Acta Chim. Slov. 2006, 53, 219–222 Short Communication
Density Dependence of the Viscosity and Excess Volume of Aqueous Solutions of Polyvinylpyrrolidone
Massoumeh Foroutan
School of Chemistry, University College of Science, University of Tehran, Tehran, Iran.
Received 06-07-2005
Abstract
A model for the estimation of density dependence of the viscosity of aqueous solutions of polyvinylpyrrolidone is presented. The proposed representation of the viscosity of PVP solutions expresses the viscosity as a function density and temperature. From the density data, the excess molar volumes of this system are also calculated.
Keywords: Density, Viscosity, Excess volume, Polyvinylpyrrolidone, Aqueous solution.
Introduction
In recent years, numerous studies have been carried out on density dependence of the viscosity of gases. Rainwater and Friend have developed a microscopically based theoretical model for the second transport virial coefficients of moderately dense gases.1-3 The viscosity r\ is expressed in terms of the second viscosity virial coefficient, B^ according to
T] = T]0(\ + B p + ...)                      (1)
where t]0 is the limiting low-density viscosity and p is the number density. In this model, Bn is dependant on the interatomic potential and temperature.
An expansion of viscosity as a power series in the density is consistent with Bogoliubov’s theory of non equilibrium statistical mechanics4, viz.
7] = 7]0(l + Bvp) + 7]2p +...,                (2)
where t]0 is the Chapman-Enskog zero-density coefficient which depict the effect of local binary collisions only and Bn and r\2, etc., are the density coefficients which account, among other things, for the effect of ternary, quaternary, and higher order collisions.
It has been suggested5 that rj2p2 needs to be replaced by rfp2 ln (p) + r\2p2. Thus, one has instead of Eq. (2) for the viscosity,
7] = ?]0 (1 + Bnp) + ?]2p2 + J]2p2 ln(/?) +...     (3)
The detailed nature of the terms beyond the quadratic term in Eq.(3) are not known but it appears that the expression contains terms which are products of powers of p and lnp.6,7 For higher gas densities, use is made of the empirical observations that the viscosity shows a regular dependence on density. The
dependence of a ^(p,T) can always be written8 as the sum of three contributions
7](T,p) = 7]0(T) + A7](T,p) + A?]C(T,p)     (4)
where tj0(T) and At](T,p) are the zero-density limit of viscosity and the excess viscosity which represents the increase in the viscosity at elevated density over the dilute gas value at same temperature, and At]c(p ,T) a critical enhancement accounting for the increase in viscosity in the immediate vicinity of the critical point.
The excess viscosity contribution describes how the viscosity of the fluid behaves as a function of density outside of the critical region. The current state of knowledge does not provide a satisfactory theory for the calculation of the excess viscosity over the whole phase space. However; in practice power series expansion in density for the representation of the excess viscosity has often been found to be adequate.9-12
The series adopted here is of the following form,
n
Arj(p,T) = 2]bi(T)pi                 (5)
i=\
where r\, p and T are viscosity, density and absolute temperature. It is customary to represent the temperature dependence of the density coefficients, bi(T), by a functional relationship of the form
m
bi = 2_Jdij IT                               (6)
j=\
where T* is the reduced temperature and defined as T* =KT/e and K/e is the energy scaling parameter. The coefficient dij are to be determined by fitting the experimental data for the viscosity to Eqs. (5) and (6). Our research has the aim to determine density dependence of the viscosity of aqueous solution of
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Table 1. Densities ?/(g.cm-3) and viscosities ?/(mPa.s) of PVP (aq) as a function of the polymer mass fraction w2 and temperature T.
w2	T =298.15 K		T =308.15 K		T =318.15 K		T =328.15 K	
	p                              T]                              p			n	P	n	P	n
0.0530	1.007011	1.3498	1.006599	1.2001	1.002569	0.9813	0.997068	0.7711
0.0621	1.009865	1.4981	1.009573	1.3011	1.005487	1.1119	1.000744	0.8748
0.0945	1.015689	1.6989	1.023001	2.0015	1.014552	1.3102	1.009541	1.1245
0.1595	1.030706	3.6862	1.027007	2.8544	1.022612	1.8102	1.017592	1.7807
0.2005	1.039827	5.6529	1.035925	4.1287	1.031333	3.0892	1.026191	2.5353
0.2636	1.054257	10.5298	1.041257	5.5831	1.040124	4.4567	1.033245	3.5124
0.2981	1.062412	15.9273	1.050015	7.7427	1.045159	5.8795	1.039754	4.5242
0.3168	1.066833	19.8458	1.057964	11.4453	1.052923	8.4904	1.039754	4.5242
0.3389	1.075001	27.4865	1.062272	13.8645	1.057132	10.4501	1.047342	6.2011
0.3712	1.079968	34.5897	1.070012	19.4562	1.062001	12.4001	1.052112	7.5412
0.3802	1.083546	39.8976	1.076167	24.9656	1.066021	15.1201	1.056451	9.2057
0.3944	1.085835	44.8655	1.080767	30.7981	1.070716	17.8233	1.064813	13.0603
0.4195	1.091457	57.2503	1.084021	36.4578	1.075211	21.3067	1.069212	15.8261
0.4367	1.093001	62.3145	1.087094	42.3916	1.079012	25.4687	1.073001	18.9029
0.4409	1.096101	69.8974	1.091011	49.2456	1.081388	29.4858	1.075261	20.7472
0.4522	1.098001	77.9987	1.093457	55.1246	1.086521	35.9871	1.079121	25.1451
0.4649	1.104001	96.8017	1.096001	61.7012	1.089911	42.0841	1.083091	30.0779
polyvinylpyrrolidone (PVP). This polymer has many uses in biotechnology. In this work, also we have calculated the excess volume of aqueous solutions of PVP.
Experimental
Materials
Polyvinylpyrrolidone with an average molecular weight of 10000 (K 15) was obtained from Aldrich and was used without further purification.
Apparatus and procedures
The solutions are prepared by weighing an appropriate amount of polymer using an analytical balance with an accuracy of + 1.10-4 g. Densities of solutions were measured with a precision of +1.10-6 g.cm-3 using a vibrating-tube densimeter (Anton Paar DSA 5000, Austria) which measures oscillating periods, calculates density and controls temperature with a precision of 0.001 C. The densimeter was calibrated with dry air and with double distilled and degasified water.
Viscosity measurements were carried out with an Ostwald-type viscometer, suspended in a thermostat at required temperature (+ 0.001 oC). The solutions were kept about 20 minutes prior the measurements for temperature equilibrium. The efflux time was determined with an accuracy of + 0.1 s.
Results and discussion
The density, viscosity and excess molar volumes of various PVP aqueous solutions determined at T= (298.15, 308.15, 318.15 and 328.15) K are given in Tables 1 and 2.
Viscosity
For determining the density dependence of the viscosity we use the excess viscosity in Eq.(4). The excess viscosity contribution describes how the viscosity of the fluid behaves as a function of density outside of the critical region. The treatment of the viscosity in the zero-density limit and the critical enhancement will not be considered because no information is available.
Table 2. Experimental excess molar volumes VE (cm3.mol-1) of PVP (aq) as a function of the polymer mole fraction x2 and temperature T.
X2	T=298.15	T=308.15	T=318.15	T=328.15
	(K)	(K)	(K)	(K)
0.0000003	-0.00947	-0.02558	-0.02467	-0.00866
0.0000011	-0.02632	-0.04205	-0.04097	-0.02473
0.0000031	-0.06876	-0.08360	-0.08191	-0.06531
0.0000041	-0.09076	-0.10517	-0.10310	-0.08629
0.0000051	-0.10955	-0.12354	-0.12130	-0.10432
0.0000063	-0.13705	-0.15046	-0.14777	-0.13047
0.0000073	-0.15983	-0.17277	-0.16978	-0.15221
0.0000086	-0.18895	-0.20129	-0.19775	-0.20293
0.0000111	-0.24168	-0.25291	-0.24863	-0.23015
0.0000135	-0.29394	-0.30404	-0.29891	-0.27992
0.0000161	-0.35077	-0.35961	-0.35362	-0.33399
0.0000213	-0.46537	-0.47160	-0.46371	-0.44275
0.0000311	-0.68835	-0.68919	-0.67733	-0.65365
0.0000374	-0.83436	-0.83141	-0.81648	-0.78999
0.0000438	-0.98448	-0.97754	-0.95971	-0.93182
0.0000484	-1.09768	-1.08770	-1.06768	-1.03800
0.0000549	-1.25346	-1.23952	-1.21628	-1.18436
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Following this method which has been used for the calculation of the viscosity of carbon dioxide in the customary way as a function of density and temperature13, a power series expansion in density for the representation of viscosity of aqueous solutions of PVP was considered
/7(An=I>;(n/?;
where bi(T) has been defined as
(7)
(8)
In examining the ability of the mentioned method to calculate the viscosity of an aqueous solution of polymer, PVP serves as our primary test, because of the abundance of available viscosity data over a wide range of temperature and concentration.
As shown in figure 1, there is good agreement between calculated and experimental viscosity of aqueous solutions of PVP at different temperatures.
The resulting coefficients of Eq. (8) at each temperature are given in Table 3.
Figure 1. Calculated and experimental viscosity of aqueous solutions of PVP at different isotherms. The calculated values are based on Eq. (8). (—) 298.15 K, (---) 308.15 K, (...) 318.15 K and (.-.-.-) 328.15 K. [p in g.cm-3and r\ in mPa.s].
Table 3. Coefficients of the	representation	of the viscosity of
the PVP aqueous solutions, Eq. (9). [T in K		p in g.cm-3and r|
in mPa.s].		
tin            0.123020x10	d12	-0.110805x10
d21           -0.489076x10	d22	0.439807x10
d31            0.732171xl010	d32	0.657483xl013
4^            0.119518xl010	*-H2	-0.106911xl013
d13            0.331510x10	di4	-0.329037x10
d23           -1.313459x10	d24	1.300967x10
d33            1.960402xl015	d34	1.938184xl017
d43            0.317373xl015	"44	-0.312178xl017
Excess molar volumes
The values of excess molar volumes (VE) can be calculated from the experimental data according to the following equation:
t,xiMi
V
P
i^;
(9)
where p is the mixture density and xi, Mi and Vi are the mole fraction, the molecular weight and molar volume of the ith component of the mixture, respectively. The equation V1=M1/p1 describes the molar volume of pure water, while that of PVP is described by V2=®2°° where ®2°° is the apparent specific volume at infinite dilution of PVP, which must be determined on the basis of data on apparent specific volume.
The apparent specific volume of the polymer, ®2 is computed from the density of the solution by using the following equation
<D2= —(--------------------
w2   p       px
)
(10)
where w2 is the polymer mass fraction. The ?2 values can be fitted to the following equation14
o=or
m
m
(11)
where bv and bvv are the empirical parameters which depend on solute, solvent and temperature, and mw2 is the weight molality of polymer.
Recently the above equation has been used for the fitting of ®2 values for aqueous PVP solutions at different temperature and the values of infinite dilution apparent specific volume of the polymer at T=298.15, 308.15, 318.15 and 328.15 K were reported as 0.7986, 0.8044, 0.8102 and 0.8160 cm3g-1 respectively.15 We will apply these values in our calculations. Once the apparently specific volumes at finite dilution of PVP are known, as well as the molar volume of water and the density of the binary mixture, values VE are determined by means of Eq. (9). Values of VE are listed in the Table 2. Figure 2 shows experimental excess molar volumes of aqueous solutions of PVP at different temperatures. By observing the values for the excess volumes at a given temperature, it can be observed that VE increases at the concentration of PVP increase.
As shown in this figure, the excess molar volumes for the binary systems in different temperatures are negative. This would indicate that molecular interactions between molecules of water and polymer are stronger than interactions between molecules in their pure states and attractive forces dominate the behavior of the solutions.16
;=i
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Acta Chim. Slov. 2006, 53, 219–222
0
-0.2 -0.4 -0.6 -0.8 -
-1
-1.2
0.00000     0.00001     0.00002     0.00003     0.00004     0.00005     0.00006
x2
Figure 2. Experimental excess molar volumes of aqueous solutions of PVP at (?) 298.15 K, ( ) 308.15 K, (?) 318.15 K and (?) 328.15 K.
Acknowledgment
The author gratefully acknowledges financial support given by the University of Tehran.
References
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15.  R. Sadeghi, M. T. Zafarani-Moattar, J.Chem. Thermodynamics 2004, 36, 665–670.
16.  J. M. Prausnitz, R. N. Lichtenthaler, E. Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, 2nd Eds., Prentice Hall, Englewood Cliffs, NJ, USA, 1986, Ch. 10.
Povzetek
¦
V članku predstavljamo model za določanje viskoznosti vodnih raztopin polivinilpirolidona kot funkcijo gostote I   in temperature raztopine. Iz gostot raztopine smo izračunali tudi presežne molske prostornine.


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Foroutan   Density Dependence of the Viscosity and Excess Volume of Aqueous .....