UMR ChemLabs Thermodynamics of Rubber Elasticity PCh 5-99
Gary L. Bertrand, Professor of Chemistry
Thermodynamics in Two Dimensions - Surface Tension:1 Thermodynamic treatments generally consider only three-dimensional work (P, pressure - V, volume): (DWrev)3D = - PdV , and occasionally with work in one-dimension (t , tension - L, length), as in the stretching of a spring or a rubber band: (DWrev)1D = t dL . The deformation of a surface involves two-dimensional (g, surface tension - s , area) work: rev (DW )2D = gd s . The thermodynamic description of a system must include these contributions if the length or area of the system is subject to change. For changes in one dimension, dG = - SdT + VdP + t dL ; t = (¶G/¶L)T,P (1a) dA = - SdT - PdV + t dL ; t = (¶A/¶L)T,V (1b) The temperature dependence of the tension at constant length is related to the entropy: (¶t /¶T)P,L = - (¶S/¶L)T,P ; (¶t /¶T)V,L = - (¶S/¶L)T,V. (1c) The inflation of a balloon or a bubble involves both three-dimensional work and two- dimensional work. The free energy functions then become:
dG = - SdT + VdP + gds ; g = (¶G/¶s )T,P (2a) dA = - SdT - PdV + gds ; g = (¶A/¶s )T,V (2b) The entropy is related to the temperature derivative of surface tension,
(¶ g/¶T)P,s = - (¶S/¶s )T,P ; (¶g/¶T)V,s = - (¶S/¶s )T,V (2c) The surface tension is often considered as the Gibbs Free Energy per unit area of the surface (Gs), and its temperature derivative as the negative of the entropy per unit area (Ss). These are related to the enthalpy per unit area (Hs) s s s H = G + TS = g - T(¶g/¶T)P,s (3) One of the most fundamental relationships for surfaces can be derived by considering a bubble or a balloon suspended within a rigid, thermostatted box at constant temperature and total volume. The system has three parts: the material (air) inside the bubble or balloon, the material outside (air), and the surface. dAsystem = dAinside + dAoutside + dAsurface dAinside = -PinsidedVinside ; dAoutside = -PoutsidedVoutside ; dAsurface = g ds There is no interaction with the rest of the universe outside of the box, so there can be no work, and
-PinsidedVinside -PoutsidedVoutside + g ds = dAT £ 0 . However, since the volume of the system and the surface do not change, dVinside = -dVoutside and
-(Pinside - Poutside)dVinside + g ds £ 0 . The equilibrium condition is
1 DP = g(ds /dV) = (Pinside - Poutside) . (4) This is the Law of Young and LaPlace, which describes how the curvature of a deformable surface responds to a pressure differential across it. This relationship provides the operational definition of surface tension. The quantity (ds /dV) can be calculated for a sphere: surface area: s = 4pr2 ; d s = 8prdr volume: V = (4/3)pr3 ; dV = 4pr2dr ds /dV = 2/r . The curvature of a complex surface is described with two radii of curvature in mutually perpendicular planes:
d s /dV = 1/r1 + 1/r2 (r1 and r2 are identical for a spherical surface) , and DP = g(1/r1 + 1/r2) (general case) (5a) DP = 2 g/r (spherical case) . (5b) This relationship allows experimental determination of surface tension for soap bubbles, balloons or other polymer films, and even for the infinitesimally thin surface of a liquid.
Theory of Polymer Elasticity:2,3 Statistical thermodynamic treatment of a polymer network of crosslinked segments of randomly coiled chains leads to an equation for the entropy change on deformation: S(a X,a Y,a Z) - S0 = (Nk/2){ln (a X a Y a Z) -a X2 -a Y2 -a Z2 + 3} , (6) in which N is the number of chain segments between crosslinks (irrespective of the length of the segment), k is the Boltzmann constant, and a X,a Y,a Z are the deformations in the X-,Y-,Z- directions:
a X = LX/LX ,0 , a Y = LY/LY,0 , a Z = LZ/LZ ,0 (7) The sample may be deformed in one direction as in stretching a rubber band in the Z- direction, or in two directions as in simultaneously stretching a film in the X,Y- directions, without an appreciable change in volume,
a X a Y a Z = 1 (constant volume) . If the deformations are symmetric in the X,Y-directions, a X = a Y = (1/a Z)1/2 . (8) For stretching in the Z-direction. Eq (6) becomes S(a Z) - S0 = - (Nk/2)(a Z2 + 2/a Z + 3) (9a) and for stretching in the X,Y-directions (a XY = a X a Y =1/a Z) S(a XY) - S0 = - (Nk/2)(2 a XY + 1/a XY2 + 3) . (9b) These entropy relationships may be differentiated to obtain the temperature derivatives of tension and surface tension, as in Equations (1c,2c):
(¶t /¶T)P,L = - (¶S/¶L)T,P = - (1/LZ ,0)(¶S/¶a Z)T,P = (Nk/LZ ,0)(a Z - 1/a Z2) (10a) and 3 (¶g/¶T)P,s = - (¶S/¶s )T,P = - (1/A0)(¶S/¶a XY)T,P = (Nk/A0)(1 - 1/a XY ) (10B) in which LZ ,0 is the equilibrium (unstretched) length, and A0 is the equilibrium area. An ideal rubber is defined (similarly to an ideal gas) as one in which the energy (and enthalpy) depend only on temperature (Hs = 0). Referring to Eq (3), then
2 3 g ideal = T(¶g ideal/¶T)P,s = (NkT/A0)(1 - 1/aXY ) (11) To relate this equation to macroscopic properties, we define a quantity V* as the volume of a mole of chain segments. Then N = N0(VS/V*) ; k = R/N0 (N0 is Avogadro’s number) and VS is the actual volume of the rubber sample being stretched, and g ideal = (RT/V*)(VS/A0)(1 - 1/a XY3) . (12) The quantity (VS/A0) can be measured as the thickness of the surface. Measurement of surface tension as a function of the distortion of the surface then allows determination of V* as a physical characteristic of the polymer. Combined Relationships: Equation (12) shows that the surface tension of an ideal rubber surface will increase rapidly from initially zero to some limiting value. This type of behavior is generally observed, but there are complications in these observations. The rubber is not a pure crosslinked material, but contains fillers and elasticizing oils. These lead to the phenomenon of creep , in which the structure is permanently distorted. As aXY is increased above a value near 4, the surface tension begins to increase with an increased rate of permanent distortion, and the surface is prone to rupture at values above 6. In this experiment, a rubber balloon is inflated with simultaneous measurement of the volume and the pressure difference. By assuming that the balloon is spherical, the radius and area may be approximated from the volume: r = (3V/4p)1/3 ; A = 4pr2 = (36 p)1/3V2/3 = 4.84V2/3 allowing calculation of the surface tension and the area for each observation. The volume of the balloon at zero pressure difference is used to calculate the undistorted area (A0), allowing calculation of a XY = A/A0 for each observation. The validity of the mathematical form of Equation (12) may be tested with this data. The temperature dependence of the surface tension is determined to further test the validity of Equation (12). It is anticipated that the surface tension will vary linearly with temperature over a range of 20-30 °C. However, Equation (12) predicts that the surface tension will increase in direct proportion to the temperature. This prediction, which is somewhat counter to intuition, is easily tested. Finally, the thickness of the rubber is measured by weighing a sample and measuring its area. The volume is calculated from the mass using the density of natural rubber (0.970 g/cm3). The molar volume of the polymer segment between crosslinks (V*) can be expressed as the number of -CH2- units using 16.1 cm3/mole as the volume of a -CH2- group.
3 Apparatus: The balloon is attached with rubber bands to the body of a buret with a removable tip. Try to get the rounded part of the balloon as close as possible to the 50-mL mark of the buret.
Procedure:
1. Determine a factor (F) for converting buret readings (VB) to height: DP(cm H2O) = F(50 - VB) ; DP(Pa)= pgh = (998 kg/m3)(9.8 m/sec2)F(50 - VB) Obtain a rectangular piece of a balloon of the same type that is being used in the experiment. Use a ruler to determine the area (AX), then weigh the sample on an analytical balance. Calculate the volume (VX) using the density of natural rubber, then calculate the average thickness (VS/A0 = VX/AX). 2. Assemble a conditioned4 balloon. Fill a 100-mL graduated cylinder with distilled water. Add this to the buret assembly in increments of 10 mL until the water level inside is visible just above the outside level when the balloon is immersed to the 50-mL mark. It may be necessary to remove air from the balloon by squeezing it around the buret, then add more water. Assemble the apparatus so that the 50-mL mark of the buret is at the overflow level of the water in the beaker. Record the volume of water added ( VA), and the volume reading of the buret (VB°). The volume of the balloon (V) is the volume added minus the volume appearing in the buret (V = VA + VB - 50). This initial value will allow calculation of the area of the undeformed rubber (A°) (the initial volume is on the order of 100 ml or 100x10-6 m3). 3. Add the remainder of the water in the graduated cylinder to the buret. Wait 30 seconds for the buret reading to become relatively stable (it will not completely stabilize). Record the buret reading (VB) and the total volume added . 4. Add water to the buret in 50-mL increments to a total volume of 300 mL. Observe and record the buret readings and total volume added.
4 5. Repeat step #4 with 100-mL increments to a total volume of 600 mL. Note: To save time, steps 6-8 will be performed with a second apparatus which will be set up in a sink. At the START of the period, record the buret reading and heat the sample as in #6 below. Take temperature + buret readings throughout the laboratory period. 6. To save time, a second apparatus with a similar, but different balloon will be used to determine temperature effects. This apparatus containing 500 mL of water will be assembled in a sink, and its surrounding water will be heated to between 50- 60°C. Take several measurements of water temperature vs buret reading as the system cools to about 30°C. 7. After the system has cooled, empty the balloon and measure the undistorted volume (V0) of this balloon.
Calculations: Refer to the Spreadsheet furnished with this Exp. 1. Transfer all data to a speadsheet in the form
References:
1 Atkins, Peter Physical Chemistry, 6th Ed., Freeman, NY, 1998, p. 154-6. 2 Flory, P. J., Principles of Polymer Chemistry, Cornell UP, Ithaca, NY, 1953, Chapter 11. 3 The one-dimensional stretching of rubber is discussed in a number of texts: Noggle, J. H., Physical Chemistry 3rd Ed, Harper-Collins, New York, 1996, Chapter 3. Rosen, S. L., Fundamental Principles of Polymeric Materials 2nd Ed, Wiley, New York, 1993, Chapter 14. 4 The balloon should be pre-conditioned by inflating to about 800 mL (preferably assembled to the apparatus), and equilibrating for several minutes.
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