oxidation, respectively. The orders of the carbon oxidation n were close t o 1.0 for the CAL carbon and 0.55 for the bone char. The diffusional factor F is significant in low percent oxygen atmosphere. The factor, f(C,), is a fundamental property of carbon, such as the physical and chemical nature of carbon, the distribution of carbon, and the pore structure of the adsorbent. The present work also demonstrates how the various techniques were developed to characterize the nature of the carbon of bone chars. The results give some insight into the characteristics of carbon and contribute to a fundamental understanding of the behavior of bone char in practical applications.
literature Cited
Blyholder, G., Eyring, H., J . Phys. Chem., 61, 682 (1957). Chou, C. C., Proc. 1970 Technical Session on Cane Sugar Refining Research, Cane Sugar Refining Research Project, Inc., pp 16-27, New Orleans, LA, 1970. Chou, C. C., Hanson, K. R., I n d . Eng. Chern. Prod. Res. Develop,. 10, 2 (1971).
Christman, D. R., Day, N. E., Hansell, P. R., Anderson, R. C., Anal. Chem., 27, 1935 (1955).
Ergun, S., Rlentser, &I.,in “Chemistry and Physics of Carbon,” Vol 1, P. L. Walker, Jr., Ed., p 203, Marcel Dekker, New York, NY, 1965. Gallagher, J. T., Harker, H., Carbon, 2, 163-731/1964). Glasstone, S., Laidler, K. R., Eyring, H., Theory of Rate Processes,” McGraw-Hill, New York, NY, 1941. Hawtin, P., Gibson, J. A., Carbon, 4, 489-500 (1966). Laidler, K. J., Glasstone, S., Eyring, H., J . Chem. Phys., 8 , 65976 (1940).
Laine, N. R., Vastola, F. J., Walker, Jr., P. L., J . Phys. Chem., 67, 2030-34 (1963).
Acknowledgment
Loebenstein, W. V., Gleysteen, L. F., Deitz, V. R., J . Res. iYat.
The author expresses thanks to Kenneth R. Hanson, the Director of Research, Research and Development Division, h m s t a r Corp., for his inspiration for this work, to Juan Guell for his assistance in analytical work, and to the management of Amstar Corp. for permission t o publish this work.
Thomas, J. X , Carbon, 8 , 413 (1970). Walker, Jr., P. L., Amer. Sci., 50,259-93 (June 1962). Walker, Jr., P. L., Rusinko,. F.,. Austin, L. G.. Advan. Catal.,
Bur. Stand., 42, 33 ‘1949).
11, 133 (1959).
’
Wicke, E., “Fifth Symposium on Combustion,” p 245, Reinhold, New York, NY, 1955. RECEIVED for review March 31, 1972 ACCEPTED May 30, 1972
Monte Carlo Simulation of Structure of low-Density Polyethylene Pai-chuan Wu, John A. Howell, and Paul Ehrlichl Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, h‘Y 1.4914
An adequate model for the structure of low-density polyethylene (LDPE) should account for a t least the following: the number and weight averages of the molecular weight and the type, number, and distribution of structural irregularities, such as branches and unsaturated groups. Obviously, it is possible in principle to predict all these structural features from a quantitative knowledge of the rate constants for propagation, termination, and those side reactions responsible for the structural features cited. This work deals with the less ambitious problem of preand dicting only the number average molecular weight the type, concentration, and distribution of short-chain branches and the concentration of vinylidene groups in freeradical polyethylene. The weight average molecular weight is controlled by long-chain so the magnitude - branching, predicted by our model for Jfw/Jfn,i.e., 2, is of no practical significance. There is mounting evidence that the short-chain branches are not distributed uniformly in L D P E and that they are not identical in length and are often complex in structure. It therefore seemed of interest t o generate the detailed structure of
(z)
(ra)
1
To whom correspondence should be addressed.
352
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
L D P E molecules by a hfonte Carlo method, using as kinetic parameters those which would reproduce the structure of polyethylene made under known conditions. Since nearly all commercial and pilot-plant samples are made under poorly defined conditions, the model polymers chosen were laboratory samples described earlier (Woodbrey and Ehrlich, 1963) and made isothermally a t two different temperatures and over a range of pressure. Reaction Mechanism
A reaction mechanism which appears to be capable of accounting for the structure of the short-chain branches and for the number average molecular weight under conditions where radical-radical termination can be neglected is shown in Figure 1 (Woodbrey and Ehrlich, 1963; Ehrlich and Nortimer, 19f0). The key reaction steps involve the “backbiting” mechanism of Roedel (1953) , as extended by Sinipson and Turner according to Willbourn (1959), and p-scission of tertiary radicals (Nicolas, 1958). The last reaction always generates a “dead” molecule containing a vinylidene group and a linear chain radical capable of further chain propagation (Woodbrey and Ehrlich, 1963).
Since the elementary reactions cited can lead to more complex structures requiring more detailed accounting of the reaction path, we find i t convenient to define a further set of probabilities which, however, is completely determined by the first. We let 521
Figure 1.
= k,,/(2 k,,
+ k,[MI)
=
- 21) - 21)
21/(1 221/(1
+1
Elementary reaction steps
I n one of the three alternative 0-scission steps, the deactivated molecule is short and is neglected in the comlxhition of the iiumber average molecular weight. This is the intended meaning of the equations iyhich refer to the p-scission step and which are not chemically-balanced but indicate t'he method of counting polymer molecules by the computer. We have not included mutual t,ermination by chain radicals in our model. T h a t step is unimportant in our reference polymer. Wiere this simplification cannot be made, the lowering of can be readily related to t'he contribution to the termination step by radical-radical reactions (Van der Xolen, 1969). Kote that according to our model, one vinylidene group is generated per polymer molecule. This requirement is approximately, though not' exactly, satisfied i n the reference polymer. ITe must, define five probabilit,ies based on t'he rate constants show1 iii Figure 1, nhere [XI is the monomer concentration and k , [ . l f ] stands for the propagation rate per radical. K e define
-q-c-q-c-c-c-c
:
c
,
C
+
22/(1
-
52)
+1
where e, and P, are the reference temperature and pressure, respectively, and the AE, and AV,* are energies and volumes of activation. K e can now generate all possible branching structures by using these basic probabilities. Figure 2 shows diagrammatically the first several generations of such structures. The extension to more complex structures is obvious. We find that
4 4
c* c
- 21)
According to the reaction mechanism proposed, the structure of L D P E is defined completely if we can assign numbers to the five probabilities, y, 6, zl, ZZ, and T.Furthermore, if we let k t represent any of the five rate constants defined in Figure 1, the effects of synthesis temperature and pressure on the structure of L D P E can be determined from the equation
-c-c-c-c-c
9 5 :
Figure 2.
Zl/(l
c.
-c-c-c-c-c
E
d d
First several generations of branching structures Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
353
Table 1.
Structure and Kinetic Parameters of Reference LDPE Molecule
Experimental 0 = 250°C, P = 1 6 0 0 a t m ; z = 63,000; methyl/lO* CH2 =
14 vinylidene/lOa CH2 = 0.17 (simulated value 0.22). All other simulated values are identical to experimental ones Probabilities Case I,
ethyl/butyl = 1 . 2
Y b 21
x2
T
AEu - AEp = AE,, - AEp = QT
Er
KT
JT
LT
UT
Case II, ethyl/butyl = 1 .9
0.017 0.030 0.300 0.400 0.050 Activation energies, kcal/mol 2.0; AE, - AEp = 1.0; AE,, 1.0; AET - AEp = 0.5
0.014 0.084 0.420 0.504 0.031
- AEp = 2.0;
NT
-butyl
I' cct-r( 7 m QT
Figure 3. tures
RT
-
ST
ethyl v i n y l i d i n e (terminal)
-linear
Symbolism for branching and terminal struc-
of all possible lengths of the straight-chain segments which lie between branching structures. We choose, as the longest length which will be used in the computation, that Lofor which the probability of the chain segment being longer than Lo is less than 1 - PLTOT (PLTOT = 0.999). The simulation now consists of assigning a group of integers to the possible straight-chain segment lengths and to each possible branching structure. The number of integers assigned to each group is proportional to the probability of occurrence of that particulslr straight-chain segment length or branch type. We now select integers by using a randomnumber generator. The first number corresponds to a chain segment, and the next to a branching structure; this is repeated until we obtain a terminal branching structure. The simulated molecule is now complete and can be printed out. Figure 4 shows the computer logic. Simulation of Reference Molecule
+ 8
IT (P
?IND AND STORE
I
Figure 4.
Computer logic
53 such structures allow us to represent a t least 95 w t % of all L D P E generated by our scheme, when the five basic probabilities are assigned possible sets of values consistent with the structure of the reference L D P E molecules. We define PBTOT as the probability that any branching structure in the molecule will be one of the 53 listed. Figure 3 shows these branch structures and assigns the symbols by which they may be identified in the computer printout. Monte Carlo Method
The simulation procedure is a simple Markov chain process. Using y only, we can generate the individual probabilities 354 lnd. Eng. Chem. Prod. Res. Develop., Vol. 11, No. 3, 1972
The experimentally observed parameters which our model must reproduce are (or the concentration of vinylidene groups) and the total concentration of branches or chain ends. There is general agreement that these consist almost entirely of ethyl and butyl groups in a ratio of between one and two in favor of the former. The precise ratio and its dependence on polymerization conditions are not well-known, and we shall only require that the simulated molecule shall have this ratio in the proper range. Clearly, no unique assignment of the five probabilities which determine the kinetics is possible. However, certain ones of these probabilities, as will be shown, can be assigned with greater certainty than others, and the same is true of the activation energies and volumes which determine the dependence of structure and Ill,on the temperature and pressure of polymerization, the latter according to transition-state theory. Table I makes two such possible assignments. The computer printout of an individual LDPE molecule (Case I, Table I) is shown in Table 11, and its structure is shown in a more conventional representation in Figure 5. Most noteworthy ars the clustering and the complexity of the branches. Such a molecule could be represented only crudely by an average length of primary chains between branch points and by an average branch structure, One of the chief merits of the Monte Carlo method is that these structural details and their variation from molecule to molecule can be visualized.
LDPE Molecule
Table II. Computer Printout of Reference
Molecular weight = 70,770; methyl/103 CH2 = 16.4; vinylidene/103 CHZ = 0.20; ethyl/butyl = 1.22a 17 A 21 A 17 A 54 A 22 A 850 -%A 57 A 42 AA 168 A 9A 14 A 13 ABA 14 AAB 130 EE 107 AAA 20 A 12 AA 13 X 35 AAA 11 A 98 A 84 AAA 2 AAA 10 A 44 A 84 AAA 23 A 25 AX 24 C 19 EE 87 A 76 HH 511 5A [c171 DDD 22 DDD 89 A 8 -4 89 A 101 EEE 15 AAA 140 Ail 52 -iA 34 A 35 d 39 A 146 AA 13 FT]' a Structure of an individual molecule need not correspond closely to the average structure (Monte Carlo simulation). Numbers refer to monomer units between branch points. c [ ] Brackets enclose part of molecule shown structurally in Figure 5.
_171
I 30 t $aN 40
e-T-0.1
35
A--T-0.3
Y -
$i25
20
e ..
z! 2" 2Y l f 10
Figure 5. Structure o f terminal 48 wt yo o f reference LDPE molecule, shown symbolically in Table II. Numbers
-
5 I
0 0.00
represent monomer units in linear chains between branch points
The type of detail shown in Table I1 and Figure 5 is largely beyond that' which can be tested experimentally, although there appears to be some hope of resolving certain details of branch structure by infrared met'hods (Ehrlich and 1Iortimer; 1970). The amount of detail generated by this method may nonetheless be relevant, for example, in testing "weak-link" theories of polymer degradation and ult'imately in designing polymer of either higher or lower t'hermal st'abilit>-.
x
I
I
0.02
0.01
x2/xl
=
I
I
0.04
0.03
0.05
0.7
1.5
General Inferences from Computer Trials
I t is of interest to inquire whether any of the experimentally accessible structural variables are determined predominantly hl- only some of the five independent' probabilities; if so, simplified algebraic expressions could be written which would relate the structural feature in questions t'o one, or perhaps two or three, of the basic five probabilities. Figure 6 shows that the total branch concentration is determined primarily by y with xl, xZ,and T making a secondary contribution only. (The experimental points represent the value of the independent variable a t which the simulations were carried out; the small amount of scattering of t'he dat'a points with respect to the smooth curves results from t8he .\toc.habtic nature of the Monte Carlo simulations.) Figure 7 shows that. the primary det'erminants of the ethyl/ butyl ratio are X I and XZ; ZIplays a larger role since the process represented corresponds to the creation of a n ethyl branch with the siniultaneou:; destruction of a butyl branch, whereas x z merely represents the creat,ion of an additional ethyl branch (Figure 1). The proper simulation of (or vinylidene concentration) iiivolves all five probabilities, but' only y is well-determined (Figure 2 ) . I3ecause of our somewhat' uncert,ain information about the et,h>-l/butyl ratio, we are allowed a certain amount of freedom
0.2
I0 .'3
I
I
0.4
I
0.6
0.5 x1
+
I
0.7
0.8
x2
Figure 7. Simulation of ethyl/butyl ratio ( y = 0.017; b = = 0.050)
0.030;T
of choice in the selection of values for X I , XZ,and T (Cases I and 11, Table I). The variety of terminal structures (Figures 3 and 5 and Table 11) furthermore show that it would be imor the possible to write a simple algebraic expression for vinylidene concentration in terms of the five basic probabilities. I t follows that y can be determined with good accuracy and next best, perhaps, the sum X I $2. b is very poorly defined, but it is reasonable to keep its magnitude between that of y and z1 and x2, since abstraction of a tertiary hydrogen is involved which should raise the value above that of y, b u t the steric factors which favor the "back-biting" by ethyl groups and make X I and x2 large are absent (Ehrlich and Mortimer, 1970). These considerations were central in assigning the values i o the kinetic parameters which represent Cases I and I1 of Table I.
+
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
355
Table 111. Dependence of Structure on Synthesis Temperature and Pressurea
Reference probabilities correspond to Case I, Table I
Pressure, atm
Temp, 'C
800
250
Total branches (cH,/l O3 CHn) Simulation Exptl
Branch type, ethyl/butyl, simulation
Vinylidene (C = C / l O3 C H J Simulation Exptl
~
MRx Simulation
10-3 Exptl
42 36 2.5 0.70 32 1200 22 20 1.6 0.30 48 1600 14.2 14 1.2 0.22 0.17 63 63 2000 12 11.2 1.0 0.11 84 2400 10.5 10 0.9 0.07 108 130 14.1 800 14 1.2 0.085 1200 7.3 8 0.7 0.065 1600 7 5.7 0.6 0,038 0.045 368 -300 2000 6 5.5 0.6 0.030 2400 5.1 5 0.025 0.6 In the absence of a solvent and/or chain transfer agent such as propane, pressures below about 1800 atm at 130°C and 1000 atm at 250°C would lead t o phase separation. Extrapolation of these results into the two-phase region should not be attempted.
85
0 250'C
01
I
1000
600
Figure 8. AV,*
-
I
I
1400 1800 PRESSURE, a m
I
2200
I 2600
AV,* as function of synthesis pressure
Dependence on Temperature and Pressure of Polymerization
Since the total branch concentration is determined predominantly by the single parameter y, successful simulation of the effect of polymerization temperature on the branch concentration determines AE, - AE,. This value is small as expected (2 kcal), and this also defines AEzl - AE,, AEz2 AE,, and AEb - AE, which should all be eimilar, allowing only for the fact that some of the reactions involve the abstraction of secondary and others of tertiary hydrogen atoms. AET - AE,, which also is small, is limited by the effect of or vinylidene coricentration (Table I). temperature on Similarly, the effect of polymerization pressure on branching determines largely AVv* - AV,*. Although none of the other differences in AV* has a substantial effect on the branch concentration, some assumption must be made. The analysis of pressure effects on the kinetics of vinyl polymerizaticns in the liquid phase, as well as of ethylene (Ehrlich and llortimer, 1970; Weale, 1967), suggests that the dominant term is AT',*. I n view of this and the rather trivial consequences of an in356
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
correct assumpt,ion on the calculated magnitude of t,he branch concentration, we assume that all AVt* - AI7,* are identical. The correct iiumerical simulation of the branch concent,ration wit'h pressure in our reference LDPE made a t 130' and 25OOC then allow the calculation of AVv* - AT7,* (Figure 8). Clearly, this parameter must be allowed to vary with pressure, and this conclusion, together with the fact that the absolute magnitude is quite large a t the low end of the pressure range, is consistent with thermodynamic considerations involving the behavior of the partial molar volume in supercritical mixtures (Ehrlich, 1971). Because of t,he uncertainties i n t'he assignments for the parameters other than y, we do not attempt to assign realistic values to t'he ot'her AV,* - AV,* terms or to calculate the effects of polymerization pressure on molecular weight or vinylidene concentration. The effects of temperature and pressure on the relevant structural parameters in t'he reference polymer and according to the Monte Carlo simulation are shown in Table 111, by use of kinetic parameters, which at' the reference t,emperature and pressure, assume bhe values represented by Case I of Table I. Conclusions
Regardless of any insufficiencies in the reaction mechanism, it is likely that use of the model together with the values proposed for the kinetic parameters will lead to reasonable predictions of t'he concentration of short-chain branches in LDPE made a t constant temperature and pressure, as well as a t changing, but knonn values cf these synthesis parameters. A better estimate of t,he kinet,ic parameters other than y and the corresponding activation energies and volumes should be possible by use of t,he model with a larger number of and better-characterized LDPE samples. The choice of kinetic parameters proposed here-particularly t.hose corresponding to Case 11-would also reproduce the structure of the LDPE synthesized in continuous stirredtank reactors by Van der Molen (1969), and the values of AVu* - AI',* derived here from t,he analysis of the branch concentration are a t least qualitatively consistent ivith the value of AT.',* derived from rate studies by Symcox and Ehrlich (1962). The detailed simulation of molecular structure made possible by the I\Iont,e Carlo method, although a t least a t present unverifiable experimentally, suggests further exploration and
testing of theories of degradation dependent on the presence of weak links, e.g., tri or tetrafunctional branch points. Although the high values of z1 and x 2 , lvhich had to be a+ sumed in the present simulation and which are responsible for the clustering and the complexity of the branches, can be readily justified on steric grounds (Ehrlich and l\Iortimer, 1970), the low value n-hich must be assumed for A E , seems puzzling. It provides perhaps the major reason why the reaction mechanism used in these calculat’ions cannot be accepted without reservation.
Literature Cited
Ehrlich, P., J . Mucromol. Sci., Chem., A5, 1271 (1971). Ehrlich, p., Alortimer, G. A., Advun. Polym. Sci., 7, 386 (1970). Nicolas, L., J . Chim. Phys., 177 (1958). Roedel, hi,J , , J , A ~Chem. ~ sot., ~ 75,. 6110 (1953). Symcox, R. O., Ehrlich, P., ibid., 84, 531 (1962). Van der Molen, T. J., IUPAC Conference, Preprint 777, Bud&pest, Hungary, 1969. Weale, K. E., “Chemical Reaction a t High Pressures,” Spon, London, England, 1967. Willbourn, A. H., J . Polym. Sci., 34, 569 (1959). J. c,, Ehrlich, p., J . Chem, Sot,, 85, 1580 (1963).
Strain Softening and Yield of Polycarbonate-Moire-Grid Biaxial-Strain Analysis Morton H. Littl and Svenning Torp Division of JIacromolecular Science, Case Western Reserve University, Cleveland, OH 44106
When an external stress field is applied to a glassy polymer, it responds as follows. With increase in stress level, there is a gradual decrease in differential modulus (strain softening) until the glass fails (brittle failure), or i t becomes zero at which point the material is said to yield. Differential modulus is the slope of the stress-strain curve. h third, unrelated mode of failure is by crazing, which sometimes takes place in tension a t higher temperatures. Strain softening, yield, and crazing have received considerable attention in the literature. The effect of the type of stress field on the yield stress of a glassy polymer has been investigated by several workers. R h i t n e y and A n d r e w (1967) and Sternstein and Ongchin (1969) fitted their yield stress data to a Mohr and ColumbNavier-type pressure-sensitive, shear-stress yield criterion. The effect of normal stress on yield wis further investigated by Rabinoivitz et al. (1970). They ran experiments up to 7 K bar hydrostatic pressure and found t h a t shear yield stress had a linear dependence on pressure as predicted by the 1Iohr-type yield criteria. The effect of temperature and strain rate on the yield stress has been investigated by several workers. Holt (1968) found yield stress for P N M A to increase linearly with logarithmic strain rate over a n eight-decade range. Ekvall and Low (1964) and Robertson (1963) found that for PC the yield stress decreased almost linearly with increasing temperature, a t least away from the transitions. Lohr (1965a,b) also carried out experiments a t different strain rates and temperatures and then combined these results to construct yieldstress master curves by a method similar in concept to the construction of time-temperature master curves. The master curves obtained were linear, the slope being a material characteristic. The effect of temperature and stress level and of stress field on craze format ion was investigated by Sternstein and Ongchiii (1969) and Sternstein et al. (1968). They found t h a t the craze is generated by the normal stress component of the applied stress field and that the magnitude of this component required to generate a craze decreased linearly with increasing temperature. To whom correspondence should be addressed.
One common explanation for yield, first suggested by Bryant (1961) and later rephrased by Litt and Koch (1967) and Andrews and Kazama (1967), is to describe it as a stressinduced glass transition.
I n this context T ois not a temperature but a phenomenon dependent on temperature, stress, and time. With increase in temperature, stress, or time, T o will approach the test temperature, and the modulus of the glass will decrease. The glasses studied must be well characterized to determine the proper dependence of To on the above parameters. Isotropic materials undergoing reversible deformation can be fully characterized by two constants: shear modulus and bulk modulus or Young’s modulus and Poisson’s ratio. We will discuss the latter two. Most of the literature today contains uniaxial tension data from which Young’s modulus can be obtained, but little has been done about Poisson’s ratio. Nielseii (1965) determined Poisson’s ratio for some glassy polymers by a stepmise load-increment tensional test. His values are ambiguous a t higher strains owing to the method and difficult to analyze as the strain rate is not constant. Whitney and Andrews (1967) have also studied some glassy polymers by a uniaxial compression dilatometer. However, there is still considerable ambiguity about Poisson’s ratio and its dependence on strain, temperature, and strain rate. With only the knowledge of Young’s modulus, the material is only partly characterized. A detailed knowledge of Poisson’s ratio should give additional information about the yielding process. The change in Poisson’s ratio, or volume, with stress in a uniaxial stress test seemed to be the simplest obtained for a n isotropic material by axial and transverse strain measurements. There are several strain-measurement devices available on the market today, and we decided to employ a newly developed one-by moire interference patterns. This method was chosen because it yields biaxial strain to a high degree of accuracy in the desired strain range, experimental technique Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
357
