I I
F. Rodriguez School of Chernicol Engineering Cornell Universitv Ithoco, New York 14850
I
Demonstrating Rubber Elasticity
Rubber elasticity is a particularly satisfying topic for the teacher of polymer behavior because it combines the rigor and purity of statistical thermodynamics with the familiarity and homeliness of rubber hands and birthday party balloons. The interplay of thermodynamics with rubber goes hack to 1806. Flory ( I ) gives a fascinating history of the early work and the suhsequent developments which led eventually to a theory that incorporates the results of statistical mechanics. Treloar's book (2) is quite comprehensive in its treatment of ruhher elasticity although several briefer treatments (I, 3, 4) can he recommended also. In the present paper, demonstrations are suggested whereby the theoretical predictions can be illustrated to large groups either in a qualitative or a quantitative manner using the overhead projector. Of course, small groups can carry out the experiments without such optical magnification. Background Equations and Assumptions For crosslinked natural rubber and many other mhherlike materials, it can be shown that the retractive force on a sample (Fig. 1) is supported by changes in the arrangement of polymer chains in space (entropy changes) and not hy bending and stretching of bonds (internal energy changes). The spatial arrangement is altered only by rotation around successive single bonds in the polymer chain. In the ideal case, the rotations involve no energy harriers. The differential work d W done by moving a force f through a distance dr can he equated with the change in entropy d S a t constant temperature T. dW
=
fdr
=
-TdS
(1)
Next, Boltzmann's formulation that the difference in entropy S between two states is equal to a constant k times the difference in the logarithms of the probability in the two states is combined with a Gaussian distribution of chain dimensions. In the unstrained state an average chain has one end at coordinates 0,0,0, and the other end a t xl,yl,zl. In the strained state, the coordinates become xz = a,xl, etc. For a single chain then, the change in entropy AS' is given by AS'
= S:
-
S{
=
+
-(h/2)(ar2 a,'
+
a";
31 (2)
where k is the Boltzmann constant, the gas constant per molecule. Real rubber-like materials change very little in volume on being deformed so that the assumption of constant volume is reasonable. It can he expressed by the condition of Another important assumption is that the macroscopic deformation of the sample represented by a,, as,and a, is identical with the deformation of the average individual molecular chain. The total entropy change for the system A S will he AS' times the number of molecular chains in the system. If N is the chain density (moleJcm3) and the gas constant per mole, R, is used in place of k , the result is 764
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Journal of Chemical Education
Bioxiol stretching Figure 1. Deformation of rubber in rectangular coordinates
Uniaxial Stretching If the sample is elongated with force f in the x direction (Fig. 1) there will he a shrinkage in the y and z dimensions in order to keep the volume constant. In this case, A is defined as A
=
a, = L/Lo
(5)
Because as and a, are indistinguishable, eqn. (3) gives a = a = A-l"
(6)
Now using eqns. (I), (4), and (5) with dr = dL, it is possible to get The subscript indicates that the process takes place a t constant temperature and volume. Defining a as the tensile stress, the force per unit original cross-sectional area (woto), eqns. (5) and ( 6 ) are used to put the deformation in terms of A
+
(NRT/2)[a(AZ ~-9/ax],, Carrying out the indicated differentiation yields o = f/(u~& =
(8)
Demonstration An acrylic sheet a t least 5 in. thick is fitted with a post or hook for holding one end of a common ruhher hand. A pulley is mounted in line with the horizontal axis (Fig. 2). Scales for a (in arbitrary units) and A are laid out together with the transformation chart at the top. The chart is a
Figure 2. One end of a rubbe; band is held on a post at the intersection of the axes. The other end initially is at A = 1. As weights of 57.5 g each are added, the crosses with upward arrows mark the elongation. Crosses with downward arrows mark the elongation an removing the weights. The average elongation ( 0 ) is converted to (A - A-2) as illustrated for the
Figure 3. The pressure within a balloon is balanced by the stress in the skin.
third point.
device for converting h to (A - k 2 ) in view of the audience. The lines a t a constant angle guide the transformation because they are attenuated a t the corresponding values of (A - A-2). In the example (Fig. 21, a rubber hand (wo = 0.40 cm, to = 2 x 0.080 cm, Lo = 6.2 cm) was elongated by equal increments of 57.5 g each. The midpoint of elongation between loading and unloading curves gives a clearly non-linear trace. However, each point can be converted to (A - A-2) as illustrated for A = 1.4 which is transformed to (A - P 2 ) = 0.89. The transformed data inevitably show reasonable linearity up to X = 2.5 or so. In practice it is good to use a disposable transparent sheet over the acrylic sheet together with various colored marking crayons or felt-tip pens for marking the points. If an alligator clip is used a t the fixed end of the rubber band, the length of any hand can he adjusted to a constant value so that the same h scale can be reused. Biaxial Stretching In the situation where a, = a, = A, than a, = A-2 from the condition of eqn. (3). Corresponding to eqn. ( I ) , we have work done by equal forces in two directions so that dW = f,dL
f,dw
=
2fdL
(10)
where dL = dw, and f, = f, = f. Considering only the force in the x direction we have f/(&&t,,= (NRT/4)[a(W'
+ h-4$/dA]T,v
(U
a = f/(woto) = NRT(X (12) Stein (5) has used the inflation of a balloon as an example of biaxial stretching. When a round balloon is inflated, the skin thickness t corresponds to the 2 axis and the other two dimensions stretch equally so that
h = DID, where D is the diameter of the balloon and =
tltO
(13) (14)
T o relate the pressure inside the balloon P (above atmospheric pressure) to the diameter i t is necessary to halance the force due to pressure, PrD2/4 against the force supported by the stress in the skin in the x direction (Fig. 3). But wo = =DO approximately. Using this with eqns. (14) and (15) gives
Atomize1 bulb Overhead projector Manometer Figure 4. Pressure ( P ) on deflating the balloon is read from the manometer, Balloon diameter I D ) can be measured with a ruler an the biackboard or with atransparent scale on the projector table.
P
=
4NRT(t,/D,)(A-'
-
A')
(16)
When dP/dh = 0, P goes through a maximum. This occurs a t X = (7Yi6 = 1.38 (17) Demonstration A toy balloon (about l-in. diameter uninflated) is fastened to the end of a 0.25-in. copper tube so that its shadow is in focus on the overhead projection (Fig. 4). The tube is connected to valves, a manometer, and a rubber atomizer bulb. When air is pushed into the balloon by the bulb, the manometer indicates the pressure. For the common hallons used in this experiment, a manometer filled with trichloroethylene containing a red dye gives a convenient maximum reading of about 50 cm. Although one can measure pressure and diameter during inflation, the readings tend to drift rapidly. By inflating to about A = 3, rather stable readings are obtained on deflation. The shadow of the balloon is projected onto a blackboard (or sheet of paper). The diameter is recorded with chalk and a student reads the pressure which also is noted on the board. After each reading, the upper mirror of the projector is tilted slightly to give a new line for diameter and pressure. It becomes apparent from the raw data that the model is qualitatively correct. Actual calculations and plotting can be assigned as homework. Typical results are shown in Figure 5. A rapid, qualitative demonstration consists of letting the balloon deflate from h = 3 with a constant bleed of air and to focus student attention on the manometer. It will stay almost constant until X reaches about 1.5, then climb noticeably (usually several cm) before dropping rapidly. Volume 50, Number 77, November 7973
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765
10 Constrained hbricoted
No. of Wts
Figure 5. A toy balloon being deflated gave the experimental points (0) expressed here as pressure P divided by maximum pressure P,,, and diameter D divided by initial diameter DO. About 10 sec elapsed between SUeCeSSive measurements. The theoretical line is eqn. (16)
0
0
X,
I cm.
2
Figure 7. For the system of Figure 6, the elimination of buckling gives a Slope which is 25% higher than before. With each weight = 57.5 g, wo = 0.016 cm, lo = 4.4 am, and Lo = 9.0 cm, the constrained case gives G = 5.1 X l o e dyn/cmz.
U/
6.Constrained bu lubricated plates
Figure 6. Simple shear. A) The alligator clip (a) holds the strips of masking tape ( b ) which are attached to the rubber sheet. 8 ) To prevent buckling, plates are added above and below the rubber sheet. A solution of glycerol in water is applied to all sliding surfaces.
Simple Shear When a force is exerted on the top of a rectangular specimen in the plane of the top surface (Fig. I), the shear stress 7 is given by f/Lowo. The differential shear strain d y is given by the motion dx of the top plane divided by the thickness to dy
=
(l/tJdx
(18)
The nature of the deformation is such that the width w and the thickness t do not change. The work done is dW
=
fdx
=
ft,dy
=
-TdS
(19)
In considering the entropy in two states, we take into account that t h e y and z coordinates do not change, only x changes. Also, x z = 7x1. For a single chain AS'
=
-(k/2)(yZ - 1)
(20)
Using eqns. (4) and (19) with (20) we get (at constant temperature and volume) that
f
=
(NRTLowo/2)(a~2/a~)~
(21)
and, finally i
=
f/(L,w,)
=
NRTy
(22)
This allows us to identify NRT with the shear modulus G. It comes as something of a surprise to find that ideal rubber is Hookean (stress is directly proportional to strain) in shear even though it is non-Hookean in tension. '"Dental dam," amber, medium weight, semi-transparent gum ruhher, available from most scientific supply houses. 766
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Journal of Chemical Education
Demonstration A sheet of rubber1 about 0.007 in. thick is supported along two edges (Fig. 6). I t is helpful to fasten the rubber hy means of paper tape which has adhesive on both surfaces in addition to screws or clips. Narrow strips of masking tape give a good bond to rubber and allow the application of the shearing force. The pulley arrangement from the extension experiment can be used here on the overhead projector. Although the stress-strain behavior is linear (Fig. I ) , and thus needs no replotting some huckling of the sheet does occur. The buckling causes the experimentally measured modulus to be lower than the true value. In a more elaborate arrangement, glass or acrylic plastic sheets are placed above and below the ruhher sheets (Fig. 6 ) after the rubber surface has been lubricated with glycerin-water or soap solution. Surface tension usually is enough to keep the sheet from buckling up to y = 0.5. The measured modulus on such a restrained system then is comparable to that obtained from an experiment in tension (Fig. 7). Some Questions for the Students 1) What happens to pressure and diameter when a balloon is heated? 2) If a catapult is made from a ~ h b e hand, r how far will it propel a projectile of known weight? 3) If a material were Hookean, that is, o = k(a - 1). in elongation, what would the pressure-diameter relation be for a balloon? ~
~
Some Further Experiments 1)The effect of temperature on the length of a stressed ~ b b e r band. This can be done directly (6) or amplified by a lever-arm as illustrated by F. H. Window in the film "Physical Chemistry of Polymers" (7). 2) The effect of internal lubricants on the modulus of rubber bands (8). The theoretical background for swollen ruhher is described in references ( I ) through (4). 3) Pure shear. This can be carried out by expanding a balloon (preferablya sausage-shapedone) insidb a lubricated glass tube.
Laerature Cited (11 Flory, P. J., "Ptinciples of Polymer Chemistry," Carncll Univenity Press. Ithaca. N.Y..1953. ChapterXI. 12) Tmlosr. L. R. G.. "The PhysicnofRubberElanticity.'' Oxford, NewYork, ,958. (31 Mullins, L. and Thomas, A. G.. in "The Chemisfn. and Physics of nubber~like Substsnces." (Ed'ditor:Bateman. L.1. John Wiley & Sons, Inc., New York. 1963,
.--. ..
"-. h~" 7
(41 Stein, R. S., "The Optical and Mechanical Properties of High Poiymen," %search Rwmt High Polymer Series No. 14, Quartermaster%s. & Eng. Center, Pioneering ReroarchDiv.. March. 1960. (51 Stein, R.S.,J.CHF.M. EDUC. 35.203l19581. (61 Laswick, P.H.. J. CHEM.EDUC. 49,469119721. (71 "Physical Chemistry of Polymcm." 16 mm mund film, h i 1Telephone Laboratories, Murray Hill. N. J. (81 Rodriguez, F.. "Principle8 of Polymer Syotems," McCraw-Hill Book Co., New York, 1970.p. 511.
