The measurements of coexistence curves and critical behavior of a binary mixture with a high molecular weight polymer

Journal of Molecular Liquids 161 (2011) 115–119

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Journal of Molecular Liquids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m o l l i q

The measurements of coexistence curves and critical behavior of a binary mixture with a high molecular weight polymer Jinshou Wang ⁎, Youmeng Dan, Yan Yang, Yan Wang, Yuanfang Hu, Yan Xie Department of Chemistry, Hubei University for nationalities, Enshi, Hubei 445000, China

a r t i c l e

i n f o

Article history: Received 16 October 2010 Received in revised form 29 April 2011 Accepted 5 May 2011 Available online 19 May 2011 Keywords: Crossover Polymer

a b s t r a c t The critical behavior of solutions of poly (diallydimethylammonium chloride) (PDDAC, Mw = 3.2 × 10 5) in the critical binary mixture of isobutyric acid (I) + water (W) was studied by the refractive index measurements. The measurements were performed at three different PDDAC concentrations near and far away from the critical point. We observed that the critical temperatures increase linearly with increasing the concentration of PDDAC. As the distance from the critical point increases, the system exhibits a crossover from the renormalized Ising critical behavior to the mean-field one. For the solutions with the highest PDDAC concentration, experiments suggest a crossover to tricritical behavior. © 2011 Elsevier B.V. All rights reserved.

1. Introduction It is commonly accepted that all fluids, simple and complex, belong to the same universality class, namely that of the 3D-Ising model near the critical point [1]. Critical-point universality originates from the long-range nature of the fluctuations of the order parameter. Sufficiently close to the critical point, the correlation length of the critical fluctuations becomes so large that a microscopic and even a mesoscopic structure of fluids become unimportant. However, in practice, the range of pure asymptotic critical behavior is hardly accessible. Even in simple fluids such as xenon and helium, the physical properties show a change in behavior, from Ising to meanfield critical behaviors, upon departure from the critical temperature. This trend (“crossover”) depends on the microscopic structure of the system, namely, on the range of interaction forces and on a molecularsize “cutoff.” In simple fluids [2–4], crossover to mean-field critical behavior is smooth and never completed within the critical domain. The physical reason is that the cutoff length and the range of interactions in simple fluids are too short. In contrast, in complex fluids [5–14] the crossover is unusually sharp, more pronounced, and sometimes even nonmonotonic. A natural phenomenological approach to the crossover problem in complex fluids is to assume that the critical behavior in such systems is affected by a competition between the correlation length ξ of the critical fluctuations and an additional length ξD associated with a supramolecular structure or/and with long-range interparticle interactions. When the correlation length dominates, the system enters the Ising critical regime. When the additional length ξD is larger than ξ, the critical fluctuations

⁎ Corresponding author. Tel.: + 86 718 8437531. E-mail address: [email protected] (J. Wang). 0167-7322/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2011.05.002

are not important and the crossover from Ising-like behavior to classical one should be expected. The additional length ξD will diverge at a tricritical or a multicritical point of some kind that emerges as a result of coupling between the order parameters. With small logarithmic corrections, the tricritical behavior is almost mean-fieldlike. Thus, if the system is close to the tricritical point, one should expect a crossover to mean-field tricriticality. The crossover to meanfield tricriticality has been observed in polymer solutions by theoretical and experimental measurements [15–19]. It is natural to assume that in polymer solutions the additional length ξD is proportional to the radius of gyration Rg. Rg can be tuned to increase by increasing the molecular weight Mw. Hence, in polymer solutions the limit Mw → ∞ means ξD → ∞, and the polymer solutions can be accessed experimentally to observe actual crossover to mean-field tricritical behavior. The behavior of long chain molecules in binary solvents [20–35] have been explored in the past three decades from both fundamental and industrial points of view. It is well known that when a long-chain polymer is dissolved in a bicomponent mixture close to the critical temperature, there exists a long-range indirect interaction between two monomers of the polymer chain. The interaction range of such attraction is the spontaneous composition fluctuation in solvent, which diverges when approaching the critical point of the solvent. On the other hand, the polymer in the critical binary fluid may behave as force centers that preferentially adsorb one of the solvent components and change the properties of the system, including the local fluctuations of the concentration, and critical exponents [30,31]. To and Choi [32] studied the critical behavior of polyacrylic acid (Mw = 7.5 × 10 5) + water + 2,6-lutidine by dynamic light scattering measurements. It was found that the critical exponent ν of the correlation length is 0.44 ± 0.03 at temperature close to the critical temperature, which is lower than ν = 0.6 of pure water + 2,6-lutidine.

J. Wang et al. / Journal of Molecular Liquids 161 (2011) 115–119

2. Experimental section PDDAC (20% aqueous solution) with Mw = 3.2 × 10 5 was provided by Aldrich Chemical Co. Isobutyric acid was purchased from Sigma Chemical Co. Twice distilled water was used in all experiments. The critical composition was determined by the technique of “equal volume”. The volume ratio of upper and lower coexistence phases was sensitive to the composition of the prepared solution. It was found that the volumes of upper and lower coexistence phases of the critical composition are equal within ±0.03% in the mass fraction at the phase separation temperature. Samples with the critical composition were prepared in a rectangular fluorimeter cell and were settled for 1 day before use due to the long equilibration time of polymer. In the process of measuring the coexistence curves, the cell was placed in a holder inside a water bath. The temperature in the bath was measured by a Keithley 2700 digital multimeter with a platinum resistance thermometer with an uncertainty of ±0.001 K. The phase separation temperature was carefully determined and taken as the critical temperature. It was observed that the mixtures nominally of the same composition had different critical temperatures, differing by as much as 0.05 K. This might have been a result of uncontrollable moisture or other impurities introduced into the mixtures in the process of preparation. However, it did not affect the final results because only one sample was used throughout the determination of the whole coexistence curve, and only temperature difference (Tc − T) was important to obtain the critical parameters. The uncertainty in measurement of (Tc − T) was ±0.003 K. The coexistence curve was determined by measuring the refractive indices of the two coexisting phases by using a technique of “minimum deviation angle”. The apparatus and the experiment procedure for measurements of refractive index have been described previously [39]. A 2-mW He–Ne laser was used as light source. The

Table 1 Properties of the I + W + PDDAC samples. m is the critical mass fraction. sample

m(PDDAC)

m(H2O)

m(isobutyric

1 2 3

0.0010 0.0030 0.0050

0.6291 0.6192 0.6139

0.3699 0.3778 0.3811

acid)

Tc 302.622 305.925 308.984

screen, the water bath, the cell holder, and the laser were adjusted so that the laser beam was normal to all the surfaces of screen and glass walls. The position of the angle of minimum deviation was determined by rotating the cell to minimize the distance between two spots of incident and refracted beams projected on the screen. Measurements were taken through the four apex angles (corners) of each cell and were averaged. This averaging canceled the errors introduced by the variations in the angles. The uncertainty in measurement of refractive index of each coexisting phase was ±0.0001.

3. Results and discussion The critical mass fractions and critical temperatures of I + W + PDDCA were carefully measured before we started refractive index measurements. The results are listed in Table 1 and shown in Fig. 1. When a low concentration of a third component is added to a binary liquid mixture with a consolute point, the critical temperature and the critical composition can change [32–35,40]. The effect of the third component can be understood both in an intuitive manner [41] and from thermodynamic arguments based on the Gibbs-Duhem [42] equation. If the third component is equally soluble in both liquids of the binary mixture, then we expect that third component to increase the mutual solubility of the two components and thus decrease an upper critical solution temperature (UCST) and increase a lower critical solution temperature (LCST). If the third component is more

310

a

308 306

Tc

Subsequently, they studied the same system by refractive index measurements [33]. The critical exponent β was found to be 0.41 ± 0.02, which is higher than (0.31 ± 0.02) that of pure water + 2,6lutidine at temperature close to the critical temperature. In a recent work, Venkatesu [34] reported the critical behavior of nitroethane + 3-methylpentane + polyethylene oxide (Mw = 9 × 10 5) by refractive index measurements. At temperatures close enough to critical temperatures, the refractive index experiments displayed the values of critical exponent β are higher than the Fisher renormalized Ising value (0.365), lower than the mean-field one (0.5), and those values decrease with increasing the concentration of polyethylene oxide. Similar critical behavior has also been observed in isobutyric acid + water + polyethylene oxide (Mw = 9 × 10 5) [35], and polystyrene (Mw = 7.19 × 10 5) + polystyrene (Mw = 1.72 × 10 5) + methylcyclohexane [36,37]. Hence, it is possible that a binary liquid mixture with dissolved high molecular weight polymers may belong to a new universality class with a set of critical exponents which are different from those predicted by 3D-Ising model, mean-field model or the crossover theory proposed by Anisimov and Sengers [38]. In order to check whether the critical exponent is universal or not for all binary mixtures containing high molecular weight polymer, we obtained the effective critical exponents βeff of a critical mixture I + W with three different PDDAC concentrations (mass fraction m = 0.0010, 0.0030 and 0.0050) by refractive index measurements near and far away from the critical point. We found that βeff of I + W + PDDAC shows the crossover from the renormalized Ising critical value [36,37] to the mean-field one as the distance from the critical point increases. For the solutions with the highest PDDAC concentration, experiments suggest a crossover to tricritical behavior. This behavior is different from that of pure I + W, which is consistent with the 3D-Ising model. From these results we observed that the value of β for the binary critical mixture with a high molecular weight polymer may not be universal.

304 302 300 298 .630 .628

b

.626 .624

mH2O

116

.622 .620 .618 .616 .614 .612 0.000

.001

.002

.003

.004

.005

.006

mPDDAC Fig. 1. Polymer effect on the critical temperature Tc and the critical mass fraction of m as a function of PDDAC concentration in I + W.

J. Wang et al. / Journal of Molecular Liquids 161 (2011) 115–119 Table 2 Coexistence curve data for I + W with PDDAC concentration m = 0.0010 as a function of temperature T.

117

Table 4 Coexistence curve data for I + W with PDDAC concentration m = 0.0050 as a function of temperature T.

T

n1

n2

T

n1

n2

T

n1

n2

T

n1

n2

302.612 302.608 302.551 302.508 302.502 302.377 302.328 302.101 301.823 301.657 300.950 300.376 299.881 298.991 298.308

1.3608 1.3610 1.3621 1.3625 1.3624 1.3631 1.3631 1.3645 1.3657 1.3656 1.3681 1.3688 1.3694 1.3699 1.3724

1.3592 1.3590 1.3588 1.3586 1.3583 1.3579 1.3575 1.3574 1.3571 1.3566 1.3558 1.3551 1.3545 1.3541 1.3534

296.866 295.568 294.373 293.680 293.075 291.686 290.799 289.744 288.836 287.885 286.970 286.151 285.285 283.807

1.3743 1.3756 1.3769 1.3776 1.3784 1.3795 1.3811 1.3824 1.3830 1.3840 1.3849 1.3857 1.3865 1.3880

1.3527 1.3523 1.3518 1.3516 1.3515 1.3512 1.3510 1.3509 1.3508 1.3508 1.3507 1.3507 1.3507 1.3506

308.975 308.933 308.875 308.761 308.656 308.480 308.301 308.104 307.941 307.478 307.009 306.529 306.152 305.659 305.175 304.133

1.3596 1.3604 1.3607 1.3608 1.3616 1.3618 1.3624 1.3622 1.3626 1.3634 1.3648 1.3650 1.3656 1.3665 1.3669 1.3680

1.3584 1.3582 1.3579 1.3572 1.3573 1.3569 1.3566 1.3558 1.3558 1.3555 1.3551 1.3548 1.3544 1.3543 1.3538 1.3535

302.916 301.845 300.798 299.930 298.929 297.400 295.783 294.213 292.649 291.460 290.204 288.074 286.958 284.712 282.907

1.3697 1.3705 1.3714 1.3720 1.3731 1.3751 1.3770 1.3786 1.3802 1.3810 1.3820 1.3839 1.3859 1.3884 1.3887

1.3530 1.3527 1.3523 1.3518 1.3508 1.3516 1.3512 1.3510 1.3510 1.3508 1.3504 1.3506 1.3507 1.3510 1.3506

soluble in one liquid component than in the other, then we expect that the mutual solubility will be decreased, thus increasing a UCST and decreasing a LCST. These generalizations, based in thermodynamics, are sometimes called the Timmerman rules [43]. On the other hand, Timmerman rules have been found to be violated in some cases [34,43,44]. PDDAC is much more soluble in pure W than in pure I. The Timmerman rules would predict that PDDAC will increase the critical temperature of I + W. Fig. 1a demonstrates that the critical temperature increases linearly with the PDDAC concentration with a rate of about 1.6/0.1 wt.%. We conclude that on the addition of PDDAC, a polymer that is more soluble in W than in I, the two solvents become mutually less soluble and this raises the critical temperature. Like the critical temperature which changed linearly with PDDAC added, the critical mass fraction of W (mH2O) decreased linearly with PDDAC added. The results are listed in Table 1 and shown in Fig. 1b. When more PDDAC was added, mH2O decreased with increasing PDDAC concentration at a rate of 0.0038/0.1 wt.%. A linear dependence of the critical concentration with increasing of impurity was also observed in other systems. Tveekrem [45] found that adding 0.1 vol.% water impurity to methanol + cyclohexane caused the critical volume fraction of methanol to increase linearly by 0.004. The refractive indices n2 and n1 of two coexistence phases were measured as a function of the temperature near and far away from the critical point. The results are listed in Tables 2, 3, and 4 and shown in Fig. 2. It can be seen from Fig. 2 that the coexistence curves shift up with increasing PDDAC concentration. This variation induces a modification of the phase transition region affected with the addition of PDDAC to I + W mixture. Such a feature has been observed in many other systems and both the direction and the amount of shift depend

on the particular systems [32–35,40]. In order to analyze the experimental data, we obtained the plot of ln(n2 − n1)vs ln(τ) (τ = |(T − Tc)/Tc|) in Fig. 3, where the uncertainties of ln(n2 − n1)are represented by the error bars. It may be clearly seen that the data of each group do not fall on a straight line. To confirm this conclusion, we arranged the data of each group into two groups: the first group is near the critical point (Tc − T b 1 K), while the second group is far away from the critical point (Tc − T N 1 K). Then the values of n2 − n1 were fitted to Eq. n2 − n1= Bτ β for each group by using a least-square method to obtain the values of critical exponent β. The values of β were obtained to be 0.389 ± 0.018, 0.379 ± 0.024 and 0.377 ± 0.011 near the critical point, and increased to 0.478 ± 0.007, 0.522 ± 0.008 and 0.567 ± 0.007 when temperatures were far away from the critical point for the three critical samples, respectively, which indicates that the effective critical exponent of the refractive index which is defined as βeff=dln|n2-n1|/dlnτ depends on the temperature and possibly undergoes a crossover from renormalized Ising critical value to the mean-field one when temperature is far enough away from the critical point. Moreover, the critical exponent β of the sample with PDDAC concentration m = 0.0050 is higher than the mean-field value far away from the critical point, which indicates that a crossover to the tricritical value β=1 might be observed. Obviously, the critical behavior of I + W with PDDCA is different from that of pure I + W, which is consistent with the 3D-Ising model in a wide temperature range around the critical point [46,47]. Hence, it is possible that the

310

T

n1

n2

T

n1

n2

305.918 305.888 305.865 305.806 305.679 305.505 305.394 305.147 304.892 304.091 303.454 302.780 302.110 301.323

1.3603 1.3614 1.3617 1.3620 1.3626 1.3630 1.3634 1.3640 1.3652 1.3664 1.3668 1.3688 1.3697 1.3701

1.3589 1.3588 1.3588 1.3584 1.3579 1.3575 1.3573 1.3569 1.3564 1.3559 1.3550 1.3548 1.3542 1.3538

299.606 298.917 297.558 296.696 296.295 294.552 293.633 292.548 291.530 290.797 290.064 288.248 287.151 286.479

1.3716 1.3724 1.3743 1.3745 1.3760 1.3779 1.3790 1.3797 1.3803 1.3810 1.3820 1.3838 1.3849 1.3857

1.3531 1.3529 1.3523 1.3520 1.3520 1.3520 1.3515 1.3515 1.3511 1.3510 1.3511 1.3510 1.3510 1.3508

300

Τ

Table 3 Coexistence curve data for I + W with PDDAC concentration m = 0.0030 as a function of temperature T.

290

280 1.34

1.35

1.36

1.37

1.38

1.39

1.40

n Fig. 2. Coexistence curve from the temperature dependence of the refractive indices of I+W with different concentrations of PDDAC. (●) m=0.0010 of PDDAC in I+W; (▲) m=0.0030 of PDDAC in I+W; (+) m=0.0050 of PDDAC in I+W.

J. Wang et al. / Journal of Molecular Liquids 161 (2011) 115–119

-3.0

.65

-3.5

.60

-4.0

.55

-4.5

βeff

ln(n2-n1)

118

-5.0

.50 .45 .40

-5.5

.35 -6.0 .30 -7.0

-6.5 -7.0 -12

-10

-8

-6

-4

-2

ln(τ ) Fig. 3. Logarithmic representation of the refractive index difference between the two coexisting phases with respect to the distance from the critical temperature. For the sake of simplicity and clarity, we did not show the error bars and fits for PDDAC concentrations m = 0.0030 and 0.0050 in I + W systems. (●) m = 0.0010 of PDDAC in I + W; (▲) m = 0.0030 of PDDAC in I+ W; (+) m = 0.0050 of PDDAC in I + W.

I + W sample with and without PDDAC may belong to different universality classes. The effective critical exponent of the refractive index may be obtained by numerical derivative from the spline fit of the experimental data of n2 − n1. This treatment is shown as points in Fig. 4 and the results of spline fit of n2 − n1 are shown as solid line in Fig. 3, which are in good agreement with the experimental ones. It can be seen from Fig. 4 the crossovers from renormalized Ising critical value to mean-filed one have been observed. The crossover behavior becomes more pronounced with increasing the concentration of PDDAC. For the solutions with the highest PDDAC concentration, experiments suggest a crossover to tricritical behavior. In all these cases the crossover is sharp and pronounced and additional length ξD is revealed. As mentioned before, it is natural to assume that in polymer solutions the additional length ξD is proportional to the radius of gyration Rg, which can be tuned to increase by increasing the molecular weight Mw. When the additional length ξD is larger than ξ, the critical fluctuations are not important and the crossover from Isinglike behavior to mean-field or tricritical behavior should be expected. The origin of the additional length ξD in the system PDDAC + I + W is not clearly understood. However, by analogy with the crossover phenomena in polymer solutions, we assume that the additional length may stem from the supramolecular structure or/and the longrange interparticle interactions of the wetting layer [21,34,35] or cloud. Thirty years ago, Brochard and de Gennes [24] made an interesting prediction for a polymer in a mixture of two good solvents, where the affinities of the macromolecule for the two solvent components differ substantially. The better solvent, which is preferentially adsorbed onto the polymer, creates a cloud of good solvent around itself. This cloud attracts other segments of the polymer, thereby constituting an indirect long-range attraction among the distant monomers within the chain and attraction between polymer chain and solvent particles. When we added the PDDAC to I + W mixture, presumably, W completely wets PDDAC molecules, and a wetting layer or cloud was formed close to the PDDAC molecules because W is a good solvent of PDDAC than I. However, the puzzling questions which arise from the work are why the crossover behavior becomes more pronounced with increasing the concentration of PDDAC and why experiments suggest a crossover to tricritical behavior. Schröer [48] concluded that the crossover behavior of some systems may be false phenomena due to the absence

-6.5

-6.0

-5.5

-5.0

ln(τ )

-4.5

-4.0

-3.5

-3.0

Fig. 4. The relations of the effective exponent βeff of the refractive index as a function of ln(τ) for I + W + PDDAC. (●) m = 0.0010 of PDDAC in I + W; (▲) m = 0.0030 of PDDAC in I + W; (+) m = 0.0050 of PDDAC in I + W.

of corrections. A similar phenomena may arise in the system I + W + PDDAC because the regular effects are not accounted for at τN10 − 2. On the other hand, we wish to address one interesting comparison between the critical exponents of I + W containing PDDCA and those of other binary mixtures containing high molecular weight polymer. It can be seen from Fig. 4 that the values of β are consistent with the Fisher-renormalized Ising exponent in the asymptotic region, and the effective exponent βeff increase beyond the mean-field value for τ N 10 − 2 for I + W with PDDAC mass fraction m = 0.0030 and 0.0050. It is clearly different from what To and Choi [32] found for high molecular weight polymer polyacrylic acid (Mw=7.5 × 10 5) dissolved in lutidine + water mixture, β = 0.41, independently of the concentration of the polymer, and also different from the results of Venkatesu [34,35] who reports the critical exponents of I + W containing high molecular weight polymer polyethylene oxide (PEO, Mw=9 × 10 5) and nitroethane + 3-methylpentane containing PEO (Mw= 9 × 10 5) higher than the Ising value depending on the concentration of polymer. These observations suggest that the value of β may not be universal in binary mixtures containing high molecular weight polymer. Of course a rigorous conclusion requires more motivated experimental studies.

4. Conclusions In this paper we have given an overview of how the phase behavior of I + W is affected by an addition of PDDAC with three different concentrations by the refractive index measurements. Our results reveal that the critical temperatures increase linearly with increasing the concentration of PDDAC. As the distance from the critical point increases, the system exhibits a crossover from the renormalized Ising critical behavior to the mean-field one, which may be related to the supramolecular structure or/and the long-range interparticle interactions of the wetting layer or cloud. For the solutions with the highest PDDAC concentration, experiments suggest a crossover to tricritical behavior. However, a rigorous conclusion requires more motivated experimental studies because the regular effects were not accounted for at τ N 10 − 2. Moreover, we observed that the value of β may not be universal in binary mixtures containing high molecular weight polymer.

Acknowledgements This work was supported by the Foundation of Hubei Provincial Education Department (Q20111905) and Hubei University for Nationalities (MY2009B013).

J. Wang et al. / Journal of Molecular Liquids 161 (2011) 115–119

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