10.0
-
J. Phys. Chem. 1990,94, 961-968
-..._ .....
9.5
-
Y
-
0
-
-
9.0
-..... -.. ...-.. ....
0.5
0
0:2
0:4
0:6
J T Figure 3. Ionic strength dependence of association rate (k)for the three varieties of SOD. The ordinate is in log base 10. The squares correspond to Mn and the diamonds to Fe SOD. The vertical bars represent standard deviations in the simulated rates. The dashed line represents simulated rates for Cu/Zn SOD.
formations could develop which allow penetration by superoxide. This would occur, for example, if tyrosine 36 were to flip out of the active site cleft. Figure 2 is identical with the previous figure except that tyrosine 36 has been removed, thereby exposing Mn. A completely analogous situation exists for the crystal structure of the Fe enzyme, although the blocking residue is tyrosine 34. To determine, in an approximate way, the effect of tyrosine 36 (or 34) placement in the active site region on the ability of superoxide to reach Mn (or Fe), simulations have been carried out with this residue simply removed. In these cases, electrostatic potentials are recomputed since the low dielectric region occupied by tyrosine is replaced with high-dielectic water. As expected, a substantial association rate results. Shown in Figure 3 are rate constants for Mn SOD (squares) and Fe SOD (diamonds) as a function of ionic strength. The dashed line denotes the corresponding rates for Cu/Zn SOD obtained previous1y.lI Experimental turnover rates for Fe12 and Mn SOD' are 3 X lo8 (M s)-I at I = 0.1 M and 5.5 X IO8 (M s)-I at I = 0.002 M, respectively. Although the actual rates are smaller than the simulated rates by factors of 5.3 (Fe SOD) and 14 (Mn SOD), the fact that they agree as well as they do indicates a low activation barrier. A simple mechanism whereby the active site is blocked by the tyrosine most but not all of the time would, by itself, largely reconcile simulated and experimental rates.20 The simulations, of course, do not prove that this simple mechanism is necessarily (20) McCammon, J. A.; Northrup, S.H.Nature 1981, 293, 316.
961
the actual mechanism. Electron transfer through the tyrosine side chain is among other possible mechanisms that could account for the observed kinetics. The salt dependence of the simulated rates merits some discussion. For Mn SOD, where the net charge on the enzyme is 0, the decrease in rate with increasing salt indicates that there is a net electrostatic attraction of the superoxide toward the active site. If the net electrostatic interaction were repulsive, the opposite salt dependence would be observed. Similar behavior has been previously reported for Cu/Zn SOD where the net protein charge is -4. Except at the lowest salt, there also appears to be a net electrostatic attraction for Fe SOD as well, despite the fact the net enzyme charge is -12. The rate at the lowest salt ( I = 0.005 M) for Fe SOD may be somewhat unreliable since the final potentials and consequently the rate constants depend on the relaxation parameter'' used in the finite-difference method. However, this uncertainty is small (about 10% in k ) and limited to this one particular case. In conclusion, all three SODSexhibit electrostatic attraction of superoxide toward the active site despite significant differences in net enzyme charge. Summary
On the basis of the crystal structures, the active sites of both the Fe and Mn SOD enzymes appear to be inaccessible to superoxide. Fluctuations in the solution structure, such as the movement of a specific tyrosine residue, may be required to expose the active site metal centers prior to reaction with superoxide. An alternative mechanism might involve direct electron transfer through amino acid residues. Simple removal of active site tyrosines results in predicted diffusion-controlled rate constants that are 5-15 times larger than experimental rates. Like the Cu/Zn enzyme, Fe and Mn SODS appear to carry charge distributions that attract superoxide toward the active sites. This result is somewhat surprising considering the significant differences in net charge of the three enzyme varieties. Since protein fluctuations appear to play a significant role in these reactions, we plan to use molecular dynamics to study atomic motions of the active site regions of Fe and Mn SOD in the near future. Acknowledgment. We thank Dagmar Ringe and Gregory Petsko at MIT for providing us with the Fe SOD coordinate and Martha Ludwig's lab at the University of Michigan for providing the Mn SOD coordinates. S.A.A. acknowledges an N S F Presidential Young Investigator Award. J.A.M. is the recipient of the 1987 George H. Hitchings Award from the Burroughs Wellcome Fund. His research is supported in part by grants from NIH, the Robert A. Welch Foundation, and the Texas Advanced Research Program. Registry No. SOD,9054-89-1; 02-,11062-77-4.
A Continuum Diffuslon Model for Viscoelastic Materials Y.Weitsman? Mechanics and Materials Center, Civil Engineering Department, Texas A & M University, College Station, Texas 77843-3136 (Received: November 14, 1988; In Final Form: June 6. 1989)
A model for diffusion in polymers is established from basic principles of irreversible thermodynamics, employing the methodology of continuum mechanics. The polymeric materials are considered to respond viscoelastically,with ageing. It is shown that effects of stress on diffusion and certain anomalies in the moisture sorption process can be explained by the present model.
l. Introduction The processes of moisture transport and sorption in polymers have been studied by numerous investigators for more than 50 Present address: Department of Engineering Science and Mechanics, The University of Tennessee, Knoxville, TN 37996-2030. 0022-3654/90/2094-0961$02.50/0
years. Most of these studies, which involved experimental, analytical, and materials science aspects, were conducted by researchers in the fields of Physical and polymer chemistry. These investigations were mostly focused toward applications in membrane twhnologY and therefore c ~ ~ c e r n aspects ed of permeability and seepage. It is far beyong the scope of the present article to 0 1990 American Chemical Society
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The Journal of Physical Chemistry, Vol. 94, No. 2, 1990
Weitsman
list the many significant contributions made to this field. References 1-9 list some of the many outstanding review articles on the subject. During the past 15 years the transport of moisture in polymers raised concern and interest in composite materials technology. Moisture plays a significant role in affecting the mechanical response of fiber-reinforced polymeric composites and in influencing the behavior of adhesive joints which are increasingly employed in composite structures. These concerns, which focus mainly on mechanical effects, produced more than 500 publications on the subject. A review and summary of some of these works is given in ref 10 and 11. In spite of many overlaps, the investigationsin the two foregoing categories reveal distinctions that reflect the disparate scientific disciplines from which they derive-polymer science in one case and applied mechanics in the other. It may thus be worthwhile to present an approach to theory of diffusion that employs the methodology of continuum mechanics and its well-developed constitutive formalism. This approach is especially useful in the modeling of coupling effects, such as between stress and diffusion, as detailed in a recent work.12 It should be noted that the most crucial issue associated with diffusing substances in polymeric composites is their detrimental effects on structural strength, integrity, and durability. In certain circumstances, diffusing substances were noted to damage the composites by debonding the fiber/matrix interfaces. This issue is a topic of current research and its modeling is hampered by uncertainties regarding the damage mechanisms. Several investigations concerning moisture-induced damage in composites were reviewed in ref 1 1 and an attempt at a continuum-level modeling of the phenomenon is provided in ref 13. The issue of damage will not be discussed in the present paper.
sidering each element in thermodynamic equilibrium with a reservoir containing vapor at pressure p , density 5, and internal energy and entropy densities ii and S, respe~tively.l~-'~ Conservation of the solid and vapor masses gives
2. Basic Equations
ii is the enthalpy of the vapor in the hypowhere = @ / p ) thetical reservoir and gi = T,i. Elimination of qi,i between (5) and (6) yields the following expression for the "reduced entropy inequalitywi2
Consider a solid body B occupying a material volume V bounded by a surface A. Let the solid, of mass density, ps, absorb vapor thrdugh its boundary and let m denote the vapor mass per unit volume of the solid. Also, let x be the position of a solid mass particle in the deformed configuration that corresponds to the place X in the undeformed state, and let f and q denote fluxes of vapor mass and heat, respectively, and v the velocity of the solid particles. I n addition, let u and s be the internal energy and entropy densities of the solid/vapor mixture per unit solid mass, and let ulj and T denote the components of the Cauchy stress due to mechanically applied loads and temperature, respectively. A proper accounting of the state of the solid/vapor mixture, which is a thermodynamically open system, is obtained by con( I ) Rogers, C. E. Solubility and Diffusivity. In The Physics and Chemistry of the Organic Solid State; Fox, D., Labes, M. M., Weissberger, A,, Eds.; Interscience: New York, 1965; pp 509-635. (2) Stannett, V.; Hopfenberg, H. B. Diffusion in Polymers. In MTP International Review of Sciences. Vol. 8: Macromolecular Science; Bawn, C. E. H., Ed.; Butterworth: London, 1972; pp 329-369. (3) Hopfenberg, H. B.; Stannett, V. The Diffusion and Sorption of Gases and Vapours in Glassy Polymers. In The Physics of Glassy Polymers; Howard, R. N., Ed.; Applied Science: New York, 1973; pp 504-547. (4) Vieth, W.R.; Howell, J. M.; Hsieh, J. H. J . Membr. Sci. 1976, I, 177. ( 5 ) Stannett, V. T.; Koros, W. J.; Paul, D. R.; Lonsdale, H. K.; Baker, R. W . Ado. Polym. Sci. 1979, 32, 69. (6) Frisch, H. L. Polym. Eng. Sci. 1980, 20, 2. (7) Frisch, H. L.; Stern, S . A. CRC Crit. Rev. Solid State Mater. Sci. 1983, II(2), 123. (8) Rogers, C. E. Permeation of Gases and Vapours in Polymers. In Polymer Permeability; Comyn, J., Ed.; Elsevier Applied Science: New York, 1985; pp 11-73, (9) Windle, A. H. Case I1 Sorption. In Polymer Permeability; Comyn, J . , Ed.; Elsevier Applied Science: New York, 1985; pp 75-1 18. (IO) Komorowski, J. P. 'Hygrothermal Effects in Continuous Fiber Reinforced Composites": Aeronautical Notes NAE-AN-4, IO, 1 I , and 12. National Research Council Canada, 1983. ( I I ) Weitsman, Y. Moisture in Composites: Sorption and Damage. In Fatigue of Composite Materials; Reifsnider, K. L., Ed.; Elsevier: New York, in press. (12) Weitsman, Y. J. Mech. Phys. Solids 1987, 35(1), 73. (13) Weitsman, Y. In?. J . Solids Struct. 1987, 23(7), 1003.
p,
+ p,V.v = 0
(1)
m = -C.f
(2)
Conservation of energy over B reads
-s
d p,u d V = dt v
The third integral on the right side of (3) expresses the mechanical power due to vapor flux, observing t h a t h / b corresponds to vapor velocity. The last integral in (3) expresses the rate of vapor-borne energy. In eq 3 and the sequel, repeated indices imply summation, unless stated otherwise. The entropy inequality reads dt v
p,s
d V L l - ( q i / T ) n i dA A
- L S j & dA
(4)
where the last integral in (4)expresses the rate of vapor-borne entropy. Application of Green's theorem to (3) and (4), and employment of ( 2 ) , yieldsf2 psU
=
-
U rJ. U1% . J.
- qi,i - h,iJ
-
+ hm
(5)
and psTS L -qi,j
+ (q;/7,)gi - TS,if.+ TSfi
(6)
+
-ps04 - psos?- -
uijcij - ( q i / T ) g i
+ pm -hp,;- Sg,J
I0
(7)
In17) 4 = u - Ts - ( 1 /pso)ui,tijis the Gibbs free energy and p = h - TS is the chemical potential of the vapor in the hypothetical reservoir. In eq 7 psodenotes the initial mass density of the solid and infinitesimal deformation has been assumed, whereby tij
0 and are components of a semipositive definite, symmetric matrix. As noted earlier, the times T and t in eq 14 and 15 are distinct because the quench time T usually occurs earlier than the exposure time t , namely t = T - t,. Equations 11, 13, and 14 give
E,
dur
PsO4uoeUf
+b=0 dr
(16)
whereby and
uf = ufOe-rlTu -R,j., - Rdf - (qj/ T)gi - Sgjf;. 1 0
(10)
with d{ = dT/b and
In eq 10, the affinities R, and R are defined by
The aging process is associated with the spontaneous collapse from an initial value Om, which is created of the free volume by the cooling of a polymer across its glass transition temperature Tgdown to a temperature To. In addition, the subjection of a polymer externally to ui., m, or AT = T - To, will trigger an irreversible process which will cause the internal variables y r to The two drift spontaneously toward their equilibrium values 7,”. above-mentioned processes are assumed independent of each ~ t h e r . ~ ~Consequently, .~’ while both 4, and Df in eq 10 denote derivatives with respect to time, the time lapse t for y, is not necessarily equal to the time span T for up Assume y,,uf
