Comment on Monte Carlo Slmulatlons of Polymer Conflguratlons
To the Editor: The paper by Engstr6m and Lindberg on Monte Carlo simulations of polymer conformations (1)interested us, as we hoped to add a computer simulation experiment to our polymer synthesis and characterization laboratory course. We requested and received from the authors the BASIC source code and object code. However, the description and program documentation left some questions as to the actual choice of narameters for inout data to the orogram. Below we desc;ibe the set of parameters that we chose and a brief descriotion of their derivation. We also describe our use of the program in our polymer laboratory course. Background
The oroeram bv Enestrom and Lindbere uses a statistical approach calcilate ;he size and shape o? a polymer molecule in solution. The polymer solution is modeled using a lattice theory. The polymer chain is assumed to occupy a set of adiacent lattice sites. one site for each monomer unit. The remiining sites around the polymer chain are occupied by solvent molecules. The computer program chooses a monomer unit in the chain at random and computes the change of energy that results from moving that monomer unit to an adjacent lattice site. The change in energy is calculated by considering all monomer-monomer, monomer-solvent, and aolvenholvent contacts before and after the move. If the new chain conformation is of lower enerev -- than the initial eonformation, then the monomer unit is moved to the new position and the orocess is reoeated. If the new conformation is hieher in energy than the initial conformation, then the chiin still has a chance to adopt the new conformation if its thermal energy (a random number) is large enough to overcome the energy barrier. In this way, the chain avoids being locked into local minima. After computing thousands of conformations in this way, the program calculates average values for mean square end-to-end distance, mean square radius of gyration, persistence length, and principal moments of inertia. The input data for the simulation are the number of monomers in the chain and the interaction energies between monomer-monomer, monomer-solvent, and solvent-solvent. Each cornouted orooertv is the mean of several subaverages (macros), each sibav&age being the average over a large number of iterations of the molecular conformation (mkros). Thus, the program also requires as input the number of macros to be used in the calculation of eeometric parameters and the number of micros to he included in each suhaverape. These parameters determine the precision of the results. Calculatlon of lnteradlon Parameters
The energetic interaction parameters for monomermonomer, monomer-solvent, and solven&aolvent interactions are not tabulated. However, these parameters can be estimated from several sources. Below we describe our estimates of these parameters (2). The molar energy of vaporization of a species i is given hy
dination number (number of nearest neighbors) of the species, and wii is the interaction energy between two molecules of species i. Since the molar energy of vaporization is related to the enthalpy of vaporization by
the interaction parameter of a solvent molecule with itself can be determined from the enthalpy of vaporization as
Because of the lattice chosen for this simulation, the coordination number z, is equal to 8, so solvent-solvent interaction parameters can be calculated from tabulated values of enthalpies of vaporization. Since enereies or enthaloies of vaoorization of oolvmers are unavailabie due to the i&olatilit;of iheinteraction oarameters must be calculated in a differentwav. The energyof vaporization per unit volume of a liquid, W;IV~,is known as the "cohesive energy density" of the liquid, and its square root 6i as the "Hildebrand solubility parameter". The heat of mixing two components, given in terms of their solubility parameters, is
where Vis the total volume and mi are the volume fractions of the two components. The solubility parameters of polymers can be estimated as approximately equal to the solubility parameters of solvents with which the heat of mixing is small. The solubility parameters of many common polymers have been tabulated (3). Using the tabulated solubility parameter for the polymer, along with its density (d) and repeat unit molecular weight (M),the monomer-monomer interaction parameter can be estimated:
Finally, an estimate of the monomer-solvent interaction is required. It was pointed out by van Laar and Lorenz (4) that the forces between two dissimilar particles will, in general, be approximated by the geometric mean of the forces that each of the particles would exert on a second particle of its own kind: DAB =
is the molar energy of vaporization, z is the coor-
(6)
Hildebrand and Scott analyzed the physical basis for this assumption and pointed out that the postulate of the geometric mean accurately expressed interactions of permanent dipoles and is a close approximation to the energetic interactions due to London dispersion forces (5).The expression is lnteractlon Parametem Used In Monb Carlo Slmulatlon Substance Polynyrene cyclohe~ane toluene hexane THF
where
$KG
W8
w..b
(kcallmol)
(kcallmol)
-2.25 -1.82 -2.10 -1.75 -1.68
Source
-
&
-2.02 -2.17 -1.98 -1.94
b!
A& An. 6,
Reference 3 6 6 6 3
' q m m J or 8s-sJ in ihe nolation of Engsmm. qE(m In the notation of Engsnbm.
Volume 68
Number 7
July 1991
621
unsatisfactory for the description of forces between permanent and induced dipoles hut acceptable for description of the mixing of relatively nonpolar molecules. Since dispersion forces are always attractive, the negative square root is used. In this treatment, therefore, the total interaction paw,,) is always rameter of Engstrom, w = w,, - 1/2(w,, positive. With an acceptable method of calculating the various energetir interactions required for the program, we can proceed to experiments. The table contains a short list of interaction parameters calculated from puhlished data.
+
Student Explorations We asked our students to look a t a variety of effects on a polystyrene chain in solution. Students conducted simulations while systematically varying the polymer chain length, solvent, and temperature (as reflected in the dimensions of the interactions Darameters.. eiven in units of kT). Our microcomputers were not equipped with the floating-point coDrocessor recommended bv Ennstrom. so our students limited their investigations tochains of 10 monomer units or less. The leneth of time for the calculations was less than an hour for most simulations, and students were able to conduct several calculations a t one time by using several computers each in our microcomputer laboratory. We also found it useful to coordinate the investigative efforts of different groups of students. Each group examined a different experimental variable, and the results from all the students were combined and made available to the entire class.
-
Conclusions The Monte Carlo simulation by Engstrom and Lindherg is a meaningful addition to a polymer solution theory or polymer laboratory course. The program illustrates several important concepts of polymer shape and provides an introduction to statistical methods for modeling polymer systems. I t is clear that the atudents benefited from the opportunit y to conduct Monte Carlo simulation experiments. Manv siudenta are unfamiliar with statistical cafculations, even if they are familiar with the solution theories. The students gained additional insight into the approximations of the theory and the Monte Carlo method by watching as the computer explored different polymer conformations on the CRT screen. This paper addressed some ambiguities in the original description of the Monte Carlo program by Engstrom and provides a rational basis for deriving energetic parameters for use as input to the program. Our students found the results of the modeling to be more meaningful when they were able to derive the energetic parameters used as input to the program from experimental data representing real chemical systems.
Acknowledgment The author wishes to thank Paul Rasmussen for helpful discussions and to acknowledge suggestions from Sven Engstrom. Literature Cited 1. EngstrDm, s.;Lindher g,M. J. Charn.Educ. 1988,65,973. inSolution; Krieger: 2. For s moreeomploted,~rivation,~Morawetz.H.Moeromolaculas Malshar. FL, 1983: pp 39-12. 3. Brsndrup, J.; Immorgut, H.. Eda. Polymer Handbook Wiley: New York. 1975. I. Van Larr. J.J.: Lurenz, R. 2. Anore.. Al&ern. Chem. 19'25. I(6.442 5. Hildebrand, J. H.: Scott. R. L. The SoluUlily of None bcfrolytes; American Chemical Society Monograph No. 17; Reinhold: NewYork:Ch apfen 7 . 8 . 6. Handbook o/ Chemistry and Physics; Wesst, R. C., Ed.; CRC. Boca Raton, FL, 1980: 6lst ed.
622
Journal of Chemical Education
Altltude and Bolllng Point To the Editor: Boyd Earl [1990,67,45] has done a great service to teachers of thermodynamics. In presenting his formula relating the boiling temperature of a liquid to elevation, he has combined in a useful and interesting context three thermodynamic relations, a t least two of which are of great importance: the (isothermal) barometric formula, the relation for the adiabatic expansion of an ideal gas, and the ClausiusClapeyron equation. While this in itself is remarkable, in addition the derivation is straightforward and elegant, and, as the author points out, it provides an instructive illustration of the use of approximations. Beyond all this is the surprising conclusion of the derivation: in fair approximation, a t least up to 10 km (misprint in the table), the decrease in the boiling temperature of water from its value a t sea level is directly proportional to the elevation. In fact, this lowering is given by where h is the elevation in kilometers, which places the boiling point on the top of Mt. Everest a t 70 "C. I am so moved by this article that I wish to make the following modest contribution. Since there are 3.28 ft in a meter, ATb (in K) is numerically equal to the elevation in thousands of feet to within the accuracy allowed by the approximations involved. Unfortunately, neither scientist nor lay person will he a t ease with this mix of units. However, everyone should be impressed that in OF and miles There is a 10 "Fd r o in ~ the boiline tem~eratureofwater for every mile of elevation, a t least u p to 6mi. In fact the value 10 provides closer agreement With the more accurate formula ( ~ a r l ' seq 4) t h k d o e s the approximate one (eq 6) in this range of elevations. I look forward to using this in my thermodynamics classes as a means of instilling life into the barometric formula, as a practical application of the dreaded (by students) adiabatic ideal gas relations, and as an extension beyond the usual applications of the Clausius-Clapeyron equation. Donald Peterson California State Universiiy Hayward. C A 94542
Paper Model of a Cuboctahedron To the Editor: The articles by Shukichi Yamana, "An Easily Constructed Cuboctahedron Model" (1985, 62, 1088) and "An Easily Constructed Model of a Coordination Polyhedron that Represents the Cubic ClosestPacked Structure"(1987,64,1040) describe nicely the methods of getting the paper models from a sealed, empty envelope. I have followed both the methods in making the models. The models formed finally, however, are identical. The same polyhedron, if seen from different angles, can look as is shown in the f i r e s of the respective papers. I would like to know whether the author and other readers are getting similar experience.
Davld S. Allan
V. D. Kelker
Northwestern University Evanston, IL 60208
Unlvenity of Poona Pune 411 007, India
