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Rheological Behavior of Binary Aqueous Solutions of Poly(ethylene glycol) of 1500 g·mol−1 as Affected by Temperature and Polymer Concentration Bernardo de Sá Costa,†,‡ Jane Sélia dos Reis Coimbra,†,* Márcio Arêdes Martins,§ Edwin Elard Garcia-Rojas,‡ Javier Telis-Romero,⊥ and Eduardo Basílio De Oliveira† †

Departamento de Tecnologia de Alimentos, Universidade Federal de Viçosa, Campus Universitário, s/n, CEP 36570-000, Viçosa, MG, Brazil ‡ Departamento de Engenharia de Agronegócios, Universidade Federal Fluminense, Avenida dos Trabalhadores, 420, CEP 27225-250, Volta Redonda, RJ, Brazil § Departamento de Engenharia Agrícola, Universidade Federal de Viçosa, Campus Universitário, s/n, CEP 36570-000, Viçosa, MG, Brazil ⊥ Departamento de Tecnologia e Engenharia de Alimentos, Instituto de Biociências, Letras e Ciências Exatas, Universidade Estadual Paulista “Julio de Mesquita Filho”, CEP 15054-000, São José do Rio Preto, SP, Brazil ABSTRACT: The rheological behavior of poly(ethylene glycol) of 1500 g·mol−1 (PEG1500) aqueous solutions with various polymer concentrations (w = 0.05, 0.10, 0.15, 0.20 and 0.25) was studied at different temperatures (T = 283.15, 288.15, 293.15, 298.15 and 303.15) K. The analyses were carried out considering shear rates ranging from (20 to 350) s−1, using a cone-and-plate rheometer under controlled stress and temperature. Classical rheological models (Newton, Bingham, Power Law, Casson, and Herschel−Bulkley) were tested. The Power Law model was shown suitable to mathematically represent the rheological behavior of these solutions. Well-adjusted empirical models were derived for consistency index variations in function of temperature (Arrheniustype model; R2 > 0.96), polymer concentration (exponential model; R2 > 0.99) or the combination of both (R2 > 0.99). Additionally, linear models were used to represent the variations of behavior index in the functions of temperature (R2 > 0.83) and concentration (R2 > 0.87).

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INTRODUCTION Knowledge of rheological properties of polymers in aqueous solution is necessary for the design of equipment used in the chemical and biochemical industry.1 Such properties affect fluid behavior when polymeric solutions are submitted to shear conditions, for example, during pumping and transport through pipelines. Therefore, accurate rheological data for solutions of biotechnologically relevant polymers are essential tools to industries or academic laboratories working in the bioseparation fields.1−3 Poly(ethylene glycol) [HO−CH2−CH2−(O−CH2−CH2)n−1− OH], commonly known as PEG or PEO, consists of nonbranched polymers, normally soluble in water and in various organic solvents. In general, their solubilities, as well as the rheology of the resulting solutions, are markedly influenced by type of solvent, polymer molar mass, dissolved polymer concentration, and temperature of the medium.4 Among other technological applications, PEGs of a wide range of molar masses, mixed with other polymers or inorganic compounds (mainly salts), are often employed to obtain aqueous two-phase system (ATPS) used as partitioning systems for biomolecules.5−13 In such systems, the rheological characteristics of the dissolved polymers affect both the time of phase separation before equilibrium is reached and the © 2013 American Chemical Society

speed of displacement of solutes within the polymer-rich phase in partitioning experiments. Despite the recognized technological importance of PEGs, the number of studies available in the literature addressing the rheological behavior of these polymers is quite small.14−21 In most of these studies, either PEG melts or solutions exhibited a non-Newtonian behavior, meaning that their viscosities do not remain constant as the velocity gradient varies. Commonly, the viscosity of these fluids decreases as the shear rate increases; that is, they present a shear thinning (pseudoplastic) behavior.22 Simpled stated, this can be attributed to the fact that under shear conditions an alignment of the nonbranched polymer molecules occurs, which eases the sliding of the chains one over another.23,24 This work is intended to generate and to make available accurate data on the rheological behavior of poly(ethylene glycol) with molar mass of 1500 g·mol−1 (PEG1500). This molar mass value was selected since rheological data for PEG1500 are scarce in literature, despite the wide applicability range of this polymer Received: June 27, 2012 Accepted: March 2, 2013 Published: March 13, 2013 838

dx.doi.org/10.1021/je300712j | J. Chem. Eng. Data 2013, 58, 838−844

Journal of Chemical & Engineering Data

Article

In this equation, k0 is a constant to be determined for each fluid in a specific temperature range; R is the universal gas constant (R = 8.314 J·mol−1·K−1); Ea is the activation of energy for flow (J·mol−1). In an analogous manner, at each temperature, the effect of PEG1500 concentration (w) on the consistency index (K) was modeled, using two well-known empirical models: an exponential model (eq 7), and a power model (eq 8):25,30,34,35

(mainly in ATPS used for the partition of proteins and other relevant biomolecules5,7,8,13,26−29). Furthermore, empirical models to describe the effects of temperature and polymer concentration on these rheological data were adjusted to the experimental results, analyzed, and discussed.

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MATERIALS AND METHODS Poly(ethylene glycol) of average molar mass 1500 g·mol−1 (PEG1500; purity > 0.98) and sodium hydroxide (NaOH; purity > 0.99) were both supplied by Vetec Quimica Fina (Brazil). Double distilled and deionized water (electrical resistivity ≈ 18.2 MΩ·cm) was used in the experiments (Milli-Q, Millipore Inc., USA). Preparation of PEG1500 Solutions. A stock solution of PEG1500 (w = 0.50), with pH adjusted to 8.0 (usual value in ATPS for separation of several molecules with biotechnological interest) was first prepared. To obtain the desired concentrations (w = 0.05, 0.10, 0.15, 0.20 and 0.25), appropriate amounts of the stock solution were diluted using double distilled and deionized water, mixing and manually stirring the resulting solutions in amber glass bottles with 200 mL capacity. In all cases, pH was monitored (digital pHmeter, Gehaka, PG 100, Brazil) and adjusted by dropping NaOH 1 mol·dm−3 solution. All the mass measurements were performed using an analytical balance with accuracy of ± 10−4 g (Denver Instruments, M-310, USA). Rheological Analyses. Rheological measurements were carried out using an AR 2000 rheometer (TA Instruments, New Castle, USA) with cone-and-plate geometry (60 mm disc, 2° 1′ 23″ angle) under controlled shear stress and temperature. For each studied polymer concentration, shear stress values (τ) were recorded for shear rates (γ̇) ranging from (20 to 350) s−1, at T = (283.15, 288.15, 293.15, 298.15 or 303.15) K. All the measurements were repeated three times for each polymer concentration, at each temperature, and highly reproducible data were obtained. Previously, experiments were performed with double distilled water at the temperature range studied. Data Modeling. Five classical rheological models were fitted to the experimentally obtained τ = f(γ̇) data: Newtonian (eq 1), Bingham (eq 2), Power Law (eq 3), Casson (eq 4), and Herschel−Bulkley (eq 5). τ = η(γ )̇

(1)

τ = τ0 + η(γ )̇

(2)

τ = K(γ )̇ n

(3)

τ 0.5 = τ0 0.5 + KC(γ )̇ 0.5

(4)

τ = τ0 + K(γ )̇ n

K = k1 exp(a′w)

(7)

K = k 2wb ′

(8)

in which k1, a′, k2, and b′ are constants to be determined from experimental data. Finally, the variation of the consistency index as a function of both temperature and PEG1500 concentration K = f(T,w) was also expressed by a single equation (eq 9).24,29,33,34 ⎛E ⎞ K = k 3 exp⎜ a + b″w⎟ ⎝ RT ⎠

(9)

in which k3 and b″ are constants to be adjusted from experimental data and R is the universal gas constant. All model fittings were carried out using the SAS statistical package v.9. The quality of fitting was evaluated in terms of the coefficient of determination (R2).

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RESULTS AND DISCUSSION Rheological Behavior of PEG1500 Solutions. The solutions presenting different polymer concentrations, w = (0.05, 0.10, 0.15, 0.20 or 0.25) were studied at five temperatures, T = (283.15, 288.15, 293.15, 298.15 or 303.15) K. The corresponding rheograms are given in Figures 1 to 5. Each point

(5) −1

In eqs 1 to 5, τ = shear stress (mPa), γ̇ = shear rate (s ), τ0 = threshold stress needed for the flow to occur (mPa; τ0 = 0 for Newtonian and power law fluids), K = consistency index (mPa·sn), KC = plastic viscosity of Casson (mPa·s0.5) and the dimensionless n = flow index (n > 1 for dilatants fluids, n < 1 for pseudoplastic ones, n = 1 for Newtonian fluids. In this last case, K ≡ η, which is the dynamic viscosity).32,33 Considering each polymer concentration, the influence of temperature (T) on the consistency index of the solutions (K) was modeled by an Arrhenius-type equation (eq 6):31,32

⎛E ⎞ K = k 0 exp⎜ a ⎟ ⎝ RT ⎠

Figure 1. Rheogram obtained for PEG1500 solution with mass fraction w = 0.05, at different temperatures: ⧫, 283.15 K; □, 288.15 K; ▲, 293.15 K; × , 298.15 K; ○, 303.15 K.

represent the average value of shear stress obtained with three measuremets for each shear rate and the vertical error bars correspond to the standard deviations. Data represented in these graphs indicated that, for a given polymer concentration, solutions were more viscous at lower temperatures. Conversely, when analyzing data for a given temperature, more concentrated solutions were more viscous. It was also observed that the more concentrated the solutions, the more pronounced the

(6) 839



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slope variations in the curves obtained at different temperatures. Indeed, at w = 0.05, the solution is very diluted so that the system can be considered as very similar to pure water). Conversely, for w = 0.25 (solutions five times more concentrated in polymer), the temperature differences had a more pronounced effect on the shear stress at higher shear rates, causing a neater separation of the curves in such conditions. It is not possible to determine which rheological model is the most reliable to describe the flow behavior of a fluid by simple visual inspection of the rheograms. Therefore, regression analyses were performed to help accurately determine the best-fitting model (Newton, Bingham, Power Law, Casson, or Herschel−Bulkley; eqs 1 to 5) to mathematically represent these data for PEG1500 solutions. The adjusted equations parameters found, along with the corresponding coefficient of determination (R2) and standard errors (s), are given in Table 1. A preliminary analysis of these R2 values can suggest that all models fitted well to the experimental data. However, the Bingham, Casson, and Herschel−Bulkley models contain the parameter τ0 (yielding stress for flow) different from zero. On the other hand, the inspection of rheograms in Figures 1 to 5 suggests that, by extrapolating the data toward the Y axis, the resulting curves would cross the axis origin, meaning that τ0 ≈ 0. Thus, these three models were not considered in the subsequent modeling analysis any more. Both Newton and Power Law models displayed good fitting to the experimental rheograms (R2 > 0.99 in both cases); so the two models might be assumed to correctly represent the rheological behavior of these PEG1500 solutions within the considered shear rate range. However, the Power Law model was adopted, and this choice can be justified by the fact that the Newton model can be seen as a particular case of the Power Law one, in which n = 1 and the consistency index (K) becomes the viscosity of the fluid. Therefore, with the aim to verify the suitability of the Power Law model to describe the rheological behavior of these polymer solutions, a comparison was done between experimental values of shear stress and those calculated with the adjusted Power Law expressions. Results for the more concentrated solution (w = 0.25), at the lowest and the higher studied temperatures (283.15 K and 303.15 K), are shown in Figures 6 and 7, respectively. As can be observed all points are practically aligned with a straight 45° slope indicating that calculated


Figure 4. Rheogram obtained for PEG1500 solution with mass fraction w = 0.20, at different temperatures: ⧫, 283.15 K; □, 288.15 K; ▲, 293.15 K; × , 298.15 K; ○, 303.15 K. 840


841

a

1.664

2.429

3.572

5.096

0.15

0.20

0.25

5.805

0.25

0.10

4.117

0.20

1.212

2.822

0.15

0.05

1.937

6.672

0.25

0.10

4.524

0.20

1.423

3.155

0.15

0.05

2.171

7.410

0.25

0.10

4.993

0.20

1.575

3.385

0.15

0.05

2.367

0.10

9.046

0.25

1.653

6.188

0.20

0.05

4.187

0.15

0.9997

0.9992

0.9999

0.9987

0.9992

0.9999

0.9999

0.9999

0.9996

0.9985

0.9998

0.9999

0.9997

0.9998

0.9985

0.9972

0.9994

0.9979

0.9997

0.9982

0.9999

0.9996

0.9999

0.9998

0.9996

R2

0.0095

0.0117

0.0029

0.0072

0.0041

0.0052

0.0046

0.0028

0.0046

0.0066

0.0112

0.0026

0.0064

0.0034

0.0074

0.0486

0.0141

0.0188

0.0050

0.0085

0.0039

0.0151

0.0043

0.0038

0.0047

s 1.975 2.863 4.196 6.208

−9.044 −3.710 −2.113 −4.887 1.693 2.387 3.395 5.002 7.429 1.637 2.205 3.175 4.546 6.729 1.490 1.972 2.852 4.164 5.856 1.250 1.684 2.457 3.619 5.177

−9.457 −4.936 −2.318 −2.157 −4.578 −14.609 −8.044 −4.648 −5.450 −13.496 −15.729 −8.279 −7.161 −11.036 −12.164 −8.999 −4.717 −6.891 −11.095 −19.215

0.1661 9.045

ηB

τ0

0.9992

0.9970

0.9997

0.9947

0.9978

0.9998

0.9997

0.9998

0.9987

0.9965

0.9992

0.9999

0.9988

0.9995

0.9956

0.9873

0.9976

0.9908

0.9987

0.9930

0.9999

0.9982

0.9996

0.9994

0.9987

R2

Binghamb (eq 2)

0.0172

0.0245

0.0048

0.0153

0.0072

0.0086

0.0075

0.0043

0.0088

0.0109

0.0229

0.0046

0.0136

0.0056

0.0135

0.1052

0.0305

0.0408

0.0104

0.0177

0.0085

0.0325

0.0092

0.0080

0.0087

s

4.599

6.685

2.285

1.559

1.044

5.697

3.876

2.675

1.786

1.112

6.328

4.428

3.083

2.040

1.309

7.352

5.073

3.428

2.349

1.563

9.088

6.184

4.305

2.874

1.822

K

1.017

1.014

1.010

1.011

1.025

1.009

1.010

1.009

1.013

1.042

1.009

1.003

1.003

1.010

1.031

1.000

0.996

0.997

1.000

1.008

0.999

0.999

0.994

0.997

1.009

n

0.9997

0.9988

0.9998

0.9970

0.9995

0.9999

0.9999

0.9999

0.9995

0.9986

0.9997

0.9999

0.9994

0.9997

0.9981

0.9949

0.9990

0.9964

0.9993

0.9975

0.9999

0.9992

0.9997

0.9996

0.9994

R2

Power Lawc (eq 3)

0.0019

0.0043

0.0014

0.0068

0.0035

0.0011

0.0012

0.0011

0.0026

0.0048

0.0019

0.0008

0.0028

0.0020

0.0055

0.0088

0.0038

0.0074

0.0032

0.0062

0.0009

0.0033

0.0021

0.0022

0.0030

s

0.3203

0.1505

0.0712

0.0479

0.2116

0.1115

0.1115

0.0655

0.1156

0.6464

0.1183

0.0216

0.0210

0.0924

0.4651

0.0070

0.0003

0.0007

0.0182

0.1239

0.0025

0.0057

0.0249

0.0049

0.1122

τ0

2.292

1.913

1.574

1.303

1.128

2.429

2.049

1.695

1.412

1.241

2.604

2.135

1.785

1.491

1.296

2.726

2.234

1.841

1.546

1.305

3.006

2.491

2.044

1.691

1.411

KC

0.9995

0.9981

0.9998

0.9960

0.9986

0.9999

0.9998

0.9998

0.9992

0.9976

0.9995

0.9999

0.9992

0.9996

0.9971

0.9919

0.9984

0.9941

0.9991

0.9957

0.9999

0.9988

0.9997

0.9996

0.9991

R2

Cassond (eq 4)

0.0059

0.0103

0.0025

0.0102

0.0052

0.0029

0.0030

0.0021

0.0049

0.0074

0.0067

0.0018

0.0061

0.0032

0.0086

0.0307

0.0108

0.0176

0.0055

0.0106

0.0026

0.0104

0.0040

0.0040

0.0050

s

86.50

77.12

52.52

14.7

4.70

105.20

67.95

40.75

21.30

8.36

121.80

78.34

47.86

27.52

12.00

134.80

92.20

55.80

33.45

15.31

177.00

115.00

70.70

42.71

21.97

τ0

0.6466

0.6408

0.6531

0.9292

0.7733

0.6709

0.6472

0.6484

0.6716

0.4350

1.2360

0.6681

1.1010

0.6458

0.6459

0.7349

0.6845

0.6736

0.6663

0.6754

0.7803

0.7067

0.6784

0.6594

0.6534

K

1.367

1.304

1.231

1.101

1.077

1.385

1.328

1.259

1.185

1.208

1.299

1.340

1.199

1.212

1.155

1.413

1.353

1.285

1.222

1.155

1.439

1.386

1.322

1.257

1.189

n

0.9715

0.9814

0.9910

0.9895

0.9993

0.9658

0.9777

0.9878

0.9953

0.9879

0.9640

0.9737

0.9821

0.9937

0.9984

0.9521

0.9696

0.9800

0.9913

0.9963

0.9449

0.9616

0.9759

0.9871

0.9957

R2

Herschel-Bulkleye (eq 5)

Units of the parameters. η = mPa·s. bτ0 = mPa; ηB = mPa·s. cK = mPa·sn; n = dimensionless. dτ0 = mPa; KC = mPa·s0.5. eτ0 = mPa; K = mPa·sn; n = dimensionless.

303.15

298.15

293.15

288.15

1.937

2.847

0.05

283.15

η

0.10

w

T/K

Newtona (eq 1)

Table 1. Adjusted Rheological Models for PEG1500 Aqueous Solutions at 283.15 K < T < 303.15 K and with Different Polymer Mass Fractions (0.05 < w < 0.25)

0.0292

0.0224

0.0146

0.0141

0.0034

0.0325

0.0250

0.0174

0.0101

0.0167

0.0313

0.0275

0.0202

0.0119

0.0056

0.0395

0.0299

0.0229

0.0142

0.0087

0.0434

0.0346

0.0259

0.0179

0.0096

s

Journal of Chemical & Engineering Data Article



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Figure 6. Predicted (Power Law adjusted equations) versus observed values of shear stress at shear rates from (20 to 350) s−1 for PEG1500 solutions at 283.15 K: (a) w = 0.05; (b) w = 0.25.

Figure 7. Predicted (Power Law adjusted equations) versus observed values of shear stress at shear rates from (20 to 350) s−1 for PEG1500 solutions at 303.15 K: (a) w = 0.05; (b) w = 0.25 .

and observed values did not present significant deviation at each shear rate. The agreement between measured and calculated values of shear stress confirms the robustness of the Power Law model in modeling the rheological behavior of PEG1500 solutions. Modeling Temperature and PEG1500 Concentration Effects. Empirical models for consistency index (K) were tested considering variations of temperature for constant polymer concentrations, variations of polymer concentration at constant temperatures, and simultaneous variations in both temperature and polymer concentration. For the effect of temperature on the consistency index values (K), at constant polymer concentrations, an Arrhenius-type model was used (eq 6). The regression parameters determined for polymer concentrations w = (0.05, 0.10, 0.15, 0.20, and 0.25) are given in Table 2. In all cases, fittings with R2 > 0.96 were obtained, indicating that this model described the reduction of K as the temperature increases. Regarding the effect of the polymer concentration at constant temperature on the K values, two empirical models were analyzed: an exponential model (eq 7) and a power model (eq 8). For both models, the regression parameters determined

Table 2. Adjusted Parameters of an Arrhenius-type Model for the Consistence Index (K) in Function of the Temperature (283.15 K < T < 303.15 K) of PEG1500 Aqueous Solutions, Considering Different Polymer Mass Fractions (w) Arrhenius-type model: K = k0 exp(Ea/(RT)) (eq 6) w 0.05 0.10 0.15 0.20 0.25

Ea/J·mol−1

k0 −3

1.890·10 1.058·10−3 1.474·10−3 1.779·10−3 1.339·10−3

3

16.220·10 18.524·103 18.643·103 19.112·103 20.709·103

R2

correlation coefficient

0.9774 0.9905 0.9663 0.9753 0.9803

0.9886 0.9871 0.9830 0.9875 0.9901

for T = (283.15, 288.15, 293.14, 298.15 and 303.13) K are given in Table 3. As shown in this table, R2 values for the exponential model were higher than 0.99 for all temperatures, while the power model adjustment gave R2 values comprised between 0.93 and 0.95. This suggests that, in the conditions of the present study, the exponential model was better than the power model to represent the increase of the consistency index in the function of PEG1500 concentration. 842



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increase of PEG1500 concentration. On the other hand, at a given polymer concentration, linear models for n variations in the function of temperature showed R2 between 0.83 and 0.97, which means that more than 83 % of the decrease observed for n values can be attributed to the augmentation of the temperature. However, it is important to note that in all cases 0 < n < 1, reflecting the pseudoplasticity of the solutions at the polymer concentration and temperatures covered in the present study.

Table 3. Adjusted Parameters of Exponential and Power Models for the Consistency Index (K) in Function of the Polymer Mass Fraction (0.05 < w < 0.25) of PEG1500 Aqueous Solutions, at Different Temperatures (T) exponential model: K = k1 exp(a′w) (eq 6) R2

T/K

k1

283.15 288.15 293.15 298.15 303.15

1.282 1.083 1.017 0.914 0.803

T/K

k2

b′

R2


283.15 288.15 293.15 298.15 303.15

28.789 22.309 19.491 17.047 14.454

0.950 0.922 0.900 0.892 0.880

0.9506 0.9394 0.9433 0.9470 0.9391

0.9750 0.9692 0.9996 0.9712 0.9731

a′


7.831 0.9995 0.9997 7.635 0.9960 0.9980 7.450 0.9993 0.9996 7.372 0.9994 0.9997 7.295 0.9971 0.9985 power model: K = k2wb′ (eq 7)

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CONCLUSIONS This work gives a contribution to the rheological characterization of poly(ethylene glycol) in diluted solutions. Specifically, the rheology of aqueous solutions of PEG with molar mass 1500 g·mol−1 (PEG1500) was studied at low shear rates, at different temperatures, considering different polymer concentrations. It was found that the rheological behavior of these solutions can be accurately modeled through the Power Law model. In addition, empirical models were obtained, which satisfactorily represent the variations of the solutions’ consistency index values in function of temperature, polymer concentration or the combination of both. Such information should be useful when working with PEG1500 solutions in ATPS, either during the production of such biphasic systems or during their use in partitioning experiments.

A single empirical model was also used to represent the variation of K simultaneously as a function of temperature and PEG1500 concentration (eq 9). After calculating the regression parameters, the following adjusted equation was obtained (R2 = 0.9965; correlation coefficient = 0.9982): ⎞ ⎛ 18.642·103 K = 4.781· 10−4 exp⎜ + 7.517w⎟ RT ⎠ ⎝

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Corresponding Author

(10)

*Tel.: 55 31 38991618. Fax: 55 31 38992208. E-mail: jcoimbra@ ufv.br.

Such type of model is particularly useful in industrial applications, when physical parameters must be promptly and accurately calculated as a function of different variables (concentration and temperature, in this case), considering the intervals used in the model derivation. Finally, concerning the modeling of variations of the behavior index (n) values in the functions of temperature and PEG1500 concentration, linear models (y = α+ βx) were adjusted with correlation coefficients > 0.9; the results are summarized in Table 4. At a constant temperature, linear models for the

Funding

We are thankful to the Brazilian agencies CAPES, CNPq, FAPEMIG, and FAPESP for the financial support, fellowships, and grants. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to acknowledge Mrs. Angélica Ribeiro da Costa and Mrs. Hiasmyne Silva de Medeiros for their technical support during the analyses.

Table 4. Adjusted Parameters of Linear Models for the Behavior Index (n) in the Function of the Polymer Mass Fraction (0.05 < w < 0.25) and Temperatures (283.15 K < T < 303.15 K) of PEG1500 Aqueous Solutions

■

model: n = α1 + β1w T/K

α1

283.15 288.15 293.15 298.15 303.15

0.592 0.549 0.643 0.535 0.541

1.394 0.9337 1.316 0.9306 0.612 0.8990 1.028 0.8718 0.938 0.9303 model: n = α2 + β2T

w

α2

β2

R2


0.05 0.10 0.15 0.20 0.25

3.437 1.927 2.960 3.718 1.314

−0.009 −0.004 −0.008 −0.011 −0.002

0.8412 0.8311 0.9781 0.8976 0.9108

0.9172 0.9116 0.9890 0.9474 0.9544

β1

R2

AUTHOR INFORMATION

REFERENCES

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correlation coefficient 0.9663 0.9647 0.9482 0.9337 0.9645

dependence of n with the polymer concentration presented R2 values between 0.87 and 0.93, meaning that about 90 % of the augmentation observed for n (β > 0) can be explained by the 843



Article

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