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Monte carlo study of a simple model for micelle structure
CHEMIC4L PHYSICS LETTERS
Volume 79. number 3
MONTE Steven
CARLO
STUL’Y OF A SiMPLE
W HAAN and Lawrence
MODEL
1 May 1981
FOR MECELLE STRUCTURE
R. PRATT
Department of Chenusny. Unwersrtyof Chkfomra.Berkeky. Gzitfomm 94720. USA Recerved 19 January 1981
A smple lattice model for a mxelJe ISformulated and studied by hlonte Carlo numerical methods. Results are presented for awegates of 20,30, and 50 cham molecules The density profiles are much broader than ISusually assumed in conventlonal pictures of nucelle structure and the average shapr found to be dlstmctly asphencal.
1. Introduction A molecular understandmg of surfactant aggregation in aqueous solutron has been the obJect of an enormous research effort. These systems extubrt a vanety of fascmating phenomena [ I] some of whrch have well appreciated apphcatron to crude oli recovery processes. Of course, rt IS also a wrdely held opmion that a molecular understandrng of nucelle fomiatron wti provrde an mcrsrve view rnto problems of orgaruzatron and structure of macromolecules of biological rmportance I1 321. In sprte of the obvrous motivatron and effort, there IS a long list of questlons about molecular Features of micelIes which have not been answered_ A cogent expression of this state of affairs is gwen by Menger [3]. These questions focus on such molecular features as nucelle size and shape, the degree of surface roughness and water penetratron, the degree of non-uruformity of the nacelle interior, the ab&ty to accommodate probe molecules, and the nature of the &am packmg in the interror. More broadly, rt IS unportant to understand the dehcate balance of forces which are responsrble for the relatively abrupt appearance of micelles when the surfactant concentration passes a critical micelle concentration. We have formulated a srnrple model of mrcelles whrch we believe to capture the statrstical features common to most miceliar systems, and which allows general answers to the questrons posed above. Here we present the results of a preliminary Monte Carlo 436
study of the model. Our rdea of how a model stands to a real micelIar solution is roughly the same as the relatron between the king model and a real ferromagnet, or as the relatron between some lattice models and real crystal-vapor rnterfaces [4] As Menger has shown, our understandrng of these systems IS sufficiently prmutrve that model burldrng can stih provide nnportant msrghts In the next two sections we describe our mode!, and the numerrcal methods whrch we use to study rt. In sectron 4 we present and drscuss the results of our inrtral calculations.
2. Model Here we descrrbe the model we have chosen for detarled study. Consrder a dramond lattice and denote by I the distance between near neighbor lattice pomts. A charn molecule of n - 1 bonds can be strung on the Iattree with the n atoms situated at lattrce pomts. One of the two end groups is desrgnated a head group, and the remaming are called tail groups. In more detaded models the head group would be charged or polar. AU vacant lattice points are considered to be occupied by solvent (water) molecules. We chose to study the diamond Iattrce rather than, say, a srmple cubic lattice not because normal alkane molecules fit naturally on the diamond lattice. The important consideration is that the diamond Lattice affords 12 possible orientational states for a completely
extended cham molecule, compared to 6 for the szmple cubic lattice. Thus one expects the dzamond lattzce structure to affect typical aggregate shapes less severely than would the simple cubic lattice structure. We consider a system of N such cham molecules on the lattice. Two chain molecules are designated as near neighbors 1f at least one group on one of the molecules has a group on the other molecule as a lattice near neighbor. Using ths notion of adJacency, we requzre that the entue system of N molecules form a connected aggregate. For each such aggregate we calcu;ate an energy, U, in the followmg way. If any pair of atoms occupy the same lattzce site, L1= 00. Othenvise, let %7
= total number of ar-_r group palrs which are near neighbors on the lattice.
Here cc, 7 may be h (head) total number of tad-head bors. Aiso define g = total number gate.
(1)
or t (tad). Thus nth IS the paus whch are near nezgh-
of gauche bonds
1n the aggrc(2)
Then c/= Ettrltt
+ ‘th?zth
I May 1981
CHEMICAL PHYSICS LETTERS
Volume 79, number 3
+ ehnw
+ u@ -
(3)
The probabdlty The e&y and ug are energy parameters. of observmg the configuration of the connected aggregate is proportional to exp(-CJ/kgT). For consistency the probab1hty of a configuratzon wluch disconnects the aggregate 1s set equal to zero. This IS because such a configuration is properly assigned to a different aggregate type. The lattice spacing, I, should correspond to a typlcal intermolecular datance. Since the mtramolecular near-neighbor distance m a cham molecule 1s connderably less than this, the most realistic interpretation of the model IS achieved by considering our chains of M lattice sates to model molecules of more than IZ groups. For example, with n = 4 the model chain has a length/ width ratio similar to jz-octane. The values given to the energy parameters rn eq. (3) have to be given careful thought. One of the most mpottant features of such a model 1s the poss1bihty of a clear and simple connection between these energy parameters and observed aggregation properties. Since we are not trymg to accurately represent any one particular surfactant-solvent system, but only to con-
struct a general micelle model, some soptisticat1on IS required rn estabhshmg reasonable values for these parameters. It is clear that we want Ett < 0. Eth > Ett, and ehh > 0 in order for the aggregate to be fairly stable, and to have the head groups on the outnde, on the average To estimate ett, we consider a hmlt u-t which the model can be taken as representing a bulk hydrocarbon hquld. We put Eth = ett, CZ,.,,,= eft, drop the connectivity restriction, and consider N large. Ifich normal alkane liquid the model corresponds to, and the appropnate density for the model 1s fived by taking our lattice bond length, I, to be the distance of closest approach, and the maximum iengrh of the lattice chain molecule to be the mawmum length of the correspondmg real normal alkane. Then the mtermoleculat internal energy assoczated with the model 1s est1mated by mean field theory, and Identified with the mtermolecular internal energy of the real hqu1d. The latter quantity can be obtamed from available computer slmulat1ons of n-alkane hqu1ds [5]. In thus way we estimate ett/kB = -800 K. The difference between the estimated ett and the real dispersion forces between, say, methylene groups 1s simply understood 1n terms of the restricted set of geometries available on the lattice, and to the reduced number of lnteractlorl sites in the model chams Note that the model we have described always has states of negative I/accessible to It as long as ~~~ < 0. Further )ztt WI.U generally be much larger than tzth or rzhh. Thus ett and ug will control the dominant part of the statlstical propertles of the aggregate. Superficial properties will be more sensitive to ‘zh and Ehh
3. Numerical
study
We have studled nuceLle-lke aggregates of these chains by a Monte Carlo technique. A trral move was a random choice between a reptatlon move [6] with equal probabiht1es for head and tail dlre&ons, and a move which corresponded to fhpp1ng a kmk intenor to the chain. For the latter case, a three-bond sequence III a gauche conformation was associated with d trial con figuration of the three-bond sequence as are the nurror halves of a cyclohexane ring m the chazr conformation. This type of move can change the total number of gauche bonds and the underlying Markov chain is not symmetric. Therefore, some care has to be taken zn
437
CHEMICAL PHYSICS LETTERS
Volume 79, number 3 writing out the sampling Frocedure.
However, thus err-
cumstance can be dealt with IIIa standard way [i’] . We took r? = 4. As discussed above, this means we are modeling molecules of geometry sin&r to single cham C-8 amphiphiles. The tall-Ml interaction energy was set at -3k,T, in accordance wrth the above discusson. The head-head and tad-head interactions were set at 2k,T and 0, respectively; we beheve that these somewhat arbrtrary choices do not substantively affect our conclusions_ The gauche bond energy, ug, was chosen so that a free n = 4 chain on the lattrce (wrth spacmg 2 = 3.7 A, a typical mteratomic drstance) would have the -me mean square end-toend distance as a free octane molecule with gauche bond energy kgT. The appropriate value was found to be US = --1 Sk,?-. We exanuned aggregates of 20,30, and 50 chains. h&al states were taken to be srrnple ordered arrangements of the charms, or were constructed from config.-uratrons observed during previous runs. The aggregates were typically aged for 2 X 106 attempted moves. Averages were then calculated from the following (2-4) X lo6 attempts Because these aggregates are dense, and the trial moves are large, the likeiihood of
a successful move IS relatively low. For the fifty-&am aggregate the success rate was *6%. However the strutture of the aggregates varied widely during the course of a run. Especially for the smaller aggregates, 106 attempts were sufficient to rotationally randomrze the moment of mertia tensor- In table 1 we present measured values of the energy per molecule, the root-mean-
length of the molecules, and a few other quantities. We also determined the density profiles for both head and tarl grolrps as a function of the distance from the center of mass of the system.
square
4. Results We find that the configurations assumed by our model are not descrrbed very weU by any of the conventronal concepts of rnicelle structure. Thrs is evrdent from the density profile for the head groups, presented in fig. 1. As far as we know, quantitative predictron of this type of property has not been previously attempted. However, the usual concepts of nucelle structure have the heads farrly well locahzed rn a surface region. The head density profile we observe for our model 1s broader than what might have been expected from this picture. Of course, this type of behavror depends on the mteractrons involved, but It IS undoubtedly realized rn some crrcumstances. These observations are hkely to be very important to theones for electrostatrc effects which depend on the charge drstnbutron in the aggregateThere are several factors whrch contrrbute to the breadth of these densrty profiles. Natural size and shape fluctuatxons of the micelle would broaden these distrx-
1.4 1.2
Table 1
UlNkBT
10) a) fi2) a) d3>
a)
nhh/N QhfN
nttJN
SJN b) ShfN b) w*)/tr2)o c)
1.0
N=20
N=30
-764iOO2 3.35 x 102 1.76 x 102 1.10 x 102 0 Of7 0 250 2 13 5.21 2.72 0 98
-7.90 = 6.95 x 3 51 x 207 x 0 024 0 328 2.23 4 84 2 62 0.98
N=50 0 a1 IO2 102 IO2
--804*OOI 3.29 x 103 6.45 X lo2 4.32 X IO2 0.035 0 368 2 28 4.63 2 56 0.98
aj See text for de~Ntxon_Ali lengths are N unm of I b) Defined by eqs (5)-(7). c) tr’)c is the mean square length of an w&ted chain molzcde.
438
I May 1981
i-W& 0.6 0.4 0.2 Ct.0
0.0
1.0
2.0
3.0 r
4.0
5.0
6.0
Fg 1. Density profile for heads flower curve) and total densty (upper curve) from center of mass. N = 30. lengths are m unitsof I, and p. IS the diamond lattice number density. i e. for a perfect diamond lattice p(r) = ~0.
Volume 79, number 3
CHEMICAL PHYSICS LETTERS
1 May 1981
We also found very extensrve contact between tall groups and solvent _ The average number of contacts, S, between the micelle and solvent can be expressed m terms of the tz,_ defined by eq. (1). S = N[6
+ 2(1r - 2)] - 2(n,,
+ frth + “hh)
.
(5)
Here n 1s the number of groups m each molecule. the total number of contacts, S, there are sh = 3N - 2”hh contacts St = N[3
Fig. 2 Observed amfiiuratlon for IV = 30 Spheres are drawn wxth diameter 1. and head groups are shaded.
butions even rf the shapes observed were fauly smooth. However the configuratlons observed were actually quite rough and irregular. Frg. 2 shows an arbrtrarrly chosen configuration for an aggregate of thirty molecules. As can be seen head groups can easily be found well msrde and well outside a surtably defined average spherical surface. Finally, the average shape observed durmg our calculatron was aspherical, even for the smaller micelles. To show thrs, we calculated the principal values of the moment of inertra tensor at each configuratron, ranked them in order of magmtude, and averaged over the run. Thrs information is contamed in table 1. The tensor I, referred to as the moment of mertia tensor, was defined as (4) Here r is the vector displacement from the center of mass of the aggregate, and p(r) IS the number densrty. The eigenvalues of I in descending order of size are denoted by I(r), 1c2), and I(3). R ecall that these runs are long enough that for N = 20,30 the moment of inertia tensor calculated m a laboratory fixed frame was found to be spherical because of rotational averaging_ As can be seen from table 1, for these aggregates the largest average principal moment is about twrce as large as the next largest value. For N = 50, the elongation is more extreme; when typical configurations of the fifty-molecule aggregate were displayed graphically, the micelle generally appeared as a stubby, but drstinctly rodshaped Object.
between
- rrt,, head groups
Of
(6) and solvent,
+ 2(t1 - 2)] - 2ntt - 11th = S - S,
and (7)
contacts between tail groups and soivent . These quantities are shown m table 1 For n = 4, there are 10 possible contacts per molecule, and about half of them, or five, are to solvent. Of these five, shghtly more than half are associated with the head group, and shghtly less than half with the tail portion of the molecule. SrrmIarly, about half of the surface of the mrcelle exposed to the solvent IS non-head group. A further unexpected result of our calculatron is that the average length of the molecules IS somewhat shorter m rhe aggregate than free rn solution. ti IS al. so shown in table 1. Although the avarlable experrmental determinatrons of &am molecule conformatron in mrcelles are somewhat discrepant, the experrmental results seem to indrcate an extension of the chains due to aggregation [8,9]. Such conclusrons certamly depend on whether long-range forces operate between the head groups, and also somewhat on chain size Both of these factors can be simply mvestrgated m our model, and undoubtedly m experrmental work. in conclusron, we note that because of +he simphcity of our model these calculations are much smaller in scale than usual computer expenments on flurds. T!ms. it should be quite feasrble to do the larger calculations necessary to determine thermodynamic properties such as the cluster size datributron.
Acknowledgement T& research was supported by the National Resource for Computation in Chenustry under a grant from the National Science Foundation and the Basic Energy Sciences Divrsion of the U.S. Department of Energy (Contract No. W-7405ENG48). Acknowledge439
Volume
19, number
3
CHEMICAL
PHYSICS
is made to the Donors ot the Petroleum Research Fund, adnunistered by the Amencan Chemical Society, and to the National Sciexe Foundation (NSF CHE-7826101) for partlal support of tlus work. ment
[l ] H Wennerstrom
[2]
440
C. Tanford, 1973)
and B Lmdman. Phys. Rept 52 (1979) The hydrophobic effect (Wdey, New York,
1
1 May 1981
F-M Menger, Accounts Chem Res 12 (1979) 1 II. J D. Weeks and G H. Gdmer, Advan. Chem. Phys. 40 (1979) 157. [S 1 J.P. Wckaert and A. Bellemans, Chem. Phys Letters 30 (1975) 123 [6] hf. Bishop, D. Ceperley, H L. Frlschand M.H. K~os, J. Chem Phys. 72 (1980) 3228. [7 ] J-C Owl&i and H.A. Scheraga. Chem. Phys. Letters 47 (1977) 600. [S] B a. Persson, T Drakenberg and B Lmdman, J. Phys Chem 80 (1976) 2124. [91 J B Rosenholm. K. Larsson and N. Dmk-Nguyen. Collold Polymer SCI 255 (1977) IG98. [3] [4]
References
LETTERS
Volume 79. number 3
MONTE Steven
CARLO
STUL’Y OF A SiMPLE
W HAAN and Lawrence
MODEL
1 May 1981
FOR MECELLE STRUCTURE
R. PRATT
Department of Chenusny. Unwersrtyof Chkfomra.Berkeky. Gzitfomm 94720. USA Recerved 19 January 1981
A smple lattice model for a mxelJe ISformulated and studied by hlonte Carlo numerical methods. Results are presented for awegates of 20,30, and 50 cham molecules The density profiles are much broader than ISusually assumed in conventlonal pictures of nucelle structure and the average shapr found to be dlstmctly asphencal.
1. Introduction A molecular understandmg of surfactant aggregation in aqueous solutron has been the obJect of an enormous research effort. These systems extubrt a vanety of fascmating phenomena [ I] some of whrch have well appreciated apphcatron to crude oli recovery processes. Of course, rt IS also a wrdely held opmion that a molecular understandrng of nucelle fomiatron wti provrde an mcrsrve view rnto problems of orgaruzatron and structure of macromolecules of biological rmportance I1 321. In sprte of the obvrous motivatron and effort, there IS a long list of questlons about molecular Features of micelIes which have not been answered_ A cogent expression of this state of affairs is gwen by Menger [3]. These questions focus on such molecular features as nucelle size and shape, the degree of surface roughness and water penetratron, the degree of non-uruformity of the nacelle interior, the ab&ty to accommodate probe molecules, and the nature of the &am packmg in the interror. More broadly, rt IS unportant to understand the dehcate balance of forces which are responsrble for the relatively abrupt appearance of micelles when the surfactant concentration passes a critical micelle concentration. We have formulated a srnrple model of mrcelles whrch we believe to capture the statrstical features common to most miceliar systems, and which allows general answers to the questrons posed above. Here we present the results of a preliminary Monte Carlo 436
study of the model. Our rdea of how a model stands to a real micelIar solution is roughly the same as the relatron between the king model and a real ferromagnet, or as the relatron between some lattice models and real crystal-vapor rnterfaces [4] As Menger has shown, our understandrng of these systems IS sufficiently prmutrve that model burldrng can stih provide nnportant msrghts In the next two sections we describe our mode!, and the numerrcal methods whrch we use to study rt. In sectron 4 we present and drscuss the results of our inrtral calculations.
2. Model Here we descrrbe the model we have chosen for detarled study. Consrder a dramond lattice and denote by I the distance between near neighbor lattice pomts. A charn molecule of n - 1 bonds can be strung on the Iattree with the n atoms situated at lattrce pomts. One of the two end groups is desrgnated a head group, and the remaming are called tail groups. In more detaded models the head group would be charged or polar. AU vacant lattice points are considered to be occupied by solvent (water) molecules. We chose to study the diamond Iattrce rather than, say, a srmple cubic lattice not because normal alkane molecules fit naturally on the diamond lattice. The important consideration is that the diamond Lattice affords 12 possible orientational states for a completely
extended cham molecule, compared to 6 for the szmple cubic lattice. Thus one expects the dzamond lattzce structure to affect typical aggregate shapes less severely than would the simple cubic lattice structure. We consider a system of N such cham molecules on the lattice. Two chain molecules are designated as near neighbors 1f at least one group on one of the molecules has a group on the other molecule as a lattice near neighbor. Using ths notion of adJacency, we requzre that the entue system of N molecules form a connected aggregate. For each such aggregate we calcu;ate an energy, U, in the followmg way. If any pair of atoms occupy the same lattzce site, L1= 00. Othenvise, let %7
= total number of ar-_r group palrs which are near neighbors on the lattice.
Here cc, 7 may be h (head) total number of tad-head bors. Aiso define g = total number gate.
(1)
or t (tad). Thus nth IS the paus whch are near nezgh-
of gauche bonds
1n the aggrc(2)
Then c/= Ettrltt
+ ‘th?zth
I May 1981
CHEMICAL PHYSICS LETTERS
Volume 79, number 3
+ ehnw
+ u@ -
(3)
The probabdlty The e&y and ug are energy parameters. of observmg the configuration of the connected aggregate is proportional to exp(-CJ/kgT). For consistency the probab1hty of a configuratzon wluch disconnects the aggregate 1s set equal to zero. This IS because such a configuration is properly assigned to a different aggregate type. The lattice spacing, I, should correspond to a typlcal intermolecular datance. Since the mtramolecular near-neighbor distance m a cham molecule 1s connderably less than this, the most realistic interpretation of the model IS achieved by considering our chains of M lattice sates to model molecules of more than IZ groups. For example, with n = 4 the model chain has a length/ width ratio similar to jz-octane. The values given to the energy parameters rn eq. (3) have to be given careful thought. One of the most mpottant features of such a model 1s the poss1bihty of a clear and simple connection between these energy parameters and observed aggregation properties. Since we are not trymg to accurately represent any one particular surfactant-solvent system, but only to con-
struct a general micelle model, some soptisticat1on IS required rn estabhshmg reasonable values for these parameters. It is clear that we want Ett < 0. Eth > Ett, and ehh > 0 in order for the aggregate to be fairly stable, and to have the head groups on the outnde, on the average To estimate ett, we consider a hmlt u-t which the model can be taken as representing a bulk hydrocarbon hquld. We put Eth = ett, CZ,.,,,= eft, drop the connectivity restriction, and consider N large. Ifich normal alkane liquid the model corresponds to, and the appropnate density for the model 1s fived by taking our lattice bond length, I, to be the distance of closest approach, and the maximum iengrh of the lattice chain molecule to be the mawmum length of the correspondmg real normal alkane. Then the mtermoleculat internal energy assoczated with the model 1s est1mated by mean field theory, and Identified with the mtermolecular internal energy of the real hqu1d. The latter quantity can be obtamed from available computer slmulat1ons of n-alkane hqu1ds [5]. In thus way we estimate ett/kB = -800 K. The difference between the estimated ett and the real dispersion forces between, say, methylene groups 1s simply understood 1n terms of the restricted set of geometries available on the lattice, and to the reduced number of lnteractlorl sites in the model chams Note that the model we have described always has states of negative I/accessible to It as long as ~~~ < 0. Further )ztt WI.U generally be much larger than tzth or rzhh. Thus ett and ug will control the dominant part of the statlstical propertles of the aggregate. Superficial properties will be more sensitive to ‘zh and Ehh
3. Numerical
study
We have studled nuceLle-lke aggregates of these chains by a Monte Carlo technique. A trral move was a random choice between a reptatlon move [6] with equal probabiht1es for head and tail dlre&ons, and a move which corresponded to fhpp1ng a kmk intenor to the chain. For the latter case, a three-bond sequence III a gauche conformation was associated with d trial con figuration of the three-bond sequence as are the nurror halves of a cyclohexane ring m the chazr conformation. This type of move can change the total number of gauche bonds and the underlying Markov chain is not symmetric. Therefore, some care has to be taken zn
437
CHEMICAL PHYSICS LETTERS
Volume 79, number 3 writing out the sampling Frocedure.
However, thus err-
cumstance can be dealt with IIIa standard way [i’] . We took r? = 4. As discussed above, this means we are modeling molecules of geometry sin&r to single cham C-8 amphiphiles. The tall-Ml interaction energy was set at -3k,T, in accordance wrth the above discusson. The head-head and tad-head interactions were set at 2k,T and 0, respectively; we beheve that these somewhat arbrtrary choices do not substantively affect our conclusions_ The gauche bond energy, ug, was chosen so that a free n = 4 chain on the lattrce (wrth spacmg 2 = 3.7 A, a typical mteratomic drstance) would have the -me mean square end-toend distance as a free octane molecule with gauche bond energy kgT. The appropriate value was found to be US = --1 Sk,?-. We exanuned aggregates of 20,30, and 50 chains. h&al states were taken to be srrnple ordered arrangements of the charms, or were constructed from config.-uratrons observed during previous runs. The aggregates were typically aged for 2 X 106 attempted moves. Averages were then calculated from the following (2-4) X lo6 attempts Because these aggregates are dense, and the trial moves are large, the likeiihood of
a successful move IS relatively low. For the fifty-&am aggregate the success rate was *6%. However the strutture of the aggregates varied widely during the course of a run. Especially for the smaller aggregates, 106 attempts were sufficient to rotationally randomrze the moment of mertia tensor- In table 1 we present measured values of the energy per molecule, the root-mean-
length of the molecules, and a few other quantities. We also determined the density profiles for both head and tarl grolrps as a function of the distance from the center of mass of the system.
square
4. Results We find that the configurations assumed by our model are not descrrbed very weU by any of the conventronal concepts of rnicelle structure. Thrs is evrdent from the density profile for the head groups, presented in fig. 1. As far as we know, quantitative predictron of this type of property has not been previously attempted. However, the usual concepts of nucelle structure have the heads farrly well locahzed rn a surface region. The head density profile we observe for our model 1s broader than what might have been expected from this picture. Of course, this type of behavror depends on the mteractrons involved, but It IS undoubtedly realized rn some crrcumstances. These observations are hkely to be very important to theones for electrostatrc effects which depend on the charge drstnbutron in the aggregateThere are several factors whrch contrrbute to the breadth of these densrty profiles. Natural size and shape fluctuatxons of the micelle would broaden these distrx-
1.4 1.2
Table 1
UlNkBT
10) a) fi2) a) d3>
a)
nhh/N QhfN
nttJN
SJN b) ShfN b) w*)/tr2)o c)
1.0
N=20
N=30
-764iOO2 3.35 x 102 1.76 x 102 1.10 x 102 0 Of7 0 250 2 13 5.21 2.72 0 98
-7.90 = 6.95 x 3 51 x 207 x 0 024 0 328 2.23 4 84 2 62 0.98
N=50 0 a1 IO2 102 IO2
--804*OOI 3.29 x 103 6.45 X lo2 4.32 X IO2 0.035 0 368 2 28 4.63 2 56 0.98
aj See text for de~Ntxon_Ali lengths are N unm of I b) Defined by eqs (5)-(7). c) tr’)c is the mean square length of an w&ted chain molzcde.
438
I May 1981
i-W& 0.6 0.4 0.2 Ct.0
0.0
1.0
2.0
3.0 r
4.0
5.0
6.0
Fg 1. Density profile for heads flower curve) and total densty (upper curve) from center of mass. N = 30. lengths are m unitsof I, and p. IS the diamond lattice number density. i e. for a perfect diamond lattice p(r) = ~0.
Volume 79, number 3
CHEMICAL PHYSICS LETTERS
1 May 1981
We also found very extensrve contact between tall groups and solvent _ The average number of contacts, S, between the micelle and solvent can be expressed m terms of the tz,_ defined by eq. (1). S = N[6
+ 2(1r - 2)] - 2(n,,
+ frth + “hh)
.
(5)
Here n 1s the number of groups m each molecule. the total number of contacts, S, there are sh = 3N - 2”hh contacts St = N[3
Fig. 2 Observed amfiiuratlon for IV = 30 Spheres are drawn wxth diameter 1. and head groups are shaded.
butions even rf the shapes observed were fauly smooth. However the configuratlons observed were actually quite rough and irregular. Frg. 2 shows an arbrtrarrly chosen configuration for an aggregate of thirty molecules. As can be seen head groups can easily be found well msrde and well outside a surtably defined average spherical surface. Finally, the average shape observed durmg our calculatron was aspherical, even for the smaller micelles. To show thrs, we calculated the principal values of the moment of inertra tensor at each configuratron, ranked them in order of magmtude, and averaged over the run. Thrs information is contamed in table 1. The tensor I, referred to as the moment of mertia tensor, was defined as (4) Here r is the vector displacement from the center of mass of the aggregate, and p(r) IS the number densrty. The eigenvalues of I in descending order of size are denoted by I(r), 1c2), and I(3). R ecall that these runs are long enough that for N = 20,30 the moment of inertia tensor calculated m a laboratory fixed frame was found to be spherical because of rotational averaging_ As can be seen from table 1, for these aggregates the largest average principal moment is about twrce as large as the next largest value. For N = 50, the elongation is more extreme; when typical configurations of the fifty-molecule aggregate were displayed graphically, the micelle generally appeared as a stubby, but drstinctly rodshaped Object.
between
- rrt,, head groups
Of
(6) and solvent,
+ 2(t1 - 2)] - 2ntt - 11th = S - S,
and (7)
contacts between tail groups and soivent . These quantities are shown m table 1 For n = 4, there are 10 possible contacts per molecule, and about half of them, or five, are to solvent. Of these five, shghtly more than half are associated with the head group, and shghtly less than half with the tail portion of the molecule. SrrmIarly, about half of the surface of the mrcelle exposed to the solvent IS non-head group. A further unexpected result of our calculatron is that the average length of the molecules IS somewhat shorter m rhe aggregate than free rn solution. ti IS al. so shown in table 1. Although the avarlable experrmental determinatrons of &am molecule conformatron in mrcelles are somewhat discrepant, the experrmental results seem to indrcate an extension of the chains due to aggregation [8,9]. Such conclusrons certamly depend on whether long-range forces operate between the head groups, and also somewhat on chain size Both of these factors can be simply mvestrgated m our model, and undoubtedly m experrmental work. in conclusron, we note that because of +he simphcity of our model these calculations are much smaller in scale than usual computer expenments on flurds. T!ms. it should be quite feasrble to do the larger calculations necessary to determine thermodynamic properties such as the cluster size datributron.
Acknowledgement T& research was supported by the National Resource for Computation in Chenustry under a grant from the National Science Foundation and the Basic Energy Sciences Divrsion of the U.S. Department of Energy (Contract No. W-7405ENG48). Acknowledge439
Volume
19, number
3
CHEMICAL
PHYSICS
is made to the Donors ot the Petroleum Research Fund, adnunistered by the Amencan Chemical Society, and to the National Sciexe Foundation (NSF CHE-7826101) for partlal support of tlus work. ment
[l ] H Wennerstrom
[2]
440
C. Tanford, 1973)
and B Lmdman. Phys. Rept 52 (1979) The hydrophobic effect (Wdey, New York,
1
1 May 1981
F-M Menger, Accounts Chem Res 12 (1979) 1 II. J D. Weeks and G H. Gdmer, Advan. Chem. Phys. 40 (1979) 157. [S 1 J.P. Wckaert and A. Bellemans, Chem. Phys Letters 30 (1975) 123 [6] hf. Bishop, D. Ceperley, H L. Frlschand M.H. K~os, J. Chem Phys. 72 (1980) 3228. [7 ] J-C Owl&i and H.A. Scheraga. Chem. Phys. Letters 47 (1977) 600. [S] B a. Persson, T Drakenberg and B Lmdman, J. Phys Chem 80 (1976) 2124. [91 J B Rosenholm. K. Larsson and N. Dmk-Nguyen. Collold Polymer SCI 255 (1977) IG98. [3] [4]
References
LETTERS