Macromolecules 1986,19, 139-154
NSF MRL P r o g r a m t h r o u g h the Center for Materials Research at S t a n f o r d University. R.J.L. was supported by the NIH Medical Scientist Training Program at the Stanford University School of Medicine. D.E.was supported by NIH Grant 2 R 0 1 GM 31674 and a NIH Research Career Development Award 5 KO4 AM 01353. We acknowledge the valuable assistance of Susan Sorlie and Matheen Haleem. b y the
References and Notes (1) Fredericq, E.; Houssier, C. “Electric Dichroism and Electric
Birefringence”; Clarendon Press: Oxford, 1973. (2) Krause, S., Ed. “Molecular Electro-optics”; Plenum Press: New York, 1980. (3) O’Konski, C. T., Ed. “Molecular Electro-optics”; Marcel Dekker: New York, 1976. (4) Elias, J. G.; Eden, D. Macromolecules 1981, 14, 410. (5) Elias, J. G.; Eden, D. Biopolymers 1981, 20, 2369. (6) Hagerman, P. J. Biopolymers 1981,20, 1503. (7) Diekmann, S.; Hillen, W.; Morgeneyer, B.; Wells, R. D.; Porschke, D. Biophys. Chem. 1982, 15, 263. (8) Stellwagen, N. C. Biopolymers 1981, 20, 399. (9) Rau, D. C.; Bloomfield, V. A. Biopolymers 1979, 18, 2783. (10) Stellwagen, N. C. Biophys. Chem. 1982,15, 311. (11) . , Marion. C.: Perrot. B.: Roux. B.: Berneneo. J. C. Makromol. Chem. 1984, 185, 1665. (12) Marion. C.: Roux. B.: Berneneo. J. C. Makromol. Chem. 1984, 185, 1647. (13) Yoshioka, K. J. Chem. Phys. 1983, 79 (7), 3482. (14) Diekmann, S.; Jung, M.; Teubner, M. J. Chem. Phys. 1984,80 (3), 1259. (15) Wegener, W. A.; Dowben, R. M.; Koester, V. J. J. Chem. Phys. 1979, 70 (2), 622. (16) Rau, D. C.; Charney, E. Macromolecules 1983, 16 (lo), 982. (17) Rau, D. C.; Charney, E. Biophys. Chem. 1983, 7, 35. (18) Eden, D.; Elias, J. G. In “Measurement of Suspended Particles by Quasi-ElasticLight Scattering”; Dahneke, B. E., Ed.; Wiley: New York, 1983; p 401. (19) Fixman, M. Macromolecules 1980, 13, 711. (20) Fixman. M.: Jagannathan, S. J.Chem. Phys. 1981.75 (8). 4048. (21) Hogan, M.; Daitagupta, N.; Crothers, D. M. h o c . Nutl. Acad. Sci. U.S.A.1978, 75 (11, 195. (22) Maret, G.; Weill, G. Biopolymers 1983, 22, 2727. I
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Atomistic Modeling of Mechanical Properties of Polymeric Glasses Doros N. Theodorou and Ulrich W. Suter* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Received J u n e 5, 1985 ABSTRACT Methods are developed for the prediction of the elastic constants of an amorphous glassy polymer by small-strain deformation of microscopically detailed model structures. A thermodynamic analysis shows that entropic contributions to the elastic response to deformation can be neglected in polymeric glasses. A statistical mechanical analysis further indicates that vibrational contributions of the hard degrees of freedom are not significant, so that estimates of the elastic constants can be obtained from changes in the total potential energy of static microscopic structures subjected to simple deformations. Mathematical procedures are developed for the atomistic modeling of deformation and applied to glassy atactic polypropylene. Predicted elastic constants are always within 15% of the experimental values, without the use of adjustable parameters. An estimate of the thermal expansion coefficient is also obtained. Inter- and intramolecular contributions to the mechanical properties are examined, and it is found that coexistence in the bulk reduces the effects of individual chain idiosyncrasy.
Introduction A method for the detailed, atomistic modeling of wellrelaxed amorphous glassy polymers has recently been introduced.’ The polymeric glass was pictured as an en-
* Address correspondence to MIT Chemical Engineering Department, Room 66-456, Cambridge, MA 02139. 0024-9297/86/2219-Ol39$01.50/0
semble of static microscork structures in detailed mechanical equilibrium ( b u t not i n thermodynamic equilibrium), each microscopic structure being represented b y a model cube with periodic boundaries, filled with segments from a single “parent chain”. Our modeling approach was applied to glassy atactic polypropylene with very satisfactory results. Model estimates of the cohesive energy and of Hildebrand’s solubility parameter agreed very well 0 1986 American Chemical Society
140 Theodorou and Suter
with experiment. Single-chain conformation was found to be essentially unperturbed and long-range order to be completely absent. We not turn to an investigation of the response of the detailed atomistic models to deformation and hence to the theoretical estimation of macroscopic mechanical properties of glasses from first principles. Little has been published on this subject to date. Previous publications pertain almost exclusively to the crystalline state, in which the existence of a well-defined periodic structure permits considerable simplification. A statistical mechanical treatment of crystalline polymethylene by Pastine2 led to a P-V-T equation of state that compares favorably with experiment, and Tashiro, Kobayashi, and Tadokoro3g4derived elastic constants for a variety of crystalline polymers. Mechanical properties of glassy amorphous polymers have not yet been deduced from their detailed molecular structure, but a number of correlations exist. The assumption of a Lennard-Jones-like relationship between total potential energy and the linear dimensions of a sample led Haward and MacCallum5 to a successful correlation between adiabatic compressibility and molar volume. Schuyer (p 267 of ref 6) used group contribution methods to arrive at an estimate of the compressibility; tensile and shear moduli can be obtained from the compressibility by using an estimate for the Poisson ratio. Yannas7vsconsidered displacements, on the molecular level, brought about by mechanical deformation of glassy polymers and found that the primary molecular mechanism is by rotation around skeletal bonds rather than by distortion of bond angles or bond distances. He introduced the term “strophon”for a segment three virtual bonds long. Inter- and intramolecular contributions to deformation were considered separately. Recently, an atomistic approach to plastic deformation in amorphous metals has been introduced by Maeda and Takeuchigand by Srolovitz, Vitek, and Egami.lo The work presented here, although developed completely independently and referring to bonded systems, small deformations, and nonzero temperatures, has certain aspects in common with their approach.
Thermodynamic Considerations A body can be regarded (Chapter 1 of ref 11) as a collection of particles whose positions are denoted by ro= (ro,l,r0,2,r0,3)E ( X O , yo, zo) in the reference (undeformed) state and by r = (rl, r2,r3) ( x , y, z ) in the deformed state. The vector s with components si = r, is the “displacement vector”. The material strain tensor is defined by
Macromolecules, Vol. 19, No. I , 1986 n, we write in matrix notation
d F = T T n dS (2) The terms c and 7 are symmetric and can also be represented, in Voigt notation, as vectors of six components (p 14 of ref 11). The two notations will be used interchangeably; which of the two notations is employed will be evident from the number of subscripts: €1 = €11, €2 = €22, €3 = 633, (34 €4 = 2c23, € 5 = 2631, €6 = 2€12
= 722, 73= 7 3 5 (3b) 74 7 5 = 731, 76 = 712 The fundamental thermodynamic equation for an elastic system in terms of the stress and strain tensors, at constant chemical composition, assumes the form d U = T dS + Vo C ~ L MdttM = T dS + voC.7, del (4) 71
= 711, = 723,
72
LM
I
(the subscript 0 denotes the undeformed state). For the Helmholtz energy (A = U - TS) we obtain, in differential form dA = -S d T VOCTLMd€LM = -S d T + VoC71 del (5)
+
LM
I
The (fourth-order) tensor of isothermal elastic coefficients is defined by
As a result of the Voigt symmetry relationships (p 32 of ref ll),eq 6 can be condensed into a symmetric 6 X 6 matrix,C. For an isotropic material, such as an amorphous glassy polymer, eq 6 then assumes the form h
0 O
A
2piA
0O
Ol
0
where A and p are the Lam6 constants. The tensile (Young’s) modulus, E, the shear modulus, G, the bulk modulus, B, and the Poisson ratio, v, are related to A and P by 3x + 2p E=p-
G = p
x+IL
1
B=-=A+-p KT
2 3
v=-
x + P)
(8)
The strain dependence of the entropy, at constant temperature, is given (p 34 of ref 11)by the Griineisen tensor Y
YLM =
We are only concerned with very small deformations. Terms of second order in the derivatives of displacement with respect to position will therefore be neglected in defining c, and material and spatial strain will be considered identical (Chapter 13 or ref 12). The material stress tensor is denoted by T . As for the strain, material and spatial stress are not distinguished. The element 7LM is taken equal to the force per unit area acting on an element of surface perpendicular to axis L and along the direction M. If d F is the surface force on an element of surface dS, where the exterior unit normal is
--
1
a7LM
(9)
where c, is the heat capacity per unit mass of material at constant strain. (The tensor y could again be represented in condensed notation.) For an isotropic solid
where a is the volumetric thermal expansion coefficient and y tge “Griineisen parameter”. Consider now an arbitrary elastic solid subjected to an arbitrary isothermal small deformation. Expanding the
Mechanical Properties of Polymeric Glasses 141
Macromolecules, Vol. 19, No. 1, 1986 internal energy, U , into a Taylor series around the undeformed state, to second order, and using the definitions introduced above, we obtain
U=A
+ TS = Uo + VOC[TLM+ ~ ~ c , T T L M ] c L M+
(16) and (17) for a typical glass. Their measurements on glassy PMMA at T = 20 “C, P = 1 atm, and a frequency of 4 MHz lead to aP = 18.7 X K-l, K T = 1.70 X MPa-’, G = 1.88 X lo3 MPA, (dKT/dP)IT = -1.59 X MPa-2, (dKT/a131p = 2.16 X MPa-’ K-l, (dG/dP)IT= 3.44, ( d G / d T ) J p= -4.69 MPa K-l and hence to
The first terms in the brackets are due to the strain derivatives of the Helmholtz energy; the second terms originate in the corresponding derivatives of the entropy. The quantity
is sometimes termed “internal pressure”. By analogy we will call the tensor u
=7
+po~,Ty
(12)
the “internal stress tensor”. We now focus on the second-order term in (11)in order to assess the relative importance of the contributions of internal energy and entropy to the elastic coefficients:
(13) Entropic effects are relatively unimportant if the dimensionless ratio
Temperature derivatives of the elastic coefficients at constant strain are not usually available experimentally. For an isotropic solid, and under the condition that the reference (undeformed) state is characterized by an isotropic stress distribution, (14) can be transformed to (see Appendix A)
+--KT
a c L MNKI T ] l ap
