sample was then released and cooled. In terms of a, to what extent willthe sample remain stretched?

18. You are shipwrecked on a desert island. You find some bushes that havea sticky sap. You also find the island has all kinds of minerals. How do youget off the island?

APPENDIX 9.1 GELATIN AS A PHYSICALLY CROSS-LINKED

ELASTOMER†

Introduction

Ordinary gelatin is made from the skins of animals by a partial hydrolysis oftheir collagen, an important type of protein (A1,A2). At home, a crude typeof gelatin can be prepared from the broth of cooked meats and fowl; this mate-rial also frequently gels on cooling.

When dissolved in hot water, the gelatin protein has a random coil type ofconformation. On cooling, a conformational change takes place to a partialhelical arrangement. At the same time, intermolecular hydrogen bonds form,probably involving the N—H linkage. On long standing, such gels may alsocrystallize locally. The bonds that form in gelatin are known not to be perma-nent, but rather they relax in the time frame of 103 to 106 s (A3–A5). Theamount of bonding also decreases as the temperature is raised. The purposeof this appendix is to demonstrate the counting of these bonds via modulusmeasurements.

Theory

By observing the depth of indentation of a sphere into the surface of gelatin,“indentation” modulus is easily determined. The indentation modulus yieldsits close relative,Young’s modulus.The cross-link density and thus the numberof hydrogen bonds (simple physical cross-links) are readily determined bytreating the gelatin as a hydrogen-bonded elastomer.

Young’s modulus may be determined by indentation using the Hertz (A6)equation:

(A9.1.1)

where F represents the force of sphere against the gelatin surface = mg

(dynes), h represents the depth of indentation of sphere (cm), r is the radiusof sphere (cm), g represents the gravity constant, and v is Poisson’s ratio.

Ev F

h r=

-( )3 1

4

2

3 2 1 2

APPENDIX 9.1 GELATIN AS A PHYSICALLY CROSS-LINKED ELASTOMER 497

†Reproduced in part from the G. V. Henderson, D. O. Cambell, V. Kuzmicz, and L. H. Sperling,

J. Chem. Ed., 62, 269 (1985).

The ball indentation experiment is the scientific analogue of pressing on anobject with one’s thumb to determine hardness. The less the indentation, thehigher the modulus.

Young’s modulus is related to the cross-link density through rubber elas-ticity theory; see equation (9.36):

(A9.1.2)

Assuming a tetrafunctional cross-linking mode (four chain segments ema-nating from the locus of the hydrogen bond):

(A9.1.3)

where n represents the number of active chain segments in network and m isthe cross-link density (moles of cross-links per unit volume). For this experi-ment, the gelatin was at 278.0 K, the temperature of the refrigerator employed.

Experimental

Time: About 30 minutes, the gelatin prepared previously.Principles Illustrated:

1. Helix formation and physical cross-linking in gelatin.

2. Rubber elasticity in elastomers.

3. Physical behavior of proteins.

Equipment and Supplies:

Five 150 ¥ 75 mm Pyrex® crystallizing dishes or soup dishes

Five 2-cup packets of flavored Jello® brand gelatin (8 g protein per packet)

Eighteen 2-cup packets of unflavored Knox® brand gelatin (6 g protein perpacket)

One metric ruler

One steel bearing (1.5-in. diameter and 0.226 kg—or any similar sphericalobject)

One lab bench

One knife

Five different concentrations (see Table A9.1.1) of gelatin were prepared,each in 600 ml of water, and allowed to set overnight in a refrigerator at 5.0°C.Then indentation measurements were made by placing the steel bearing in thecenter of the gelatin samples and measuring the depth of indentation, h (seeFigure A9.1.1).As it is difficult to see through the gelatin to observe this depth,

E RT= 6m

E nRT= 3

498 CROSS-LINKED POLYMERS AND RUBBER ELASTICITY

it is desirable to measure the height of the bearing from the level surface ofthe gelatin and subtract this quantity from the diameter of the bearing (FigureA9.1.1).

The measured depth of indentation, the radius of the bearing, and the forcedue to the bearing are algebraically substituted into equation (A9.1.1). Thisvalue of Young’s modulus is substituted into equation (A9.1.3) to yield hydro-gen bond cross-link density.

Results

A plot of E as a function of gelatin concentration (Figure A9.1.2) demonstratesa linear increase in Young’s modulus at low concentrations. The slight upwardcurvature at high concentrations is caused by the increasing efficiency of thenetwork. However, the line should go through the origin.

Physical cross-link concentrations were determined using equation(A9.1.3), and the results are shown in Table A9.1.2. Assuming a molecularweight of about 65,000 g/mol for the gelatin, there is about 1.2 physical bondsper molecule (see Table A9.1.2).

Using gelatin as a model cross-linked elastomer, its rubber elasticity canalso be demonstrated by a simple stretching experiment. Thin slices of the

APPENDIX 9.1 GELATIN AS A PHYSICALLY CROSS-LINKED ELASTOMER 499

Table A9.1.1 Gelatin concentrations

Dish 1 2 3 4 5

Concentrationa 3.0 2.0 1.0 0.75 0.50Jellob 1 1 1 1 1Gelatinc 8 5 2 1.25 0.5

a Concentration = number of times the normal gelatin concentration (each dish contains 600 ml

of water).b Jello = number of 2-cup packets of Jello® brand black raspberry flavored gelatin.c Gelatin = number of 2-cup packets of Knox® brand unflavored gelatin.

Figure A9.1.1 Schematic of experiment, measuring the indentations of the heavy ball in the

gelatin.

more concentrated gelatin samples were cut and stretched by hand. On releasefrom stretches up to about 50%, the sample snaps back, illustrating the rub-berlike elasticity of these materials. At greater elongation the sample breaks,however. The material is weak because the gelatin protein chains are muchdiluted with water.

Discussion

For rubbery materials,Young’s modulus is related to the number of cross-linksin the system. In this case the cross-links are of a physical nature, caused byhydrogen bonding. Measurement of the modulus via ball indentation tech-niques allows a rapid, inexpensive method of counting these bonds. TableA9.1.2 shows that the number of these bonds is of the order of 10-7 mol/cm3.The number of these bonds also was shown to increase linearly with concen-tration, except at the highest concentrations.

500 CROSS-LINKED POLYMERS AND RUBBER ELASTICITY

Figure A9.1.2 Modulus of gelatin samples versus concentration. By comparison, a rubber

band has a Young’s modulus of about 1 ¥ 106 pascals (1 pascal = 10 dynes/cm2).

Table A9.1.2 Gelatin indentations yield bond numbers

Concentration h (cm) m (mol/cm3) N

0.5 1.50 3.5 ¥ 10-7 1.200.75 1.30 4.4 ¥ 10-7 1.001.0 1.20 5.0 ¥ 10-7 1.002.0 0.80 9.1 ¥ 10-7 0.863.0 0.40 2.6 ¥ 10-6 1.70

Key: h, indentation measured at gelatin temperature of 5°C; Cx, number of hydrogen bonds;

N, number of bonds per molecule.

Table A9.1.2 also demonstrates that at each gelatin concentration, thenumber of bonds per gelatin molecule is relatively constant. This number, ofcourse, is the number of bonds taking part in three-dimensional network for-mation. Not all the gelatin chains are bound in a true tetrafunctionally cross-linked network. Many dangling chain ends exist at these low concentrations,and the network must be very imperfect.

The gelation molecule is basically composed of short a-helical segments inthe form of a triple helix with numerous intramolecular bonds at room tem-perature; see Section 9.13. The a-helical segments are interrupted by prolineand hydroxy proline functional groups. These groups disrupt the helical struc-ture, yielding intervening portions of chain that behave like random coils, andwhich may be relatively free to develop intermolecular bonds. The subject hasbeen reviewed by Djabourov (A7) and Mel’nichenko et al. (A8).

In this experiment the concentration of sugar was kept constant so as tominimize its effect on the modulus. In concluding, it must be pointed out thatif sanitary measures are maintained, the final product may be eaten at the endof the experiment. If gelation five times normal or higher is included in thestudy, the student should be prepared for his or her jaws springing open afterbiting down!

REFERENCES

A1. A. Veis, Macromolecular Chemistry of Gelatin, Academic Press, Orlando, 1964.

A2. E. M. Marks, in Encyclopedia of Chemical Technology, Kirk-Othmer, Interscience,New York, 1966, Vol. 10, p. 499.

A3. J. L. Laurent, P. A. Janmey, and J. D. Ferry, J. Rheol., 24, 87 (1980).

A4. M. Miller, J. D. Ferry, F. W. Schremp, and J. E. Eldridge, J. Phys. Colloid Chem., 55,1387 (1951).

A5. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley, New York, 1980, pp.529–539.

A6. L. H. Sperling, Interpenetrating Polymer Networks and Related Materials, PlenumPress, New York, 1981, p. 177.

A7. M. Djabourov, Contemp. Phys., 29(3), 273 (1988).

A8. Yu. Mel’nichenko, Yu. P. Gomza, V. V. Shilov, and S. I. Osipov, Polym. Intern. (Brit.

Polym. J.), 25(3), 153 (1991).

APPENDIX 9.2 ELASTIC BEHAVIOR OF A RUBBER BAND†

Stretching a rubber band makes a good demonstration of the stress–strainrelationships of cross-linked elastomers.The time required is about 30 minutes.

APPENDIX 9.2 ELASTIC BEHAVIOR OF A RUBBER BAND 501

†Reproduced in part from A. J. Etzel, S. J. Goldstein, H. J. Panabaker, D. G. Fradkin, and L. H.

Sperling, J. Chem. Ed., 63, 731 (1986).