate ion in ethylene glycol–water mixed solvent medium at 308.15 K in 0.1 mass fraction ethylene glycol (circles), 0.2 mass fraction ethylene glycol (squares), and 0.3 mass fraction ethylene glycol (triangles).
counterion transference numbers, which is a characteristic feature of solutions of polyelectrolytes, indicates that a substantial proportion of the counterions is associated with the polyions and travels with them towards the anode. Reports on the Tp values larger than
2
fps ¼
jzp jfF ; kp
ð17Þ
where zp is the number of elementary charges on the monomer unit of the completely dissociated polyion, and the other symbols have their usual significance. The results are summarized in figures 11(a)–(c) and 12. Figure 11(a)–(c) shows the dependence of fps on pffiffiffi c in 0.1, 0.2, and 0.3 mass fraction ethylene glycol–water mixtures each at 298.15, 308.15, and 318.15 K. Influence of medium on the concentration dependence of the fps values at a given temperature
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C. Das, B. Das / J. Chem. Thermodynamics 68 (2014) 293–302
a
b
8
-1
monomol )
-1
monomol )
8
-1
s
-1
6
-14
10 fps/ (g
10 fps/ (g
s
4
-14
2
0
6
4
2
0 0
0.02
0.04
0.06
0.08 1/2
-1/2
L
8
6
0.12
0
0.02
0.04
0.06
√c / (monomol
)
0.08 1/2
-1/2
L
0.10
0.12
)
-1
-1
c monomol )
√c / (monomol
0.10
10 fps / (g
s
4
-14
2
0
0
0.02
0.04
0.06
√c / (monomol
0.08 1/2
-1/2
L
0.10
0.12
)
FIGURE 11. Concentration dependence of the coefficient of friction between polystyrenesulfonate ion and the solvent (fps) in ethylene glycol–water mixed solvent medium containing (a) 0.1 mass fraction ethylene glycol, (b) 0.2 mass fraction ethylene glycol, and (c) 0.3 mass fraction ethylene glycol. Experimental temperatures are: 298.15 K (circles), 308.15 K (squares), and 318.15 K (triangles).
10-14fps / g · s -1 · monomol -1)
8
6
4
2
0
0
0.02
0.04
0.06
√c / (monomol
0.08 1/2
-1/2
L
0.10
0.12
)
FIGURE 12. Concentration dependence of the coefficient of friction between polystyrenesulfonate ion and the solvent (fps) in ethylene glycol–water mixed solvent medium at 308.15 K in 0.1 mass fraction ethylene glycol (circles), 0.2 mass fraction ethylene glycol (squares), and 0.3 mass fraction ethylene glycol (triangles).
(308.15 K) is depicted in the representative figure 12. We can see from the coefficients of friction of the polystyrenesulfonate ion (figures 11(a)–(c) and 12) that the possible conformational changes of
the molecules caused by dilution lead to slight changes in solvodynamic resistance. The effects become more prominent as the temperature is lowered in a given medium, or as the medium becomes richer in the organic solvent at a given temperature. The friction coefficients of the monomer units decrease with increasing temperature over the entire polyelectrolyte concentration range in a given mixed solvent medium thus indicating smaller sizes of the monomer units at higher temperatures. This supports the phenomenon of gradual desolvation of the monomer units inferred earlier. An increase in the friction coefficients with increasing ethylene glycol content of the mixture over the entire polyelectrolyte concentration range at a given temperature, on the other hand, reflects bigger sizes of the monomer units as the medium becomes richer in ethylene glycol. This finding is also in conformity with the fact that the polyionic sites are getting preferentially solvated by the bulkier ethylene glycol molecules over the smaller water molecules as demonstrated earlier. 5. Conclusions An investigation on the electrical conductivity of salt-free solution of an anionic polyelectrolyte – sodium polystyrenesulfonate – in ethylene glycol–water mixed solvent media has been performed as a function of polymer concentration at three different temperatures. The equivalent conductivities are found to increase with increasing temperature over the entire concentration range in a
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given mixed solvent medium whereas these are found to decrease as the relative permittivity of the medium decreases. The conductivity theory, proposed by Manning, for salt-free polyelectrolyte solutions, was applied to analyze the experimental data. The measured values of equivalent conductivity could not be quantitatively described by the Manning counterion condensation theory. This discrepancy probably arises from the fact that the polyelectrolyte solutions investigated herein are different from the model valid only at infinite dilution that underlies this theory. A recent model based on the scaling approach for the configuration of a polyelectrolyte chain in semidilute solution has, therefore, been employed to determine the fractions of uncondensed counterions. Our results suggest that the effective charge on the polyion chain, incorporating any effect of counterion condensations, depends on the polymer concentration. The influences of the temperature the medium, and the polymer concentration on (i) the fractions of uncondensed counterions, (ii) the polyion equivalent conductivities, (iii) the standard state free energies of counterion condensation, (iv) the polyion transference numbers, and (v) the coefficients of friction between the polyion and the solvent have been interpreted from the viewpoints of polyion–counterion interactions, solvation of counterions and the polyionic sites, and counterion dissociation.
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Acknowledgements This work is supported by the Department of Science and Technology, New Delhi, Government of India [SR/S1/PC-67/2010]. The authors also thank the Sikkim Government College, Tadong, Gangtok for its support while carrying out this work there. References [1] F. Oosawa, In Polyelectrolytes, Marcel Dekker, New York, 1993. [2] H. Dautzenberg, W. Jager, J. Koetz, B. Philipp, C. Seidel, D. Stscherbina, In Polyelectrolytes; Formation Characterization and Application, Hanser Publishers, Munich, 1994 (Chapter 5). [3] K.S. Schmitz, Macro-ion characterization. From dilute solutions to complex fluids, ACS Symposium Series, vol. 548, American Chemical Society, Washington, DC, 1994. [4] A.C. Chatterji, H.N. Bhargava, Z. Bhargava, Kolloid Z. 170 (1960) 116–120. [5] M. Nagasawa, I. Noda, T. Takahashi, N. Shimamoto, J. Phys. Chem. 76 (1972) 2286–2294.
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JCT 13/371
