Wypych Handbook of Solvents

680	Tai-ichi Shibuya

11.2DIELECTRIC SOLVENT EFFECTS ON THE INTENSITY OF LIGHT ABSORPTION AND THE RADIATIVE RATE CONSTANT

Tai-ichi Shibuya

Faculty of Textile Science and Technology

Shinshu University, Ueda, Japan

11.2.1 THE CHAKO FORMULA OR THE LORENTZ-LORENZ CORRECTION

The intensity of light absorption by a molecule is generally altered when the molecule is immersed in a solvent or transferred from one solvent to another. The change may be small if the solvents are inert and non-polar, but often a significant increase or decrease is observed. The first attempt to correlate such effects with the nature of the solvent was made by Chako1 in 1934. Chako’s formula reads as

f ′′	=	(n 2 + 2)2	[11.2.1]

f9n

where:

foscillator strength of an absorption band of a molecule

f ′′	apparent oscillator strength of the molecule in solution

nrefractive index of the solution at the absorbing frequency

The apparent oscillator strength is proportional to the integrated intensity under the molar absorption curve. To derive the formula, Chako followed the classical dispersion theory with the Lorentz-Lorenz relation (also known as the Clausius-Mosotti relation), assuming that the solute molecule is located at the center of the spherical cavity in the continuous dielectric medium of the solvent. Hence, the factor derived by Chako is also called the Lo- rentz-Lorenz correction. Similar derivation was also presented by Kortòm.2 The same formula was also derived by Polo and Wilson3 from a viewpoint different from Chako.

Chako’s formula always predicts an increase of the absorption intensity with the refractive index. This does not hold, for instance, for the allowed π → π* electronic transitions of cyclohexadiene and cyclopentadiene,4 and monomethyl substituted butadienes.5

11.2.2THE GENERALIZED LOCAL-FIELD FACTOR FOR THE ELLIPSOIDAL CAVITY

A natural generalization of the Chako formula was made by generalizing the spherical cavity to an ellipsoidal cavity. Such a generalization was shown by Shibuya6 in 1983. The generalized formula derived by him reads as

f ′′	=	[s(n 2	− 1) + 1]2	[11.2.2]
f			n

where:

sshape parameter which takes a value between 0 and 1

This parameter s is more generally known as the depolarization factor, whose values are listed for special cases in general textbooks.7 For the spherical cavity, s = 1/3 in any axis; for

idea11

11.2 Dielectric solvent effects

Figure 11.2.1. Dependence of f ′′ / f on the refractive index n for different values of s. [After reference 6]

681

a thin slab cavity, s = 1 in the normal direction and s = 0 in plane; and for a long cylindrical cavity, s = 0 in the longitudinal axis and s = 1/2 in the transverse direction. The shape of the ellipsoidal cavity is supposed to be primarily determined by the shape of the solute molecule. Typical cases are long polyenes and large planar aromatic hydrocarbons. One can assume s = 0 for the strong π → π* absorption bands of these molecules. For smaller molecules, however, one should assume s ≈ 1/3 regardless of the shape of the solute molecule, as the cavity shape then may be primarily determined by the solvent molecules rather than the solute molecule. Note that Eq. [11.2.2] gives the Chako formula for s = 1/3, i.e., for the spherical cavity.

For transitions whose moments are in the longitudinal axis of a long cylindrical cavity or in the plain of a thin slab cavity, Eq. [11.2.2] with s = 0 leads to f ′′ / f = 1/ n, so that the absorption intensity always decreases with the refractive index. If the transition moment is normal to a thin slab cavity, Eq. [11.2.2] with s = 1 leads to f ′′ / f = n 3 . The dependence of the ratio f ′′/f on the refractive index n according to Eq. [11.2.2] is illustrated for different values of s in Figure 11.2.1. The slope of the ratio is always positive for s > 1/4. For 0 < s < 1/4, it is negative in the region 1 ≤ n ≤(1−s) / 3s and positive in the other region.

Eq. [11.2.2] can be also written as the following form:

	=		s(n 2 − 1) +
(nf ′′)
		f		f	[11.2.3]

This equation shows a linear relationship between (nf ′′) and (n2 - 1). If a set of measured values of f ′′ vs. n are provided for a solute, the least-squares fitting to Eq. [11.2.3] of (nf ′′) against (n2 - 1) gives the values of f and s for the solute molecule. Note that f ′′ and f in Eq. [11.2.3] can be replaced by any quantities proportional to the oscillator strengths. Thus, they can be replaced by the integrated intensities or by their relative quantities.

Figure 11.2.2 shows such plots for the π → π* absorption bands of β-carotene and the n → π* absorption bands of pyrazine measured8 in various organic solvents. Here, the relative intensities f ′′ / fc′′, where fc′′ is the absorption intensity measured in cyclohexane as the reference solvent, are considered, and y = nf ′′ / fc′′ is plotted against x = n2 - 1. The least-squares fittings give s = 0 for the allowed π → π* transition of β-carotene and s = 0.29 for the vibronic n → π* transition of pyrazine. Note that in this case the least-squares fitted line gives f / fc′′ as its intercept and sf / fc′′as its slope so that s is given as the ratio of the slope divided by the intercept. A similar study was made9 on the n → π* absorption bands of acetone and cyclopentanone, giving the results s = 0.88 and f = 1.8×10-4 for acetone and s=0.72 and f = 2.2×10-4 for cyclopentanone.

A similar generalization was also made by Buckingham.10 He followed Kirkwood’s in deriving the electric moment of a dielectric specimen produced by a fixed mole-

682	Tai-ichi Shibuya

Figure 11.2.2. Plots of y = nf ′′ / f c′′ vs. x = n2 - 1 for the π → π * absorption bands of β-carotene (crosses) and the n → π * absorption bands of pyrazine (solid circles). [After reference 6]

cule in its interior and Scholte’s extension12 of the cavity field and the reaction field in the Onsager-BØttcher theory13,14 to an ellipsoidal cavity. Buckingham’s formula involves the polarizability of the solute molecule and appears quite different from Eq. [11.2.2]. It was shown6 that the Buckingham formula reduces to Eq. [11.2.2].

11.2.3DIELECTRIC SOLVENT EFFECT ON THE RADIATIVE RATE CONSTANT

The radiative rate constant is related to the absorption intensity of the transition from the ground state to the excited state under consideration. The application of Eq. [11.2.2] leads15 to

k r′′ / k r = n[s(n 2 − 1) + 1]2

[11.2.4]

where:

k′′r	apparent radiative rate constant of the solute molecule measured in a solvent of the
	refractive index n
kr	radiative rate constant of the molecule in its isolated state

Note that the local-field correction factor n[s(n2 - 1) + 1]2 varies from n to n5 as s varies from 0 to 1. For 9,10-diphenylanthracene (DPA), the correction factor was given15 as n[(0.128)(n2 - 1) +1]2, which lies between n and n2. This agrees with the observed data16 of fluorescence lifetimes of DPA in various solvents.

REFERENCES

1 N. Q. Chako, J. Chem. Phys., 2, 644 (1934).

2G. Kortòm, Z. Phys. Chem., B33, 243 (1936).

3(a) V. Henri and L. W. Pickett, J. Chem. Phys., 7, 439 (1939); (b) L. W. Pickett, E. Paddock, and E. Sackter, J. Am. Chem. Soc., 63, 1073 (1941).

4 L. E. Jacobs and J. R. Platt, J. Chem. Phys., 16, 1137 (1948).

5 S. R. Polo and M. K. Wilson, J. Chem. Phys., 23, 2376 (1955).

6T. Shibuya, J. Chem. Phys., 78, 5176 (1983).

7 C. Kittel, Introduction to Solid State Physics, 4th Ed., Wiley, New York, 1971, Chap. 13. 8 A. B. Myers and R. R. Birge, J. Chem. Phys., 73, 5314 (1980).

9T. Shibuya, Bull. Chem. Soc. Jpn. , 57, 2991 (1984).

10A. D. Buckingham, Proc. Roy. Soc. (London), A248, 169 (1958); A255, 32 (1960).

11J. G. Kirkwood, J. Chem. Phys., 7, 911 (1939).

12T. G. Scholte, Physica (Utrecht), 15, 437 (1949).

13L. Onsager, J. Am. Chem. Soc., 58, 1486 (1936).

14C. J. F. BØttcher, (a) Physica (Utrecht), 9, 937, 945 (1942); (b) Theory of Electric Polarization, Elsevier, New York, 1952; 2nd Ed., 1973, Vol. I.

15T. Shibuya, Chem. Phys. Lett., 103, 46 (1983).

16R. A. Lampert, S. R. Meech, J. Metcalfe, D. Phillips, A. P. Schaap, Chem. Phys. Lett., 94, 137 (1983).

12

Other Properties of Solvents,

Solutions, and Products

Obtained from Solutions

12.1RHEOLOGICAL PROPERTIES, AGGREGATION, PERMEABILITY, MOLECULAR STRUCTURE, CRYSTALLINITY, AND OTHER PROPERTIES AFFECTED BY SOLVENTS

George Wypych

ChemTec Laboratories, Inc., Toronto, Canada

12.1.1 RHEOLOGICAL PROPERTIES

The modification of rheological properties is one of the main reasons for adding solvents to various formulations. Rheology is also a separate complex subject which requires an in-depth understanding that can only be accomplished by consulting specialized sources such as monographic books on rheology fundamentals.1-3 Rheology is such a vast subject that the following discussion will only outline some of the important effects of solvents.

When considering the viscosity of solvent mixtures, solvents can be divided into two groups: interacting and non-interacting solvents. The viscosity of a mixture of non-interacting solvents can be predicted with good approximation by a simple additive rule rule:

i = n
logη = ∑ φi logηi	[12.1.1]

i =1

where:

ηviscosity of solvent mixture

iiteration subscript for mixture components (i = 1, 2, 3, ..., n)

φfraction of component i

ηi	viscosity of component i.

Interacting solvents contain either strong polar solvents or solvents which have the ability to form hydrogen bonds or influence each other on the basis of acid-base interaction. Solvent mixtures are complicated because of the changes in interaction that occurs with changes in the concentration of the components. Some general relationships describe vis-

684	George Wypych

cosity of such mixtures but none is sufficiently universal to replace measurement. Further details on solvent mixtures are included in Chapter 9.

The addition of solute(s) further complicates rheology because in such mixtures solvents may not only interact among themselves but also with the solute(s). There are also interactions between solutes and the effect of ionized species with and without solvent participation. Only very dilute solutions of low molecular weight substances exhibit Newtonian viscosity. In these solutions, viscosity is a constant, proportionality factor of shear rate and shear stress. The viscosity of these solutions is usually well described by the classical, Einstein’s equation:

η = ηs (1+ 2.5φ)

[12.1.2]

where:

ηs	solvent viscosity

φvolume fraction of spheres (e.g. suspended filler) or polymer fraction

If φis expressed in solute mass concentration, the following relationship is used:

	φ =	NVc	[12.1.3]
		M

where:

NAvogadro’s number

Vmolecular volume of solute ((4/3)πR3) with R - radius

csolute mass concentration

Mmolecular weight

Combination of equations [12.1.2] and [12.1.3] gives:

	η − ηs	=	2.5NV	[12.1.4]
	ηs c		M

The results of studies of polymer solutions are most frequently expressed in terms of intrinsic, specific, and relative viscosities and radius of gyration; the mathematical meaning of these and the relationships between them are given below:

η − ηs

[η] = lim

[12.1.5]

ηs c

c→0

ηsp

= [η] + k1 [η] 2 c +

[12.1.6]

c

lnηr

[

]

k ′

[

2 c

+

[12.1.7]

c

= η −

1

η]

ηr = ηsp

+ 1=

η

[12.1.8]

ηs

12.1 Rheological properties, aggregation, permeability								685
	1000					1.2
						1
Viscosity,Pa s	100
						0.8
						viscosity,Log cP
						0.6
	10
						0.4
	1	0.1	1	10	100	0.2	0	0.2
	0.001 0.01					-1 -0.8 -0.6 -0.4 -0.2
		Shear rate, s-1				Log concentration, g dl-1
Figure 12.1.1. Viscosity vs. shear rate for 10% solution						Figure 12.1.2. Viscosity of polyphenylene solution in
of polyisobutylene in pristane. [Data from						pyrilidinone. [Data from F. Motamedi, M Isomaki,
C R Schultheisz, G B McKenna, Antec ‘99, SPE, New						M S Trimmer, Antec ‘98, SPE, Atlanta, 1998, p. 1772.]
York, 1999, p 1125.]
where:
	[η]	intrinsic viscosity
	ηsp	specific viscosity
	ηr	relative viscosity
	k1	coefficient of direct interactions between pairs of molecules
	k1′	coefficient of indirect (hydrodynamic) interactions between pairs of molecules
	In Θ solvents, the radius of gyration of unperturbed Gaussian chain enters the follow-

ing relationship:

Φ R 3

[η]0 = 0 g, 0 [12.1.9]

M

where:

Φ0	coefficient of intramolecular hydrodynamic interactions = 3.16±0.5×1024
Rg,0	radius of gyration of unperturbed Gaussian chain

In good solvents, the expansion of chains causes an increase of viscosity as described by the following equation:

[η] =	Φ0	α η3Rg3, 0	[12.1.10]

M

where:

αη	= [ η]1/3 / [ η]1/0 3 is and effective chain expansion factor.

Existing theories are far from being universal and precise in prediction of experimental data. A more complex treatment of measurement data is needed to obtain characteristics of these “rheological” liquids.

Figure 12.1.1 shows that the viscosity of a solution depends on shear rate. These data comes from the development of a standard for instrument calibration by NIST to improve

686										George Wypych
	109						500
	108						400
Viscosity, Poise	107					-1
	106					Shear rate, s	300
	105						200
	104
							100
	1000
	100						0
	1000	104	105	106	107		0	5	10	15	20

Molecular weight, Daltons

Figure 12.1.3. Viscosity of 40% polystyrene in di-2-ethyl hexyl phthalate. [Data from G D J Phillies, Macromolecules, 28, No.24, 8198-208 (1995).]

-6

Molecular weight x 10

Figure 12.1.4. Shear rate of polystyrene in DOP vs. molecular weight. [Data from M Ponitsch, T Hollfelder,

J Springer, Polym. Bull., 40, No.2-3, 345-52 (1998).]

Viscosity, mPa s

	1.2						1.4
			DXN
	1.1						1.2
	1					CCl	1
						4
	0.9
						η rel	0.8
	0.8		DCE
						log
	0.7						0.6				associated
							0.4				unassociated
	0.6
			CHCl
					TOL
	0.5	THF	3				0.2
			DCM
	0.4						0
							0	0.5	1	1.5	2	2.5	3
	0	10	20	30	40	50
		PMMA concentration, %						Concentration, g dl-1

Figure 12.1.5. Viscosity of PMMA solutions in different solvents vs. PMMA concentration. Basic solvents: tetrahydrofuran, THF, and dioxane, DXN; neutral: toluene, TOL and CCl4; acidic: 1,2-dichloroethane, DCE, CHCl3, and dichloromethane, DCM. [Adapted, by permission, from M L Abel, M M Chehimi, Synthetic Metals, 66, No.3, 225-33 (1994).]

Figure 12.1.6. Relative viscosity of block copolymers with and and without segments capable of forming complexes vs. concentration. [Data from I C De Witte, B G Bogdanov, E J Goethals, Macromol. Symp., 118, 237-46 (1997).]

the accuracy of measurements by application of nonlinear liquid standards.4 Figure 12.1.2 shows the effect of polymer concentration on the viscosity of a solution of

polyphenylene in N-methyl pyrilidinone.5 Two regimes are clearly visible. The regimes are divided by a critical concentration above which viscosity increases more rapidly due to the interaction of chains leading to aggregate formation. These two sets of data show that there

12.1 Rheological properties, aggregation, permeability

687

	80							400
	70				MEK			350					PEO/PAA
					toluene								PEO
-1
								300					PAA
g	60
3							Viscosity, Poise
Intrinsic viscosity, cm	50							250
								200
	40							150
	30							100
	20							50
	10	20	40	60	80	100		0	2	4	6	8	10	12	14
	0							0
			PVAc, wt%									pH

Figure 12.1.7. Intrinsic viscosity of PS/PVAc mixtures in methyl-ethyl-ketone, MEK, and toluene. [Data from H Raval, S Devi, Angew. Makromol. Chem., 227, 27-34 (1995).]

Figure 12.1.8. Viscosity of poly(ethylene oxide), PEO, poly(acrylic acid), PAA, and their 1:1 mixture in aqueous solution vs. pH. [Adapted, by permission, from I C De Witte, B G Bogdanov, E J Goethals, Macromol. Symp., 118, 237-46 (1997).

are considerable departures from the simple predictions of the above equations because, based on them, viscosity should be a simple function of molecular weight. Figure 12.1.3 shows, in addition, that the relationship between viscosity of the solution and molecular weight is nonlinear.6 Also, the critical shear rate, at which aggregates are formed, is a nonlinear function of molecular weight (Figure 12.1.4).7

These departures from simple relationships are representative of simple solutions. The relationships for viscosities of solution become even more complex if stronger interactions are included, such as the presence of different solvents, the presence of interacting groups within polymer, combinations of polymers, or the presence of electrostatic interactions between ionized structures within the same or different chains. Figure 12.1.5 gives one example of complex behavior of a polymer in solution. The viscosity of PMMA dissolved in different solvents depends on concentration but there is not one consistent relationship (Figure 12.1.5). Instead, three separate relationships exist each for basic, neutral, and acid solvents, respectively. This shows that solvent acid-base properties have a very strong influence on viscosity.

Figure 12.1.6 shows two different behaviors for unassociated and associated block copolymers. The first type has a linear relationship between viscosity and concentration whereas with the second there is a rapid increase in viscosity as concentration increases. This is the best described as a power law function.8 Two polymers in combination have different reactions when dissolved in different solvents (Figure 12.1.7). In MEK, intrinsic viscosity increases as polymer concentration increases. In toluene, intrinsic viscosity decreases as polymer concentration increases.9 The polymer-solvent interaction term for MEK is very small (0.13) indicating a stable compatible system. The interaction term for toluene is much larger (0.58) which indicates a decreased compatibility of polymers in toluene and lowers viscosity of the mixture. Figure 12.1.8 explicitly shows that the behavior of

688								George Wypych
	106					104
			20% solvent					4 wt% CO
								2
			no solvent					no CO
Poise								2
	105				Pa s
Viscosity,					Viscosity,	1000
	104	10	100	1000		100	100	1000
	1					10
		Shear rate, s-1					Shear rate, s-1

Figure 12.1.9. Apparent melt viscosity of original PET and PET containing 20% 1-methyl naphthalene vs. shear rate. [Adapted, by permission, from S Tate,

S Chiba, K Tani, Polymer, 37, No.19, 4421-4 (1996).]

Figure 12.1.10. Viscosity behavior of PS with and without CO2. [Data from M Lee, C Tzoganikis,

C B Park, Antec ‘99, SPE, New York, 1999, p 2806.]

individual polymers does not necessarily have a bearing on the viscosity of their solutions. Both poly(ethylene oxide) and poly(propylene oxide) are not affected by solution pH but, when used in combination, they become sensitive to solution pH. A rapid increase of viscosity at a lower pH is ascribed to intermolecular complex formation. This behavior can be used for thickening of formulations.8

Figures 12.1.9 and 12.1.10 show one potential application in which a small quantity of solvents can be used to lower melt viscosity during polymer processing. Figure 12.1.9 shows that not only can melt viscosity be reduced but also that the viscosity is almost independent of shear rate.10 In environmentally friendly process supercritical fluids can be used to reduce melt viscosity.

The above data illustrate that the real behavior of solutions is much more complex than it is intuitively predicted based on simple models and relationships. The proper selection of solvent can be used to tailor the properties of formulation to the processing and application needs. Solution viscosity can be either increased or decreased to meet process technology requirements or to give the desired material properties.

REFERENCES

1A Ya Malkin, Rheology Fundamentals, ChemTec Publishing, Toronto, 1994.

2Ch W Macosko, Rheology. Principles, Measurements, and Applications, VCH Publishers, New York, 1994.

3 R I Tanner, K. Walters, Rheology: an Historical Perspective, Elsevier, Amsterdam, 1998.

4C R Schultheisz, G B McKenna, Antec ‘99, SPE, New York, 1999, p 1125.

5 F Motamedi, M Isomaki, M S Trimmer, Antec ‘98, SPE, Atlanta, 1998, p. 1772.

6G D J Phillies, Macromolecules, 28, No.24, 8198-208 (1995).

7M Ponitsch, T Hollfelder, J Springer, Polym. Bull., 40, No.2-3, 345-52 (1998).

8I C De Witte, B G Bogdanov, E J Goethals, Macromol. Symp., 118, 237-46 (1997).

9H Raval, S Devi, Angew. Makromol. Chem., 227, 27-34 (1995).

12.1 Rheological properties, aggregation, permeability

689

10S Tate, S Chiba, K Tani, Polymer, 37, No.19, 4421-4 (1996).

11M L Abel, M M Chehimi, Synthetic Metals, 66, No.3, 225-33 (1994).

12M Lee, C Tzoganikis, C B Park, Antec ‘99, SPE, New York, 1999, p 2806.

12.1.2 AGGREGATION

The development of materials with an engineered morphological structure, such as selective membranes and nanostructures, employs principles of aggregation in these interesting technical solutions. Here, we consider some basic principles of aggregation, methods of studies, and outcomes. The discipline is relatively new therefore for the most part, only exploratory findings are available now. The theoretical understanding is still to be developed and this development is essential for the control of industrial processes and development of new materials.

Methods of study and data interpretation still require further work and refinement. Several experimental techniques are used, including: microscopy (TEM, SEM);1,2 dynamic light scattering3-6 using laser sources, goniometers, and digital correlators; spectroscopic methods (UV, CD, fluorescence);7,8 fractionation; solubility and viscosity measurements;9 and acid-base interaction.10

Dynamic light scattering is the most popular method. Results are usually expressed by the radius of gyration, Rg, the second viral coefficient, A2, the association number, p, and the number of arms, f, for starlike micelles.

Zimm’s plot and equation permits Rg and A2 to be estimated:

KC

1

16

2 Rg2

2

+ 2A2C +

=

1+

π

sin

θ +

[12.1.11]

Rθ

λ20

Mw

3

where:

Koptical constant

Cpolymer concentration

Rθ		Rayleigh ratio for the solution
Mw		weight average molecular weight

0wavelength of light

θscattering angle.λ

The following equations are used to calculate p and f:

		p =	Magg			; M		= KI		/ C
			M1				agg		q→0
		Rg			= f (1−v )/ 2
		Rgarm
	where:
		Magg					mass of aggregates
		M1					mass of free copolymers

Iscattered intensity

Rgarm

radius of gyration of linear polymer

vexcluded volume exponent.

[12.1.12]

[12.1.13]