NOTES
306
Relative to the aqueous solution ammonia brings about a red shift of only -lo00 cm.-I, which is accompanied by decrease of intensity. This is the regular response of the band to solvent eflects.IO ~ P ( N O ~ - ) , , ~ is hardly affected by temperature, but the intensity is markedly affected (Figure 1). The decrease of intensity with increase of temperature also occurs in dimethylformamide and aqueous solutions, and it was ascribed to weakening of hydrogen bonding or dipolar effects.ll Some vibrational structure appears at low temperatures.
Acknowledgmnt. We gratefully acknowledge the support of this research by the U. S. Army (Contract DA-91-591-EUC 3583).
- 20
0
20 A0 temperature ("C)
(9) E. Gusarsky and A. Treinin, J . Phys. Chem., 69, 3176 (1966). (IO) E.Rotlevi and A. Treinin, ibid., 69, 2646 (1966). (11) S. J. Stickler and M. K d a , "Moleculsr Orbitals in Chemistry, Physics and Biology," Academic Press Inc., New York, N. Y.,1964, p. 241.
60
Figure 2. The variation of 1/XmSx of I- and CNS-- with temperature.
Polymer Configuration at an Adsorbing Interface by the Monte Carlo Method
'"i
by Sydney Bluestone and Charles L. Cronan' Depart& of CherniStry, F r e m State College, Freano, Cdiforniu (Received August 86, 1966)
E 121
Y
7
.c
I
I
loa
I
I
1
110
112
114
.hv
I 116
I 118
I
,kcol/mol
Figure 3. hv, of CNS- us. hvmX of I- in liquid ammonia, acetonitrile, and acetonitrilewater mixtures (the weight per cent is recorded); t = 20".
Figure 3 where hv,,, of this band in a few solvents (in which the band is clearly separated from the overlapping transition) is plotted against their c.t.t.s. values. By extrapolation we obtain for the energy in water ~Y(CNS-)~,,= 4.59 X lo4 cm.-'. For K (CNS-) the value 18.9 crn.-I deg.-l is obtained (Figure n\
AI.
A completely tSrpe Of effect exhibited by the low-intensity internal tramition of NOa-. The Journal of Physical C h i S t r u
Several r e p ~ r t s ~have - ~ been published dealing with the shape of a long chain polymer at an interface. The adsorbed polymer can be described either as a thick film4with large loops, extending far into solution, and with only a few chain segments adsorbed, or as a thin film28-sconsisting of small loops, confined rather close to the adsorbing interface, with a large fraction of the segments adsorbed. I n the present study an IBM 1620 I1 digital computer was used to perform Monte Carlo calculations in order to establish what the film thickness is under various adsorption potentials. The method employed was that of Verdier and Stockmayeld (VS), used previously by Bluestone and Vold (1) This paper is baaed on a thesis submitted by C. L. Cronan to the faculty of Freano State College in partial ful6llment of the requirements for the undergraduate course, Independent Study 190. (2) (a) H.L.Frisch, J . Phys. Chm., 59, 633 (1966); (b) W.I. Himchi, ibid., 65, 487 (1961). (3) A. Silberberg, ibid., 66, 1872, 1884 (1962). (4) R. Simha,H.L. FiSoh, and F. R. Eirich, ibid., 57, 684 (1963). (6) P. E. Verdier and W. H. Stockmayer, J . Chem. Phys., 36, 227 (1962).
NOTES
307
in another investigation6 and modified to allow B 31-segment linear molecule to be adsorbed at an interface.
Computer Program The method simulates a chain of atoms or beads which undergo random thermal collisions with solvent molecules. The movement of the chain in three dimensions is governed by the rules of VS, except that the chain beads are not allowed below a base, the x-y plane ( x = 0), which is regarded as the adsorbing interface. The beads move from one lattice point to another on a simple cubic lattice. Adsorption occurs upon contact with the base; however, all beads can move on or off the surface, excluded volume conditions being obeyed at all times. Depending upon an adsorption potential, the movement of the beads away from the adsorbing surface is biased. The computer uses the potentials as follows: bead displacements are the same as those without any interface, except if the bead is to move away from the surface and it lies within a small distance, x, from the interface. The energy of adsorption (segment) in units of kT, E,, is operative only if x is less than five segment lengths; Le., the z coordinate is not greater than four. If the x coordinate is to be changed from z to (z l),the function A, = 1 - exp(-E,) is calculated. This function, A,, is the probability of the bead being restrained in moving away from the adsorbing interface. A random number between 0 and 1 is chosen. If it is greater than the probability function A,, the bead has freedom to move to (z l),providing that the corner and nondouble occupancy rules of VS still hold. If the random number is less than A,, the bead does not move and a new machine cycle is started. The program was performed with three different sets of potentials, listed in Table I under programs 1, 2, and 3; the energy of interaction with the surface as 2 increases was assumed to behave similarly to gas-surface interaction appropriate to carbon? The value of E, = 0 for program 1 is close to that given by Frisch2&for simple hydrocarbons at 300'K. in a polar solvent. The program was tested by running a random coil relaxation calculatioq5 and the results obtained were in excellent agreement with those of VS for a chain of 32 beads.
2o
Results and Discussion The polymer codgurations, chosen to be random coils, were allowed to wiggle near (all segments initially had z coordinates ranging from 0 to 6 and about 2 / ~ of the were in the range Of the the adsorbing base. Figures 1and 2 correspond to programs 1 and 2, respectively, showing the relationship between
0
5151
01
4
0
5
I
10 Cycles X 10-8.
I
I
15
20
Figure 1. Number of adsorbed segments, N , as a function of machine cycles for program 1. The dashed line indicates the equilibrium value.
25: 20
z
15
10
+
+
tt
25
01
I
I
I
I
,
I
I
20 30 Cydes X lo-*.
10
0
40
Figure 2. Number of adsorbed segments, N , as a function of machine cycles for program 2. The dashed line indicates the equilibrium value.
Table I: Adsorption Potentials, .E8,in Units of kT (T, 300°K.) as a Function of the Distance z from the Interface -Program 2
0 1 2 3 4 55
1-
-Program El
E5
A,
0.750 0.316 0.159 0.106 0.088
0.53 0.27 0.15 0.10
0.333 0.137 0.071
0.08 0.00
0.039
0.000
0.047
0.000
2-
-Program
3-
AZ
E,
A5
0.28 0.13 0.07 0.05 0.04 0.00
0.075 0.032 0.026 0.010 0.009
0.08 0.03
o.oO0
0.02 0.01 0.01 0.00
the number of adsorbed segments and time, machine Cycles. No data W a s obtained from program 3 since the chain escaped from the range of the adsorbing poten2000 cyc1es* tial after (6) 8. Bluestone and M. J. Vold, J. C h m . Phys., 42,4175 (1966). D. F. Eggem, et gaphysicd John wiley ssd
),(
SOW
bo., New York, N. Y.,1964,p. 718.
Volume 70, Number 1 January 1966
NOTES
308
Table 11: Compilation of Results of Programs 1 and 2 Program
fno
lev
fROt
1 2
0.26 0.22
0.26 0.26
0.05 0.03
zz 68.3 82.1
Ez-
o
P(0)
pb
24.5 13.4
0.750 0.333
0.79 0.43
4.3 6.4
Table I1 gives the pertinent results collected from the computer printouts for programs 1 and 2 at the equilibrium configurations of the chain; ie., the results of program 2 were obtained from the last 25,000 cycles. fncr fev, and fPoe are the fraction of cycles terminated because of noncorner, excluded volume conditions, and influence of the adsorption potential, respectively. L2,the mean square end-to-end distance; the average number of adsorbed segments; P(O), the fraction of adsorbed segments; Pb, the average number of segthe average number of ments in a hanging loop; Rl, adsorbed loops in the polymer chain; and RIe,the average number of loops and free ends extending into space away from the adsorbing surface; are also listed in Table 11. It should be noted that the fraction of cycles tenninated by 6he influence of the potential, fpot, is small compared to fno and fev,8 demonstrating that only a relatively small bias is needed for polymer adsorption. However, the fraction of adsorbed segments on the surface, P(O), is greatly affected by the value of the potential, E,=o, and increases if the potential increases. The present results can be compared to several theoretical treatments on the adsorption problem. Frisch2a and Higuchi,2bwho considered the inhence of short-range interactions between the surface and segments in solution, obtained similar equations to calculate the fraction of adsorbed segments, P(0)
n,
+
P(0) = 2a(1 - @(l XE,o) (Tjt) where X is 1 and for the Frisch and Higuchi calculations, respectively; E,,o is the adsorption energy in units of kT at the surface; cy, the probability of a successful contact between an active site and a segment, is taken as unity in the present case; 6, the fraction of sites covered, is zero for isolated chains; f, the segment “Musion” coefficient, is assumed to be for the Monte Carlo calculations in which the beads move only about half the time5; and t is thenumber of segments in the chain (assumed to be large), which in the present case is 31. The comparison is shown in Table I11 along with an extrapolation of Higuchi’s graphical presentation of values of P(0) to 31 segThe Journal of Physical Chmistrg
-
B
Ni
1.3 2.5
-
Nxe
2.0 3.1
ments. Silberberga has used the method of statistical mechanics on this problem. His computed value P(0) for a cubic lattice with a minimum loop length of two segments (in the present model it is three) is also listed in Table 111. Table III : Monte Carlo Results of the Fraction of Segments Adsorbed, P(O), Compared to the Predicted Values of Frisch,28 EtiguchiZb,and Silberberg8 (the Adsorption Potentials, EZSo, Corresponding to Programs 1,2, and 3 Were Used in All Calculations)
Frisch, eq. 1 Higuchi, eq. 1 Higuchi, graphical Silberberg Monte Carlo
-
-
\”/
Program 1
Program 2
Program 3
0.616 0.748 0.59 0.75 0.79
0.470 0.528 0.46 0.65 0.43
0.378 0.396 0.39 0.35 0.00
Table I11 shows that the Monte Carlo results are in good agreement with the theoretical calculations. The present work further substantiates that for relatively small adsorption potentials, a large fraction of the segments are adsorbed. However, at the very small potential, E,=o = 0.075, program 3 found no adsorption on the base. It means that the Monte Carlo results predict a rather sharp rise in segments adsorbed with increasing adsorption potential and lends confirmation to a recent statistical mechanical calculation by who has shown that as the energy parameter for adsorption becomes smaller, a stage is reached in which the polymer molecule is unstable at the interface. The values of PI,, RI,and R1e from Table I1 help to establish the configuration of an average adsorbed chain. An average chain of program 1 had an adsorption loop of 4 segment lengths and a free end; the average chain of program 2 had two adsorption loops of about 6 segment lengths each and a free end. The above information, plus the average heights of the adsorption loops, one segment length for program 1 and (8)fne, fev, and Zeare also greater than the values of 0.21, 0.23, and 66.3, respectively, reported by Verdier and Stockmayer. (9) R. J. Roe, Proc. Natl. A d . Sci. U.S., 53, 60 (1966).
NOTES
z = o
z = o
Figure 3. Two-dimensional illustration of an average chain adsorbed on a surface obtained from programs 1 (bottom) and 2 (top); 0 - 4 indicates one segment length.
1.2 for program 2, allows Figure 3 to be drawn. Thk two-dimensional figure definitely gives strong support to the thin-film theory of polymer adsorption.lOJ1
Addendum Recently, Roe12 and DiMarzio and McCrackenI3 have shown theoretically that the conformation of adsorbed polymers can be classified into three types with different dependencies on the chain length. A transition region is found at a particular value of the attractive energy. In the present work, the attractive energies of programs 1 and 2 would correspond to values greater than, and program 3 to a value less than, the transition energy. (10) Fontanall has shown by infrared measurements that selective adsorption of a strongly polar segment from a copolymer can lead to thicker flms than would o t h e m k be obtained. A Monte Carlo calculation was tried in which one end of the chain waa permanently k e d to the surface and the other 31 beads could wiggle freely under no adsorption potential. The result waa that no segments were adsorbed in addition to the stationary one, indicating that probably thicker a m s will be obtained from selective adsorption. (11) B. J. Fontana, J. Phys. Chem., 67, 2360 (1963). (12) R. J. Roe, J . Chem. Phys., 43, 1691 (1966). (13) E. A. DiM'arzio and F. L. McCracken, %%id., 43, 639 (1966).
Sensitivity of Sedimentation Equilibrium Data to Solute Polydispersity'
by Hiroshi Fujita and J. W. Williams Department of C h i s t r y , University of Wisconsin, Madison, Wisconsin (Received September 16,1066)
309
the form of the boundary gradient curve that otherwise would reflect the heterogeneity of the solute. The sedimentation equilibrium experiment has the advantage that it provides such thermodynamic data as molecular weight and activity coefficient. It is coming into quite general use. As regards the sensitivity of this experiment to solute heterogeneity, however, it seems that much remains to be made clear. This note represents an effort in this direction. Solute distributions at sedimentation equilibrium are modified both by solute heterogeneity and by thermodynamic nonideality of the solution. I n the present study we shall restrict ourselves to the ideal2 solution in which the solute is heterogeneous only with respect to molecular weight. The fundamental equationa descriptive of the sedimentation equilibrium of an uncharged monodisperse solute in such a solution is dc - -- M(l - gp)w2rc dr RT where c is the solute concentration in gram per milliliter of solution, T is the radial distance from the center of rotation, p is the density of solution, @ is the partial specitic volume of solute, M is the molecular weight of solute, w is the speed of rotation, R is the molar gas constant, and T is the absolute temperature. If one assumes that the solution is so dilute that p and o may be replaced by their values at infinite dilution, po and go, eq. 1may be integrated to give
Here c, is the concentration of the given solution before ultracentrifugation, z = (+ - r12)/(r22 - ~ 1 (TI is the radial distance from the center of rotation to the meniscus in the centrifuge cell; rz is the corresponding distance to the cell bottom), and X = (1 - f10p~)(r22- r12)w2/2RT. Equation 2 may be rewritten in the form
In c It is an advantage of the ultracentrifugal techniques that one can gain information about the homogeneity of a macromolecular solute. Such information may be derived either from sedimentation velocity or from sedimentation equilibrium experiments. In principle, the transport experiment is the more advantageous from the point of view of resolution, but in practice there are complications due to the necessity of corrections for diffusion and for concentration and pressure dependences of sedimentation coeEcient, all of which modify
=
In [coXM/(eXM- l ) ]
+ AMs
(3)
which indicates that plots for In c vs. x form a straight line and the molecular weight of the solute may be evaluated from its slope. It also follows from eq. 2 (1) Presented at the 144th National Meeting of the American Chemical Society, Los Angeles, Calif., March 31-April 5, 1963. (2) By the ideal solute is meant one in which the higher terms in eq. 1 can be neglected. For linear polymer solutes this condition is obtained in B solvents. (3) H. Fujita, "Mathematical Theory of Sedimentation Analysis," Academic Press Inc., New York, N. Y., 1962, p. 237.
Volume 70,Number 1 January 1066
~ )
