Theoretical and Experimental Models on Viscosity: I. Glycerol - The

J. Phys. Chem. B 2007, 111, 9563-9570

9563

Theoretical and Experimental Models on Viscosity: I. Glycerol Salvatore Magazu` ,*,† Federica Migliardo,† Nicolay P. Malomuzh,‡ and Ivan V. Blazhnov‡ Dipartimento di Fisica, UniVersita` di Messina, P.O. Box 55, I-98166 Messina, Italy, and Department of Theoretical Physics, Odessa National UniVersity, 2 DVoryanskaya str., Odessa, 65026, Ukraine ReceiVed: March 10, 2007; In Final Form: June 1, 2007

The main aim of the present work is to show, for the case of glycerol, the connection between the macroscopic fragility parameter, defined as m ) (d log η)/(d(Tg/T))|T)T+g , and the average number of H-bonds per molecule. Furthermore, the relation between the macroscopic and microscopic definitions of fragility, which takes into account the temperature dependence of the atomic mean square displacement, is discussed.

I. Introduction liquids,1-6

It is well-known that among glass-forming hydrogenbonded systems tend to be unusual with regard to their supercooling propensity and highly complex behavior in the supercooled state.7,8 As an example, they tend to vitrify more readily than the general rule, Tb /Tm > 2.0 would indicate (e.g., for glycerol Tb /Tm ) 1.7), Tb being the boiling temperature, and in the supercooled liquid state, they have large activation energies and relatively small departures from Arrhenius behavior. In addition, the liquid ranges, defined by the ratio Tb/Tg, tend to be smaller than for molecular liquids of similar constitution but lacking H-bonds. In recent times, the departures from the Arrhenius behavior and other related properties (e.g., nonexponential and nonlinearity of relaxation) have been discussed in terms of the strongfragile liquids classification.1,4,5 In such a classification scheme, an extreme is represented by conformity to the Arrhenius equation

η ) η0 exp

( ) Ea kBT

(1)

where Ea is the activation energy and kB is the Boltzmann constant, while the other extreme is represented by a marked departure from Arrhenius behavior, which at the limit becomes a first-order transition from fluid to glass, unless a first-order fragile-liquid-to-strong-liquid transition occurs.4,5 The pattern of increasing deviations from the Arrhenius form may be accounted for the Vogel-Tamman-Fulcher (VTF) form of temperature dependence using the “strength” parameter, D, as a variable:7,8

η ) η0 exp

( ) DT0 T - T0

(2)

Here, T0, the ideal glass transition temperature, which is the temperature of viscosity divergence, lies below Tg by an amount that depends on the D parameter. Liquids in which during a temperature increase above Tg the short and medium range order of the initially glassy structure * Corresponding author. Phone: +39 0906765025. Fax: +39 090395004. E-mail: [email protected]. † Universita ` di Messina. ‡ Odessa National University.

resist to thermal disruption have a large D parameters and are called “strong”, whereas those with a small D are called “fragile”.4,5,7,8 From the slope of the Tg-scaled Arrhenius plot of viscosity or of any relaxation time, a fragility parameter m can be defined as

m)

d log η d(Tg/T)

|

(3)

T)T g+

and is related to the strength parameter D by

m ) 590(D-1 + 0.027)

(4)

Strong liquids exhibit small or undetectable increases in heat capacity at Tg, whereas fragile liquids show large changes amounting in some cases even to a doubling of the glassy state heat capacity. The fact that certain hydrogen-bonded liquids provide exceptions to this general pattern, having both large Cp(l)/Cp(g) and large D, heightens the interest in the behavior of aqueous solutions and other hydrogen-bonded liquids, such as glycerol and sugars.4,5 In terms of the Adam-Gibbs theory of relaxation for viscous liquids and in terms of the density of minima of the potential energy hypersurface characteristic of the molecular systems, the small number of minima on the surface is characteristic of a strong liquid, whereas the much more degenerate surface is descriptive of the fragile liquids.7,8 The case of hydrogen-bonded liquids is anomalous, since the surface shows the same density of minima of fragile systems due to hydrogen bonds between molecules, whereas the highenergy barriers (∆µ) between them is responsible for their relatively strong behavior (D ∼ ∆µ/∆Cp). Glycerol is considered as anomalous due to the large change in heat capacity that it experiences at the glass transition, and it occupies an intermediate position in the Angell classification. The present work is aimed at describing a model capable of evaluating the temperature dependence of viscosity and activation energy and, hence, a fragility parameter based on these calculations. Finally, the link among different micro- and macrodefinitions of fragility is discussed. II. Theoretical Background To show that the activation energy of shear viscosity, η, and hence, the macroscopic fragility parameter are connected with

10.1021/jp071949b CCC: $37.00 © 2007 American Chemical Society Published on Web 07/26/2007

9564 J. Phys. Chem. B, Vol. 111, No. 32, 2007

Magazu` et al. By using these asymptotes, we estimate a value of the Stickel temperature as TS ) 234.74 K. By the asymptotes (eqs 7.1 and 7.2), the following set of straight lines has been created;

log η ) [loghigh η0 + b(h)‚(1/T)](1 - ψ) + [loglow η0 + b(l)‚(1/T)]ψ (8) namely

log η ) [-32.28 + 8091.40‚(1/T)]‚(1 - ψ) + [-8.68 + 2547.68‚(1/T)]‚ψ (9) where ψ fulfill the requirements Figure 1. Arrhenius plot of viscosity for glycerol. Experimental data (circles) have been taken from ref 11. The dashed lines are the asymptotes for low and high viscous regions. The Stickel temperature, TS, is indicated.

the average number, n H(T), of the H-bonds per molecule, we take into consideration the Arrhenius behavior described by eq 1.9,10 In Figure 1, the Arrhenius plot of viscosity for glycerol is shown. Experimental data have been taken from ref 11. In the case of glass-forming systems, a more appropriate expression of shear viscosity has been suggested by Frenkel,12

( ( ))

Ea Tg η(T) ) exp -1 ηg kBTg T

(5)

where ηg ) η(Tg) is the value of η at the glass transition temperature, Tg. In the model we are proposing, an important parameter is the Stickel temperature, TS, whose physical meaning lies in the circumstance that it marks both a qualitative change in the evolution of the susceptibility spectra of glass formers13 and the departure of the relaxation times from the VTF function.14 To determine the Stickel temperature, TS, we consider the asymptotes of log η as a function of where 1/T in low and highly viscous regions intersect,10,12 as shown in Figure 1 (see Table 1), whose equations are

log

low

η ) log

low

η0 + b ‚(1/T) (l)

loghigh η ) loghigh η0 + b(h)‚(1/T)

ψ(T) )

1 in low viscous states T > TL 0 in high viscous states T > TS

(10)

The ψ parameter is proportional to the number of the degrees of freedom responsible for the formation of highly viscous states and manifested in R relaxation. Although at this stage the ψ parameter is treated as purely empirical, its physical meaning lies in its linear relation with the number of molecular H-bonds, as will be discussed later. By the substitution of an arbitrary set of values of the ψ function that fulfill the requirements (10) (0 < ψ < 1), the set of straight lines results (see Figure 2 and Table 2): By fitting the experimental data of the shear viscosity using a polynomial fit of second order, we obtain the following result:

log η ) 4.04 - 5398.22‚(1/T) + 1.24‚106‚(1/T)2 (11) By calculating the straight lines tangent to the curve and parallel to the straight lines (see Table 2), we obtain the temperature dependence of the ψ function (see Table 3), shown in Figure 3. Taking into account that, in the case of glycerol, Tm ) 291 K and Tg ) 186 K, we consider only the data in the 186 K < T < 291 K region. By using the following model function for the fitting,

ψ(T) )

(6.1) (6.2)

for low and high viscous regions, respectively, which in the case of glycerol has the form

{

{

[ ( )]

1 - exp 0

T - Tψ ∆ψ

n

T > Tψ

(12)

T < Tψ

we obtain the result shown in Figure 3 and for the fitting parameters, the values of Tψ ) 165.67 K and ∆y ) 82.67 K. III. Results and Discussion

loglow η ) -8.68 + 2547.68‚(1/T)

(7.1)

loghigh η ) -32.28 + 8091.40‚(1/T)

(7.2)

The introduction of the ψ parameter proportional to the effective number of the degrees of freedom that “freeze” in the

TABLE 1: Experimental Data of the Shear Viscosity of Glycerol 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1/T (K-1)

log10 η (Pa‚s)

0.003 127 0.003 153 0.003 182 0.003 208 0.003 250 0.003 271 0.003 292 0.003 301 0.003 309 0.003 360 0.003 394 0.003 415 0.003 445 0.003 475

-0.7000 -0.6667 -0.5667 -0.5000 -0.4000 -0.2667 -0.4000 -0.1333 -0.1667 0.0000 0.1333 0.2000 0.3000 0.3667

15 16 17 18 19 20 21 22 23 24 25 26 27 28

1/T (K-1)

log10 η (Pa‚s)

0.003 492 0.003 504 0.003 538 0.003 551 0.003 623 0.003 661 0.003 678 0.003 708 0.003 725 0.003 754 0.003 818 0.003 835 0.003 860 0.003 873

0.4333 0.5000 0.6000 0.8000 0.9333 1.1000 1.266 7 1.1667 1.2000 1.4667 1.5333 1.8667 1.9667 1.8333

29 30 31 32 33 34 35 36 37 38 39 40 41 42

1/T (K-1)

log10 η (Pa‚s)

0.003 886 0.003 928 0.003 958 0.003 958 0.003 987 0.004 021 0.004 034 0.004 059 0.004 110 0.004 114 0.004 153 0.004 182 0.004 220 0.004 237

1.7667 2.1667 2.1000 2.3000 2.3667 2..5667 2.4000 2.7667 2.9667 3.0000 3.1667 3.2667 3.2667 3.6000

43 44 45 46 47 48 49 50 51 52 53 54

1/T (K-1)

log10 h (Pa‚s)

0.004 258 0.004 314 0.004 331 0.004 424 0.004 500 0.004 609 0.004 710 0.004 875 0.004 975 0.005 047 0.005 195 0.005 211

3.7000 4.0333 3.8333 4.6000 5.1250 5.7500 6.3750 7.1875 7.9375 8.5625 9.5625 10.0625

Models on Viscosity

J. Phys. Chem. B, Vol. 111, No. 32, 2007 9565

Figure 2. Determination of the set of straight lines for the choice of the ψ function.

Figure 4. Activation energy as a function of temperature. Circles are the values of Ea obtained from experimental data using eq 16, and dashed lines are the asymptotic fits according to eqs 17.1 and 17.2.

b(ψ) ) b(h)(1 - ψ) + b(l)ψ + b(i)ψ(1 - ψ) + higher order terms (15) For the fitting, the higher order terms have not been taken into account. Now we calculate the values of activation energy starting from the values of the shear viscosity listed in Table 1 and linked by the following relationship:

Ea ) (3.17‚10-23)‚T‚[2 + log10 ηl] for η0 ) 10-2 Pa‚s (16)

Figure 3. Temperature dependence of the ψ function. The circles represent the ψ values directly determined from eqs 9-10, and the line represents the theoretical fit by using eq 12.

system amorphous solid states enables one to describe the temperature dependence of the shear viscosity, ηl. The viscosity of many liquids consisting of the anisotropic molecules is satisfactorily described by the formula

( )

Ea(T) ηl ) η0 exp kBT

(13)

Ea being the activation energy. Since near Tη the ψ parameter undergoes the most essential changes, the temperature dependence of Ea and b can be approximated by the expressions

Ea(ψ) ) Ea(h)(1 - ψ) + Ea(l)ψ + Ea(i)ψ(1 - ψ) + higher order terms (14)

The results of this calculation are listed in Table 5 and shown in Figure 4. Fitting the points corresponding to low and highly viscous regions by the formulas

Ea(low) ) Ea(l) + b(l)‚(TL - T) T > TS

(17.1)

Ea(high) ) Ea(h) + b(h)‚(TS - T) T < TS

(17.2)

and

where TL ) 214 K and TS ) 234.74, we obtain the following result (see Table 6 and Figure 4):

Ea(low) ) 3.92‚10-20 + 2.48‚10-22‚(214 - T) T > 234.74 K (18) ) 1.02‚10-19 + E(high) a 8.54‚10-22‚(157 - T) T < 234.74 K (19) In this procedure, the points corresponding to the activation energy have been fitted by using eq 14. The free fit parameters

TABLE 2: Set of Straight Lines for the Choice of ψ Function 1 2 3 4 5 6 7 8 9 10 11

ψ

straight line

straight line tangent

0.01 0.02 0.04 0.05 0.06 0.08 0.1 0.2 0.3 0.4 0.5

y ) -2.05 + 8035.97x y ) -1.81 + 7980.54x y ) -1.34 + 7869.66x y ) -1.10 + 7814.22x y ) -30.87 + 7758.79x y ) -30.40 + 7647.91x y ) -9.92 + 7537.04x y ) -7.56 + 6982.66x y ) -5.20 + 6428.29x y ) -2.84 + 5873.92x y ) -0.48 + 5319.55x


12 13 14 15 16 17 18 19 20 21 22

ψ

straight line

straight line tangent

0.6 0.7 0.8 0.9 0.91 0.92 0.94 0.95 0.96 0.98 0.99




Magazu` et al.

TABLE 3: Temperature Dependence of the ψ Function tangency point 1 2 3 4 5 6 7 8 9 10 11

tangency point

1/T

log η

T (K)

Ψ

0.005 39 0.005 36 0.005 32 0.005 30 0.005 28 0.005 23 0.005 190 0.004 964 0.004 742 0.004 520 0.004 297

11.149 10.971 10.619 10.445 10.272 9.930 9.592 7.978 6.488 5.120 3.876

185.656 186.425 187.983 188.772 189.567 191.179 192.817 201.451 210.894 221.266 232.721

0.01 0.02 0.04 0.05 0.06 0.08 0.1 0.2 0.3 0.4 0.5

12 13 14 15 16 17 18 19 20 21 22

1/T

log η

T (K)

Ψ

0.004 075 0.003 853 0.003 630 0.003 410 0.003 390 0.003 360 0.003 320 0.003 300 0.003 270 0.003 230 0.003 210

2.756 1.758 0.8837 0.1326 0.0640 -0.0286 -0.1330 -0.1970 -0.2590 -0.3800 -0.4380

245.404 259.562 275.454 293.419 295.345 297.297 301.278 303.310 305.368 309.570 311.715

0.6 0.7 0.8 0.9 0.91 0.92 0.94 0.95 0.96 0.98 0.99

TABLE 4: Fit Numerical Values of the Temperature Dependence of the ψ Function fit numerical values Tψ ∆ψ n

165.67 K 82.67 K 2

are E(l), E(h), and E(i), while the parameters corresponding to the ψ(T) function are kept fixed (see Table 7). The obtained results are shown in Table 8 and Figure 5). To simplify the calculation of the parameter b(i), we consider a polynomial fit of second order for fitting the points corresponding to the activation energy, obtaining the following result:

Ea ) 3.04‚10-19 - 1.68‚10-21 T + 2.42‚10-24‚T 2

(20)

Calculating the derivative respective to the temperature, we obtain the following expression:

b(T) ) 1.68‚10

-21

+ 4.85‚10

-24

T

(21)

Substituting some of the temperature values corresponding to the data of the activation energy in expression 21, we

Figure 5. Use of the ψ parameter for fitting the activation energy behavior. Circles represent the values directly determined from eq 16, and the dashed line represents the fit by using eq 20.

obtain a set of data that help to determine the temperature dependence of the parameter b(T). The results of this calculation are listed in Table 9 and shown in Figure 6. The obtained data have been fitted by using eq 15. The only free-fit parameter is b(i), whereas the parameters corresponding

TABLE 5: Values of Activation Energy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1/T (K-1)

log η (Pa‚s)

T (K)

Ea (joule) for η0 ) 10-2Pa‚s

0.003 127 0.003 153 0.003 182 0.003 208 0.003 250 0.003 271 0.003 292 0.003 301 0.003 309 0.003 360 0.003 394 0.003 415 0.003 445 0.003 475 0.003 492 0.003 504 0.003 538 0.003 551 0.003 623 0.003 661 0.003 678 0.003 708 0.003 725 0.003 754 0.003 818 0.003 835 0.003 860

-0.7000 -0.6667 -0.5667 -0.5000 -0.4000 -0.2667 -0.4000 -0.1333 -0.1667 0 0.1333 0.2000 0.3000 0.3667 0.4333 0.5000 0.6000 0.8000 0.9333 1.1000 1.2667 1.1667 1.2000 1.4667 1.5333 1.8667 1.9667

319.80 317.16 314.27 311.72 307.69 305.72 303.77 302.94 302.21 297.62 294.64 292.83 292.83 287.77 286.37 285.39 282.65 281.61 276.01 273.15 271.89 269.87 268.46 266.38 261.92 260.76 259.07

1.321 × 10-20 1.344 × 10-20 1.431 × 10-20 1.486 × 10-20 1.564 × 10-20 1.684 × 10-20 1.544 × 10-20 1.797 × 10-20 1.760 × 10-20 1.891 × 10-20 1.997 × 10-20 2.047 × 10-20 2.140 × 10-20 2.164 × 10-20 2.214 × 10-20 2.267 × 10-20 2.335 × 10-20 2.506 × 10-20 2.573 × 10-20 2.691 × 10-20 2.822 × 10-20 2.715 × 10-20 2.730 × 10-20 2.934 × 10-20 2.941 × 10-20 3.204 × 10-20 3.265 × 10-20

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

1/T (K-1)

log η (Pa‚s)

T (K)

Ea(joule) for η0)10-2Pa‚s

0.003 873 0.003 886 0.003 928 0.003 958 0.003 958 0.003 987 0.004 021 0.004 034 0.004 059 0.004 110 0.004 144 0.004 153 0.004 182 0.004 220 0.004 237 0.004 258 0.004 314 0.004 331 0.004 424 0.004 500 0.004 609 0.004 710 0.004 875 0.004 975 0.005 047 0.005 195 0.005 211

1.8333 1.7667 2.1667 2.1000 2.3000 2.3667 2..5667 2.4000 2.7667 2.9667 3.0000 3.1667 3.2667 3.2667 3.6000 3.7000 4.0333 3.8333 4.6000 5.1250 5.7500 6.3750 7.1875 7.9375 8.5625 9.5625 10.0625

258.20 257.33 254.58 252.65 252.65 250.82 248.69 247.89 246.37 243.31 241.31 240.79 239.12 236.97 236.02 234.85 231.80 230.89 226.04 222.22 216.97 212.31 205.13 202.22 198.14 192.49 191.90

3.145 × 10-20 3.080 × 10-20 3.371 × 10-20 3.291 × 10-20 3.452 × 10-20 3.480 × 10-20 3.432 × 10-20 3.466 × 10-20 3.732 × 10-20 3.840 × 10-20 3.834 × 10-20 3.953 × 10-20 4.002 × 10-20 3.966 × 10-20 4.200 × 10-20 4.254 × 10-20 4.444 × 10-20 4.280 × 10-20 4.741 × 10-20 5.031 × 10-20 5.343 × 10-20 5.650 × 10-20 5.989 × 10-20 6.386 × 10-20 6.650 × 10-20 7.072 × 10-20 7.355 × 10-20

Models on Viscosity

J. Phys. Chem. B, Vol. 111, No. 32, 2007 9567 TABLE 7: Values of ψ Parameters as Obtained by the Activation Energy Behavior fixed parameters (function ψ(T))

fit parameters E(l) E(h) E(i)

starting value (before fit) 3.9218‚10-20 (joule) 1.0215‚10-19 (joule) 2‚10-20 (joule)

Tψ ∆ψ n

value 165.67 K 82.67 K 2

TABLE 8: Fit Numerical Values as Obtained by the Activation Energy Behavior fit numerical values

Figure 6. Behavior of the b(T) parameter. Circles represent the values determined by using eq 21, and the solid line represents the fit by using eq 15.

TABLE 6: Fit Numerical Values of Activation Energy

tion and the asymptotical interpolation,

η0 ) 10-2Pa‚s

fit numerical values

E(l) b(l) E(l)/kBTm b(l)/kB E(h) b(h) E(h)/kBTm b(h)/kB

3.9218‚10-20 ( 2.4394‚10-21 (joule) 2.4818‚10-22 ( 2.4678‚10-23 (joule‚K-1) 9.8 18 1.0215‚10-19 ( 5.7548‚10-21 (joule) 8.5467‚10-22 ( 1.46‚10-22 (joule‚K-1) 25.4 62

b(l)

EL(Ψ) ) E (l)(1 - Ψ) + E (h)Ψ

(22)

at ψ ) 1/2, we have

(

(l) E(l) ) E(l) 0 + b (TS - T) T > TS

(28)

(h) E(h) ) E(h) 0 + b (TS - T) T < TS

(29)

takes place approximately at T ) TS and is equal to

da ) EL(TS) - Ea(l)(TS)

1 1 ) (E (l) + E (h)) + 2 2 1 (i) E + higher order terms (23) 4

Taking into account eqs 28-30, we get

da ≈ b(l)TS

1 1 D ) da - d ≈ da ) b(l)TS 2 2

(32)

E (i) ≈ 2b(i)TS

(33)

In addition,

and

1 1 EL ψ ) ) (E (l) + E (h)) + higher order terms (24) 2 2

)

The higher order terms have been neglected in the fitting procedure. The approximate distance between the real and linear curves at ψ ) 1/2 can be expressed as

(

d ≈ EL Ψ )

1 1 1 - E Ψ ) ) - E (i) 2 2 4

) (

)

(25)

On the other hand, the low-temperature straight-line asymptote to the curve log η as a function of 1/T, described by eq (6.1), at T ) TS is

Ea(l)(TS) ) Ea(h)(TS) ) E0(l)

(26)

The low viscous region corresponds to T ) (1.5 ÷ 2)TS; therefore,

E0(l) ≈ E (l) + b(l)TS

(27)

The maximum distance between curves of the linear interpola-

(31)

From geometrical picture, it follows that approximately d ≈ 1/2 da, so the distance D between the curve and the straightline asymptotic interpolation is

)

(

(30)

b(h)

to the ψ(T) function and the parameters and are kept fixed (see Table 10). The results are shown in Figure 6 and Table 11. From the linear dependence

Eψ)

1.9219‚10-20 ( 7.2615‚10-22 (joule) 4.8 7.8289‚10-20 ( 8.4909‚10-22 (joule) 19.5 -2.4634‚10-20 ( 3.4132‚10-22 (joule) -6

E(l) E(l)/kBTm E(h) E(h)/kBTm E(i) E(i)/kBTm

is proportional to the distance between the Stickel point and the curve ln η(T), corresponding to experimental data, and therefore takes large values for the substances that are “fragile” according to the Angell’s classification.10-12 If we consider rigorously the standard activation theory, the dependence of the activation energy on the temperature contradicts some of its basic assumptions. In fact, according to this latter approach, the activation energy can depend only on density,

E ) E(n)

(34)

and therefore, it must be constant on isochors. As a matter of fact, the states of systems that undergo a glass transition depend not only on temperature, but also on other parameters.10,12 The fragility parameter, m, is connected with the activation energy and the thermal expansion coefficient b(h) by the relation

m)

1 (E (h) + b(h)Tg) Tg ln(10) 0

(35)


Magazu` et al.

TABLE 9: Temperature Dependence of b(T) Parameter 1 2 3 4 5 6 7 8 9

T (K)

b (T)

319.80 314.27 307.69 303.77 302.21 294.64 292.83 286.37 282.65

-1.292 921 68‚10-22 -1.561 394 332‚10-22 -1.880 842 804‚10-22 -2.071 152 532‚10-22 -2.146 888 036‚10-22 -2.514 399 424‚10-22 -2.602 272 028‚10-22 -2.915 894 692‚10-22 -3.096 494 74‚10-22

10 11 12 13 14 15 16 17 18

T (K)

b (T)

276.01 271.89 268.46 261.92 259.07 257.33 252.65 250.82 247.89

-3.418 856 116‚10-22 -3.618 875 524‚10-22 -3.785 396 536‚10-22 -4.102 903 072‚10-22 -4.241 266 012‚10-22 -4.325 740 228‚10-22 -4.552 946 74‚10-22 -4.641 790 312‚10-22 -4.784 037 124‚10-22

19 20 21 22 23 24 25 26 27

T (K)

b (T)

243.31 240.79 236.97 234.85 230.89 222.22 212.31 202.22 192.49

-5.006 388 796‚10-22 -5.128 730 764‚10-22 -5.314 185 652‚10-22 -5.417 108 26‚10-22 -5.609 359 924‚10-22 -6.030 274 552‚10-22 -6.511 389 196‚10-22 -7.001 242 552‚10-22 -7.473 618 484‚10-22

TABLE 10: Values of b(T) Parameter fit parameters

fixed parameters

starting value (before fit) 4‚10-23 (joule‚K-1)

b(i)

value b(l) b(h) Tψ ∆ψ n

-2.4818‚10-22 (joule‚K-1) -8.5467‚10-22 (joule‚K-1) 165.67 K 82.67 K 2

TABLE 11: Fit Numerical Values of b(T) Parameter fit numerical values 3.6904‚10-23 ( 2.1691‚10-23 (joule‚K-1) 2.7

b(i) b(i)/kB

The quantity + g) is the invariant about the choice of a value η0, and the terms E0(h) and b(h) depend on η0. Assuming η0 ) 10-2Pa‚s for glycerol, we obtain (E0(h)

b(h)T

Figure 7. Average number of H-bonds per molecule for glycerol as a function of temperature. The circles represent the values directly determined from eq 37, and the dashed line represents the temperature dependence, nH(T) ) nH(Tg) - (nH(Tg) - nH(Tm))ψ(T).

By considering eqs 37 and 39, we determine nH(T),

E0(h)/b(h)Tg

1 ≈ 2

which means that the main contribution to the fragility parameter is given by the term b(h)Tg. It has been shown15 that, in hydrogen-bonded glass-forming systems, such as glycerol, considering Ea ) EW + EH (the van der Waals and H-bond contributions of the activation energy, respectively), the main contribution to the activation energy is due to the structural function nH(T).

EH(T) ) γ1nH(T) + ...

(37)

Thus, within this approximation, the temperature dependencies of νH(T) and EH(T) are linear in nH(T), which represents the average number of H-bonds per molecule. Taking into account eqs 14 and 37 for the activation energy, we obtain

1 - Ψ(T) ≈ κ(nH(Tg) - nH(T))

( )

(36)

(38)

Eexp(T) - EW ∂ nH(T) )∂T T γ T2 where

Eexp(T) )

∂ ln η(T) ∂(1/T)

κ)

1 nH(Tg) - nH(Tm)

(39)

indicating that the parameter (1 - Ψ(T)) is proportional to the number of H-bonds per unit volume.13

(41)

Taking into account that Eexp(T) ≈ EW + γ1nH(T), neglecting EW and supposing γ1 to be equal10kBTm, we are able to estimate the value and temperature dependence of nH(T). As an example, we evaluate the average number of H-bonds per molecule for glycerol as a function of temperature, as reported in Figure 7. The dashed line in Figure 7 shows the temperature dependence of nH(T) ) nH(Tg) - (nH(Tg) - nH(Tm))ψ(T). Here, we used for glycerol the values nH(Tg) ) 5.9 and nH(Tm) ) 2.3. Therefore, if we consider the trend of nH(T) and the number of H-bonds near the glass transition temperature, a molecule of glycerol can form a maximum of six H-bonds.15 From Figure 7, it follows that near Tg, the following approximate equality takes place,

|n′H(Tg)Tg| ≈ 0.1nH(Tg)

where

(40)

1

(42)

where n′H(T) is the temperature derivative of nH(T). Taking into account that EW e 10kBTm , γ1nH(T), we obtain for the fragility parameter of glycerol the expression

m≈

1 n (T ) Tg ln(10) H g

(43)

Models on Viscosity

J. Phys. Chem. B, Vol. 111, No. 32, 2007 9569

Figure 8. Temperature dependence of the mean square displacements for glycerol. Experimental data (circles) have been taken from ref 16, and the symbol size has been chosen by taking into account the error bars. The solid line represents the theoretical fit by using eq 44.

Figure 9. Behavior of log η versus (loc)-1. The squares represent the experimental viscosity and mean square displacement data taken from refs 11 and 16, respectively. The dashed line represents the fit according to eq 45.

which shows that m, η, and nH are interrelated, since m depends on nH through the shear viscosity, η.15 Figure 8 shows the temperature dependence of the mean square displacements for glycerol.16 The mean square displacements as a function of temperature have been determined by using elastic neutron scattering.17,18 It is evident that a dynamical transition occurs at ∼Tg ) 186 K. Below the onset temperature, the elastic intensity has the Gaussian form expected for a harmonic solid, and the mean square displacement behavior can be fitted within the framework of the harmonic approximation

〈∆u2(T)〉 )

(

h〈ν〉 h〈ν〉 coth -1 2K 2KBT

)

(44)

To characterize the anharmonic region, a new calculation for the degree of fragility has been shown.17,18 In this picture, one obtains the viscosity17,18

η ) η0 exp[u02/〈u2〉loc]

(45)

where 〈 b u2〉loc ) 〈 b u2〉anharm - 〈 b u2〉harm is the difference between the mean square displacement of the disordered phase (amorphous and liquid) characterized by an anharmonic behavior and mean square displacement of the ordered phase (crystalline) described by a harmonic trend. Equation 45 allows one to characterize the fragility of the investigated systems, since the mean square displacement is measurable even below Tg and above Tm. The linear fitting procedure of log η versus (loc)-1 furnishes through its slope the u0 parameter value. Figure 9 shows the result of this procedure for glycerol. The activation energy is linked to the mean square displacement by the formula 2

E)T

u0

(46)

〈u b2〉loc

To evaluate the “fragility” degree of the investigated systems, the proposed parameter is written:17,18

M)

|

b2>loc) d(u02/

Theoretical and Experimental Models on Viscosity: I. Glycerol - The

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