Some Aspects of Rubberlike Elasticity Useful in Teaching Basic

In the Classroom

Some Aspects of Rubberlike Elasticity Useful in Teaching Basic Concepts in Physical Chemistry1 J. E. Mark Department of Chemistry and the Polymer Research Center, The University of Cincinnati, Cincinnati, OH 45221-0172; [email protected]

There are several advantages to including polymer topics in both undergraduate and graduate physical chemistry courses. The first is to illustrate similarities and differences between systems with which the student is already familiar (such as ideal gases and metals) and more complex materials (in the present case polymeric materials having elastomeric properties). Another advantage is to provide particular examples illustrating the validity of very general results, such as the Carnot cycle analysis being independent of the working substance. In this way it is possible to use aspects of rubberlike elasticity to illustrate a number of very general molecular and thermodynamic concepts, making them more meaningful to the student. The items chosen here for this purpose are: (i) random flights of polymer chains and excluded volume effects, (ii) molecular origins of the elasticities of some types of materials, (iii) equations of state, (iv) thermodynamic non-ideality, (v) temperature changes during deformations and retractions, (vi) Carnot cycles and mechanochemistry, (vii) energy storage and hysteresis, and (viii) gel collapse. Random Paths and Excluded Volume Effects

The Random Walk Model Random-path models are important in rubberlike elasticity since the polymer chains in an elastomer are in random spatial arrangements prior to being stretched out in a mechanical deformation (1). The two-dimensional version of this problem is frequently described humorously in terms of a drunk starting a walk at a lamp post in a large parking lot. The question posed is how far the person gets after n steps, each of length l; this turns out to be proportional to the square root of the number of steps. Specifically, the “endto-end distance” r for the walk has been shown to be n1/2l, which is a much smaller distance than the value nl that would characterize a directed walk (along a straight line). A smallmolecule example in physical chemistry would be some atoms or molecules randomly skittering around on a catalyst surface. The Random Flight Model The three-dimensional statement of this problem is called the random flight, and has been used to describe a diffusing molecule, or a particle undergoing Brownian motions. The answer for the distance traveled is of the same form (1, 2), with the predicted end-to-end distance being proportional to the square root of the elapsed time, t1/2. Applying this model to the arrangements (spatial configurations) of a polymer chain is complicated by the fact that the mathematical path or trajectory characterizing the excursions of a molecule or particle are now replaced by a physical chain. The chain is made of atoms and bonds that have their usual spatial re-

quirements. This causes an expansion in the end-to-end dimensions of a polymer chain in solution. The more compact the spatial arrangement a chain tries to experience, the more likely it is to be excluded because it involves putting two atoms or segments into the same region in space (1, 3). This excluded volume effect is described in the following section.

The Nature of the Excluded Volume in Gases The simplest example of an excluded volume effect is that occurring in gases, which can range in structure from monoatomic to however many atoms the molecule can have without making it non-volatile under ambient conditions. This type of excluded volume is totally intermolecular, and thus can be suppressed by reducing the pressure to zero ( p → 0). As pointed out below, this makes the excluded volume b per molecule negligible relative to the now very large molar volume of the gas, that is, V – b ≅ V. There is of course a corresponding intermolecular excluded volume in the case of polymer molecules and, in the case of solutions, can be made negligible by going to infinite dilution (c → 0). This intermolecular part, however, is greatly augmented by an intramolecular part since polymer molecules are so long that different parts of the same molecule can interfere with one another. Such interfering segments are of course close together in space; they are, however, generally well separated along the chain trajectory and for this reason are also called “longrange interactions”. Obviously, there is no way of suppressing them because they are inherent to the long-chain structures of polymer molecules. What is done instead nicely parallels Boyle’s treatment of non-ideal gases (4, 5). In this treatment, the criterion for ideality, pV兾RT = 1, is evaluated as a function of the pressure of the gas. As p increases, this criterion deviates from ideality, with excluded volume effects increasing the pressure and intermolecular attractions decreasing it. This can be illustrated by the b and a corrections, respectively, in the van der Waals equation for non-ideal gases a    p + 2  (V − b ) = RT V

(1)

where V is the molar volume, and a and b are positive and specific for a particular gas. When this equation is rearranged to p =

RT a − 2 (V − b ) V

(2)

it becomes more obvious that a nonzero excluded volume (b ≠ 0) increases p while nonzero intermolecular attractions (a ≠ 0) decrease it. At high temperatures, the excluded volume predominates, so the temperature is lowered until the increase

JChemEd.chem.wisc.edu • Vol. 79 No. 12 December 2002 • Journal of Chemical Education

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In the Classroom

in pressure it causes is just offset by the pressure decrease from the intermolecular attractions. At this unique temperature, called the Boyle temperature for a particular gas, the excluded volume interactions are still there but are now nullified by being exactly offset by the intermolecular attractions. The real gas thus “masquerades” as an ideal gas, sometimes over a very large range in pressure.

used to establish structure-property relationships, for example, how putting para-phenylene groups into a repeat unit increases the stiffness of the corresponding chain (6, 9). The relevance here is that the network chains in an undeformed elastomer are in their unperturbed states, as described in the following section, and this is central to defining the reference state for any elastic deformation.

The Nature of the Excluded Volume in Polymers In the case of polymers, the criterion for ideality is now obtained from the van’t Hoff relationship

Excluded Volume Effects in the Bulk Amorphous State of a Polymer Although excluded volume interactions increase the dimensions of a polymer random coil in solution, they do not have this effect in the absence of solvent or diluent (i.e., in the bulk, amorphous state). This is because there is now no distinction between intramolecular and intermolecular effects; there is no advantage for the chain to expand to relieve intramolecular exclusions since this will only replace them with intermolecular ones (1). As a result the polymer chain does not expand, and thus exhibits its unperturbed dimensions in the undiluted state. This surprisingly simple prediction has been amply verified, most convincingly by elastic neutron scattering results on deuterated chains in a non-deuterated host (10). This establishes the reference state for undeformed polymer network chains, and explains the occurrence in molecular theories of rubberlike elasticity of the unperturbed mean-square end-to-end dimensions o.

πM = 1 cRT

(3)

where π is the osmotic pressure of a solution of the polymer. If a very good solvent is chosen for the polymer being characterized, the observed excluded volume effect is very large, with a correspondingly large size of the polymer random coil. One explanation for this is that in a very good solvent, the coil opens up to maximize the number of favorable interactions. An alternative is the statement that at high temperatures, the virial coefficients are large and positive, which increases the osmotic pressure. This drives more solvent molecules into the random coil, thereby expanding it. In any case, lowering the temperature makes the solvent poorer and the polymer segments prefer interacting with themselves rather than with the solvent. These enhanced attractions between polymer segments compress the random coil. The compressive effects are just sufficient to nullify the chain expansion effects from the excluded volume interactions at a unique temperate called the Θ temperature, also known as the Flory temperature or ideal temperature (1). At Θ, the excluded volume interaction is without effect and the chain acts as if it had zero cross-sectional area! Properties of chain molecules in this Θ state are said to be “unperturbed” (1, 6–8), and are

Figure 1. A typical elastomeric network structure is shown in the left portion of the figure, with the dots representing cross-links. The right portion shows a vertical deformation stretching the chains to lower entropy states.

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Molecular Origins of the Elasticities of Different Types of Materials

Rubberlike Elasticity and Network Structure Discussion of molecular features of rubberlike elasticity first requires its definition: large deformability with essentially complete recoverability. In order for a material to exhibit this type of elasticity, three molecular requirements must be met: (i) the material must consist of polymeric chains, (ii) the chains must have a high degree of flexibility and mobility, and (iii) the chains must be joined into a network structure (1, 3, 11–16). The first requirement arises from the fact that the molecules in a rubber or elastomeric material must be able to alter their arrangements and extensions in space dramatically in response to an imposed stress. Only long-chain molecules have the required large number of spatial arrangements of different extensions. The second characteristic required for rubberlike elasticity specifies that the different spatial arrangements be accessible—changes in these arrangements should not be hindered by constraints as might result from inherent rigidity of the chains, extensive chain crystallization, or the very high viscosity characteristic of the glassy state (1, 3, 10, 17–19). The last characteristic cited is required in order to obtain the elastomeric recoverability. It is obtained by joining together or cross-linking pairs of segments, approximately one out of a hundred, thereby preventing stretched polymer chains from irreversibly sliding by one another (15). This type of network structure is shown schematically in the left portion of Figure 1. Thermoelasticity Elasticity experiments involve the dependence of stress on strain, while thermoelasticity studies are concerned with

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In the Classroom

the effect of temperature on the stress–strain relationship. In fact, the earliest studies of rubberlike elasticity focused on the thermoelasticity of uncross-linked natural rubber, and were carried out by Gough in 1805 (1, 3, 17, 18, 20–22)! The discovery of vulcanization or curing of rubber into network structures by Goodyear and Hayward in 1839 was important in this regard since it permitted the preparation of samples which could be investigated with much greater reliability. More quantitative experiments of this type were carried out by Joule, in 1859. Another important experimental fact relevant to the development of these molecular ideas was that mechanical deformations of rubberlike materials generally occurred essentially at constant volume, so long as crystallization was not induced (1). (In this sense, the deformation of an elastomer and a gas are very different). Two simple thermoelastic experiments are shown schematically in Figure 2. If a stretched strip of elastomer is heated, it shrinks, as illustrated in the upper portion of the figure. This is, of course, the opposite of what one would expect from other materials, such as a stretched steel wire or quartz spring. This behavior results from the fact that stretching the elastomer reduces the entropy of the chains by increasing their end-to-end distances, as is shown schematically in the right portion of Figure 1. The decrease in entropy can be illustrated more quantitatively through the Gaussian distribution function W(r) for the end-to-end distance r, where W(r) ∝ exp(–r2), and the Boltzmann equation S = k ln , where the thermodynamic probability ∝ W(r). In the present application, the shrinkage occurs because adding thermal energy drives the system in the direction of increased entropy, which is the undeformed state. This type of elasticity is therefore entropically derived, rather than energetically based, as in the case of a stretched metal wire (10). These conclusions were reached only a few years after entropy was introduced as a concept in thermodynamics in general! The molecular interpretation of rubberlike elasticity being primarily entropic in origin had to await Staudinger’s

demonstration, in the 1920s, that polymers were covalentlybonded molecules, and not some type of association complex best studied by the colloid chemists (1, 22). In 1932, Kuhn used the observed constancy in volume to point out that the changes in entropy must therefore involve changes in orientation or shape of the network chains (10, 15, 17). The described thermoelastic behavior of the elastomer is analogous to the effect of heating a gas at constant pressure, as is illustrated in the lower portion of Figure 2. The gas expands (as is illustrated by the ideal gas law), again because of the increased driving force in the direction of maximum disorder or entropy (which now corresponds to infinite volume). Equations of State

Types of Deformation The work required to deform a material is given by W = 兰f dl, which for a gas simply involves f dl = ( f兾A)(Adl ) = pdV, where A is the area over which the force is imposed. The strain is then the reciprocal of the (molar) volume, = 1兾V. Since an elastomer is a solid, it can be deformed in a variety of ways. These deformations are characterized by three deformation ratios (relative lengths) along Cartesian coordinates: x, y, and z. In addition, since the deformation occurs at essentially constant volume, x y z = 1. Some examples are elongation α = αx

and

αy = α z =

1 α

(4)

1 α2

(5)

equi-biaxial extension (compression)

α = α x = αy

and

αz =

and shear α = α x , αy = 1, and α z =

W

W

d A = − p dV − S dT

(7)

 ∂A   ∂E   ∂S  p = − = − + T  ∂V  T  ∂V  T  ∂V  T

(8)

giving

W W

(6)

Thermodynamic Equations of State In the case of a gas, the analysis is based on the fundamental equation

heat elastomer shrinks

1 α

heat gas expands

Figure 2. Thermoelasticity experiments on a stretched elastomer and compressed gas, where W represents a weight. An increase in temperature decreases the length of the elastomer, and increases the volume of the gas.

Introduction of Maxwell’s equation (∂p兾∂T )V = (∂S兾∂V )T then gives  ∂p   ∂E  p = − + T   ∂V T  ∂T  V

(9)

For an ideal gas, (∂p兾∂T )V = R兾V = p兾T, making (∂E/∂V )T = 0, as expected.


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In the Classroom

In the case of an elastomer (23–25), eq 7 and 8 are generalized to

d A = − p dV − S dT + f dL

(10)

 ∂A   ∂E   ∂S  f = −  =   − T   ∂L  T, V  ∂L  T, V  ∂L  T, V

(11)

Introduction of Maxwell’s equation (∂fy∂T)L,V = {(∂Sy∂L)T,V then gives  ∂E   ∂f  f =   + T  ∂L  T,V  ∂T  L,V

(12)

which is analogous to eq 9. Adopting a molecular model to calculate ∆A then gives a molecular equation of state for rubberlike elasticity, specifically the force as a function of the number of network chains, temperature, and deformation, as is illustrated in the following section.

Molecular Equations of State The equation of state of an ideal gas is simply p = RT(1兾V), with deformation = 1兾V (where V is the molar volume). The simplest elastomer application assumes that the chain end-to-end distances can be described by a Gaussian distribution function and that there are no energy contributions (11, 26, 27). The earliest approach yielded f * = kT where the deformation or relative length is = L兾Li, and is the number of network chains. This relationship is an exact equivalent to the ideal gas law! It was soon recognized, however, that the condition of constant volume meant that the network was contracting in directions perpendicular to the stretching direction. This caused some chains to be compressed, which introduced an 2 term into the elastic equation of state: (13) f * = νkT α − α-2

(

)

This compression is shown schematically for the chains lying approximately horizontally in Figure 1. In the case of real gases, improvements between theory and experiment can be obtained by adopting a more sophisticated model, for example the van der Waals model described in eq 1. Analogous strategies are used in the case of elastomers. For example, introducing the unperturbed dimensions o of the network chains yields   f * = νkT  2  α − α-2  o 

(

)

 ∂E  lim   = 0 p → 0  ∂V  T

(15)

Correspondingly, an ideal elastomer has (∂E/∂L)T,V = 0, whereas a real elastomer has (∂E/∂L)T,V ≠ 0. The reasons for this non-ideality are more complex: (i) deformations generally require different conformational choices by the network chains to increase their end-to-end distances, and (ii) different conformations generally correspond to different energies (6–8). This is an intramolecular effect that cannot be removed, for example, by dilution effects paralleling the decrease in pressure of a gas (since this will remove only intermolecular effects). Thus there are no corresponding limiting conditions for achieving thermodynamic ideality in the case of elastomers. Thermal Effects during Deformations and Retractions

Temperature Changes, Refrigerators, and Heat Pumps These aspects of deformation are most simply illustrated by the compression of a gas to raise its temperature in a diesel engine. The corresponding stretching of elastomer (Gough–Joule experiment) increases its temperature analogously. This can best be explained in terms of the entropy change within what is temporarily an isolated system ∆S (Total) = ∆S (Deformation) + ∆S (Temp change) 0 (−) ( +)

(16)

(Isolated System))

The total entropy change has to be positive, and the contribution from the chain deformation is obviously negative. There must be an additional, positive contribution from a temperature change, and this change must be an increase. The stretching–retraction cycle of an elastomer can therefore be used as an alternative to the condensation–evaporation of a gas used to operate a refrigerator. There are also interesting parallels with cycles involving the demagnetization of crystals (4, 5). Carnot Cycles and Mechanochemistry

(14)

where represents the mean-square dimensions of the network chains in the undeformed state. Such chain models (10) can be made very realistic by adopting the correct structural information (bond lengths, bond angles, rotational state angles), and thermodynamic information (conformational preferences, including cooperativity) (6–8).

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Thermodynamic Non-Ideality As shown above, an ideal gas has (∂E/∂V )T = 0, whereas a real gas has (∂E/∂V )T ≠ 0. The reasons for this thermodynamic non-ideality in the case of the real gas is the presence of interparticle attractions (essential for liquefying it), and excluded volume effects (all intermolecular). Since both of these complications disappear as the pressure is decreased, there is a low pressure limit for achieving ideality for any gas,

Carnot Cycles It is also of interest to replace the usual Carnot cycle based on a gas undergoing: (i) isothermal expansion at an upper temperature T1, (ii) adiabatic expansion decreasing the temperature to a lower temperature T2, (iii) isothermal compression at T2, and (iv) adiabatic compression increasing the temperature back to T1. The efficiency, ε, is found to be 1 – T2兾T1, and is stated as being independent of working


In the Classroom

substance. This generality can be illustrated by using an elastomer as working substance, and replacing the expansions by retractions and the compressions by extensions (23–25).

Mechanochemistry The conversion of thermal or chemical energy into mechanical work has been of considerable interest (28, 29). There are a number of advantages in using an elastomer as a working substance in these applications (30–35). They include (i) a small adiabatic ∆T is useful for small differences T1 – T2, (ii) a broad range of temperatures is possible (no need for condensation or vaporization transitions), (iii) advantageous contractile transitions may be introduced using oriented fibers, (iv) no containment of gas or liquid is required, (v) stalling and starting torques are high, and finally (vi) construction is simple, with less material required (31). Energy Storage and Hysteresis

Energy Storage The swinging pendulum serves to illustrate the simplest energy storage concepts, as shown schematically in Figure 3. Point a corresponds to original potential (stored) energy, b is the point at which potential energy is converted into kinetic energy, and c is the point at which kinetic energy is converted back into potential energy. In this case, the mo-

Original Potential Energy a

Final Potential Energy c

b

Figure 3. Change in potential energy of a pendulum as it swings from its original position on the right, to a lower level corresponding to decreased potential energy on the left.

lecular origins of losses in stored energy arise from air resistance and friction at the pivot. The analogous case of a rubber ball bouncing off a surface is shown schematically in Figure 4 (36). Again, point a corresponds to original potential (stored) energy, and at point b potential energy is converted into kinetic energy. Now, however, kinetic energy is converted into elastic deformation energy upon impact with surface, at point c. At point d, elastic energy is released and converted into kinetic energy. Finally, at point e kinetic energy is converted back into potential energy. The amount of the original height recovered is a measure of the efficiency of the energy storage. In this case, in addition to minor effects from air resistance, the losses in stored energy are from viscous effects as chains change their spatial configurations from random to compressed and then back to random.

Hysteresis These hysteretic effects have parallels in small-molecule systems, for example in magnetization–demagnetization loops (4, 5). They are particularly important in elastomers since they correspond to wastage of energy, and overheating (heatbuild-up, with accompanying increases in thermal degradation). The amount of hysteresis can also be gauged from stress–strain isotherms, as shown schematically in Figure 5. The area below the upper elongation curve corresponds to the energy of deformation, and the area below the lower retraction curve corresponds to the energy recovered. The area between the two curves thus represents the energy wasted, in hysteresis. Some Relevant Biological Information Minimizing hysteresis is very important in the case of jumping insects, such as grasshoppers and fleas. In these cases, the elastomer is a protein called resilin, and the energy is stored by compressing a plug of this material (10). It is released when the insect wishes to jump, for example away from a predator, and the larger the fraction of the stored energy available the better. Release time is obviously also critically important, and is approximately 1 msec. Insects with more sluggish bioelastomers were presumably phased out by natural selection.

hysteretic loss

f*

a e

b

c

d 0.0 1.0

Figure 4. Change in potential energy of a rubber ball as it drops from its original position on the left to the impact point converting kinetic energy to elastically-stored energy, to bouncing up to a level corresponding to decreased potential energy.

α=

L Li

Figure 5. Stress-strain cycle of an elastomer at constant temperature, illustrating the occurrence of hysteresis


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In the Classroom

Resilin also is important in flying insects, such as dragonflies, where plugs under the wings smooth out the flapping by alternating being compressed and expanding. Large hysteretic effects would be bad not only because of the inefficiencies involved, but because of possible overheating of the dragonfly. The cross-linking in these bioelastomers is carefully controlled by nature. The number and spacing of the cross-links is fixed by the ribosome controlled α-amino acid sequence, since the cross-linking occurs only through the lysines (using a copper-activated enzyme call lysyl oxidase; ref 10). Particularly intriguing is that the lysine sites are preceded and succeeded by alanines (which may be in α-helical conformations). Placing these potential cross-linking sites at the ends of two stiff sequences may help control their spatial environment, for example their entangling with other protein repeat units. Trying to parallel the control nature exerts in cross-linking bioelastomers, for example by end-linking reactions, is an example of biomimicry. Other relevant examples are the use of (i) irregular copolymer sequences to suppress crystallinity, (ii) small side groups to enhance flexibility and mobility, (iii) nonpolar side groups to reduce intermolecular interactions, and (iv) plasticizers to reduce brittleness (10).

v2

lightly swollen gel

highly swollen gel

T

V

gas

Gel Collapse A final phenomenon relevant here is gel collapse (37– 39). It involves the relatively abrupt deswelling of a swollen elastomer (a gel) with small changes in some variable, such as temperature, composition, pH, ionic strength, application of an electric field, or irradiation with light. An example is shown schematically in the upper portion of Figure 6. Here, v2 is the volume fraction of polymer present in the gel, and the discontinuous shrinkage (increase in v2) occurs upon decrease in temperature. The deswelling is not complete in that the network still contains substantial amounts of diluent, but the amount expelled is enough to give very substantial changes in the dimensions of the gel. Also, the process is reversible, in that the deswollen gel can be reswollen by restoring the variable to its earlier value. If all dimensions of the gel are sizable, a considerable amount of time may be required for this deswelling or syneresis to occur, since outward diffusion of the small molecules is required. The process is, of course, much more rapid in the case of a film or fiber, because of the much larger ratios of surface area to volume when one or more dimensions of the sample are very small. The possibility of having these changes occur relatively quickly has encouraged attempts to harness the accompanying mechanical motions in a variety of devices. Examples of potential applications include actuators, switches, drug delivery systems, and artificial muscles. There are parallels and differences in the case of the condensation of gases, as is illustrated by the p –V isotherm shown in the lower part of the figure. Here, increase in pressure causes a discontinuous decrease in volume to that of the liquid. Again, there are large changes in dimensions upon condensation of a gas to the much denser liquid phase, but the isotropic nature of the phases makes this much more limited with regard to possible mechanical applications. Conclusions

liquid

p Figure 6. The upper sketch portrays gel collapse, as evidenced by the abrupt increase in volume fraction of polymer in the gel; it shrinks when the temperature drops to a critical value. This syneresis and reswelling can be exploited by harnessing the mechanical motion in a variety of devices. The lower sketch shows the corresponding situation as the pressure of a gas is increased to a critical value that causes condensation to the liquid state.

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It is useful to give one illustration, however, of how such “bioinspiration” can lead one astray. All of the early work trying to mimic the flight of birds by designing aircraft with flapping wings turned out to be disastrous! The successful approaches involving propellers or jets were probably not inspired at all by analogies with biological systems. Circular motions and jets of fluids for locomotion are relatively rare in biology, and are used in aqueous fluids, rather than in air.

There are numerous opportunities to use elastomeric systems to illustrate some of the most important basic concepts in physical chemistry. Examples can demonstrate the wide range of applicability of these principles, make the subject more interesting, and underscore the relevance of these concepts to modern technologies. Acknowledgment It is a pleasure to acknowledge the financial support provided by the National Science Foundation through Grant DMR-0075198 (Polymers Program, Division of Materials Research).


In the Classroom

Note 1. This article is based on a symposium presentation upon receiving the “Paul J. Flory Polymer Education Award”, Division of Polymer Chemistry, Inc., San Francisco ACS Meeting, March 2000. Literature Cited

19. 20. 21.

1. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. 2. Rodriguez, F. J. Chem. Educ. 1976, 53, 92. 3. Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd ed.; Clarendon Press: Oxford, 1975. 4. Moore, W. J. Basic Physical Chemistry; Prentice Hall: Englewood Cliffs, NJ, 1983. 5. Atkins, P. W. Physical Chemistry, 4th ed.; Oxford University Press: Oxford, 1990. 6. Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969. 7. Mattice, W. L.; Suter, U. W. Conformational Theory of Large Molecules. The Rotational Isomeric State Model in Macromolecular Systems; Wiley: New York, 1994. 8. Rehahn, M.; Mattice, W. L.; Suter, U. W. Adv. Polym. Sci. 1997, 131/132, 1. 9. Wang, S.; Mark, J. E. Comput. Polym. Sci. 1993, 3, 33. 10. Erman, B.; Mark, J. E. Structures and Properties of Rubberlike Networks; Oxford University Press: New York, 1997. 11. Mark, J. E. J. Chem. Educ. 1981, 58, 898. 12. Kauffman, G. B.; Seymour, R. B. J. Chem. Educ. 1990, 67, 422. 13. Kauffman, G. B.; Seymour, R. B. J. Chem. Educ. 1991, 68, 217. 14. Seymour, R. B.; Kauffman, G. B. J. Chem. Educ. 1992, 69, 967. 15. Mark, J. E. In Applied Polymer Science—21st Century; Craver, C. D.; Carraher, C. E. Jr., Eds.; American Chemical Society: Washington, 2000. 16. Mark, J. E. J. Phys. Chem. B in press. 17. Mark, J. E.; Erman, B. Rubberlike Elasticity. A Molecular Primer; Wiley-Interscience: New York, 1988. 18. Mark, J. E. In Physical Properties of Polymers, 2nd ed.; Mark,

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J. E.; Eisenberg, A.; Graessley, W. W.; Mandelkern, L.; Samulski, E. T.; Koenig, J. L.; Wignall, G. D., Eds.; American Chemical Society: Washington, DC, 1993; p 3. Mark, J. E.; Erman, B. In Polymer Networks; Stepto, R. F. T., Ed.; Blackie Academic, Chapman & Hall: Glasgow, 1998. Gough, J. Proc. Lit. Philos. Soc., Manchester, 2nd Ser. 1805, 1, 288. Mason, P. Cauchu. The Weeping Wood; Australian Broadcasting Commission: Sydney, 1979. Morawetz, H. Polymers: The Origins and Growth of a Science; Wiley-Interscience: New York, 1985. Wall, F. T. Chemical Thermodynamics, 3rd ed.; Freeman: San Francisco, 1974. Callen, H. B. Thermodynamics and an Introduction to Thermostatistics; Wiley: New York, 2000. Tschoegl, N. W. Fundamentals of Equilibrium and Steady-State Thermodynamics; Elsevier: Amsterdam, 2000. Nash, L. K. J. Chem. Educ. 1979, 56, 363. Queslel, J. P.; Mark, J. E. J. Chem. Educ. 1987, 64, 491. Mandelkern, L. Crystallization of Polymers; McGraw Hill: New York, 1964. Mandelkern, L. An Introduction to Macromolecules, 2nd ed.; Springer-Verlag: New York, 1983. Strong, C. L. Sci. Am. 1971, 224 (4), 118. Farris, R. J. Polym. Eng. Sci. 1977, 17, 737. Mandelkern, L. J. Chem. Educ. 1978, 55, 177. Farris, R. J.; Lyon, R. E. Materials and Society 1984, 8 (2), 239–256. Farris, R. J.; Lyon, R. E.; Wang, D. X.; MacKnight, W. J. J. Appl. Polym. Sci. 1984, 29, 2857. Farris, R. J.; Lyon, R. E. Polymer 1987, 28, 1127. Kauffman, G. B.; Mason, S. W.; Seymour, R. B. J. Chem. Educ. 1990, 67, 198. Tanaka, T. In Polyelectrolyte Gels; Harland, R. S.; Prud’homme, R. K., Eds.; American Chemical Society: Washington, 1992; Vol. 480, p 1. Li, Y.; Tanaka, T. Annu. Rev. Mater. Sci. 1992, 22, 243. Grinberg, V. Y.; Dubovik, A. S.; Kuznetsov, D. V.; Grinberg, N. V.; Grosberg, A. Y.; Tanaka, T. Macromolecules 2000, 33, 8685.


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Some Aspects of Rubberlike Elasticity Useful in Teaching Basic

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