Free Energy Change of Micelle Formation for Sodium Dodecyl

物理化学学报

Acta Phys. -Chim. Sin. 2018, 34 (10), 1163–1170 1163

Received: December 25, 2017; Revised: February 2, 2018; Accepted: February 19, 2018; Published online: February 27, 2018. *Corresponding authors. Email: [email protected]; Tel./Fax: +81-52-788-6213 (N.Y.). Email: [email protected];

Tel./Fax: +81-52-789-5829 (S.O.).

This work was supported by FLAGSHIP2020, MEXT within Priority Study 5 (Development of New Fundamental Technologies for High-Efficiency Energy

Creation, Conversion/Storage and Use) Using Computational Resources of the K Computer Provided by the RIKEN Advanced Institute for Computational

Science through the HPCI System Research Project (hp170241). This work was also funded by MEXT KAKENHI Grant Number 17K04758 (N.Y.).

© Editorial office of Acta Physico-Chimica Sinica

[Article] doi: 10.3866/PKU.WHXB201802271 www.whxb.pku.edu.cn

Free Energy Change of Micelle Formation for Sodium Dodecyl Sulfate from a Dispersed State in Solution to Complete Micelles along Its Aggregation Pathways Evaluated by Chemical Species Model Combined with Molecular Dynamics Calculations

YOSHII Noriyuki 1,2,*, KOMORI Mika 2, KAWADA Shinji 2, TAKABAYASHI Hiroaki 2, FUJIMOTO Kazushi 2, OKAZAKI Susumu 1,2,*

1 Center for Computational Science, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan. 2 Department of Applied Chemistry, Nagoya University, Nagoya 464-8603, Japan.

Abstract: Surfactant molecules, when dispersed in solution, have been shown

to spontaneously form aggregates. Our previous studies on molecular dynamics

(MD) calculations have shown that ionic sodium dodecyl sulfate molecules

quickly aggregated even when the aggregation number is small. The

aggregation rate, however, decreased for larger aggregation numbers. In

addition, studies have shown that micelle formation was not completed even

after a 100 ns-long MD run (Chem. Phys. Lett. 2016, 646, 36). Herein, we

analyze the free energy change of micelle formation based on chemical species

model combined with molecular dynamics calculations. First, the free energy

landscape of the aggregation, ∆G†i+j, where two aggregates with sizes i and j

associate to form the (i + j)-mer, was investigated using the free energy of

micelle formation of the i-mer, Gi†, which was obtained through MD calculations.

The calculated ∆G†i+j was negative for all the aggregations where the sum of DS

ions in the two aggregates was 60 or less. From the viewpoint of chemical equilibrium, aggregation to the stable micelle is

desired. Further, the free energy profile along possible aggregation pathways was investigated, starting from small

aggregates and ending with the complete thermodynamically stable micelles in solution. The free energy profiles, G(l, k), of

the aggregates at l-th aggregation path and k-th state were evaluated by the formation free energy i ii

n Gl,k † and the

free energy of mixing i i

i

n k Tlnl,k l,k ln ,n kB( ) ( ( ) / ( )) , where ni(l, k) is the number of i-mer in the system at the l-th

aggregation path and k-th state, with i

i

l,k ln ,k= n . All the aggregation pathways were obtained from the initial

state of 12 pentamers to the stable micelle with i = 60. All the calculated G(l, k) values monotonically decreased with

increasing k. This indicates that there are no free energy barriers along the pathways. Hence, the slowdown is not due to

the thermodynamic stability of the aggregates, but rather the kinetics that inhibit the association of the fragments. The time

required for a collision between aggregates, one of the kinetic factors, was evaluated using the fast passage time, tFPT.

The calculated tFPT was about 20 ns for the aggregates with N = 31. Therefore, if aggregation is a diffusion-controlled

process, it should be completed within the 100 ns-simulation. However, aggregation does not occur due to the free energy

barrier between the aggregates, that is, the repulsive force acting on them. This may be caused by electrostatic repulsions

produced by the overlap of the electric double layers, which are formed by the negative charge of the hydrophilic

1164 Acta Physico-Chimica Sinica Vol. 34

groups and counter sodium ions on the surface of the aggregates.

Key Words: Free energy change; Aggregation pathway; SDS; Micelle; Molecular dynamics calculation

1 Introduction Stability of micelles in solution is of great interest in basic

chemistry including the fields of amphiphilic molecular

aggregates, such as biomolecules. A large number of theoretical

and experimental studies have been conducted so far 1–6. First,

regarding the stability of micelles, Tanford 1 and Israelachivili

et al. 2 proposed models based on the attractive force between

hydrophobic groups and repulsive force between hydrophilic

groups. Further, Everette developed a model 3 that incorporates

micellar surface contributions. In addition to these studies,

Blankschtein et al. 4, Oxtoby et al. 5, and Chandler et al. 6

proposed detailed thermodynamic and statistical mechanical

models that include intermolecular interactions of surfactant

molecules. It is now possible to accurately reproduce the free

energy, aggregation number, and critical micelle concentration

(CMC) of the micelles.

From the viewpoint of computational science, the free energy

of micelle formation and micelle size distribution have also

been investigated by molecular dynamics (MD) calculations 7.

Furthermore, using all-atomistic and coarse-grained MD/Monte

Carlo calculations, the aggregation process of surfactant

molecules dispersed in water was examined to evaluate the size

distribution of the aggregates 8–22. We also performed

non-equilibrium MD calculations for anionic sodium dodecyl

sulfate (SDS), cationic dodecyltrimethylammonium chloride

(DTAC), zwitterionic dodecyldimethylamine oxide (DDAO),

and nonionic octaethylene glycol monododecylether (C12E8)

surfactant molecules with an alkyl chain of 12 carbon atoms

and n-dodecane molecules 23,24. The aggregation number, i, of

C12E8 and n-dodecane increased in proportion to the elapsed

time t. This indicates that the aggregation obeys the

Lifshitz-Slyozov (LS) law 25 and is a diffusion-controlled

process. In contrast, the aggregation showed non-LS behavior

for SDS and DTAC (I ∝ t0.3), and for DDAO (I ∝ t0.6).

Among them, formation of stable micelles was completed

smoothly for nonionic C12E8 within 50 ns. However, it was not

completed for ionic SDS and DTAC in spite of the 100-ns-long

MD calculations where a few aggregates composed of several

tens of surfactant molecules remained unassociated with each

other in solution.

In this study, thermodynamic stability is evaluated for the

SDS aggregates of intermediate size on the way to a complete

micelle. A molecular structure of the SDS is shown in Fig. 1. If

there are a high-free-energy aggregate on the way to the

complete micelle, it takes long time for the aggregates to reach

the complete micelle. In order to investigate possibility of such

unstable aggregates, all aggregation pathways were examined

and free energy changes along these pathways were evaluated

based on the chemical species model. There, we use free energy

of formation of the aggregates which were obtained in our

previous study 7 based on thermodynamic integration method

and MD calculations. We also investigated a kinetic factor in

the association process. For the association to proceed, the

aggregates need to collide with each other by diffusion. In the

present study, the collision rate was estimated by the fast

passage time 26 based on the Smolchowski equation. By

evaluating the thermodynamic stability of the aggregates

together with the collision rate of the aggregates, we can obtain

the microscopic rate determining factors that control the

association process of the aggregates.

2 Theory In our previous study, free energy of formation was

evaluated as a function of aggregation number 7 for SDS

aggregate using MD calculations. In the present study, the free

energy of aggregation was evaluated for all possible SDS

fragments using our previous result.

2.1 Chemical species model

Free energy of aggregation of i surfactant molecules (an

“i-mer”) from monomers can be evaluated based on the

chemical species model. In this model, aggregates with

different aggregation numbers are regarded as different

chemical species. An isolated surfactant molecule in vacuum

was regarded as the reference state. Let μi0 be the chemical

potential of the fully hydrated i-mer in the infinite dilute

solution. Since there are no aggregates other than i-mer, the

mole fraction Xi of the i-mer is unity. Then, the chemical

potential, μi, of the i-mer can be expressed by

0B lni i i ik T X (1)

where kB is the Boltzmann constant, T the absolute temperature,

and γi the activity coefficient.

For the system composed of Ni i-mers and Nw water

molecules, free energy of the total system can be described by

w wi i

i i

G N N (2)

Fig. 1 The molecular structure of SDS.

Yellow: sulfur atom, red: oxygen atom, cyan: carbon atom, gray:

hydrogen atom, and blue: sodium ion. Color online.

No. 10 doi: 10.3866/PKU.WHXB201802271 1165

where μw is the chemical potential of water.

2.2 Equilibrium mole fraction of i-mer

We assume that the aggregation reaction of one (i − 1)-mer

and one monomer to form one new i-mer occurs reversibly. We

also assume that other aggregates in solution do not change.

Then, the free energy change, ΔG, is expressed using Eqs. (1)

and (2) as

0B B

1 1 1 1ln lni i

ii i

XG k T k T

X X

(3)

where Δμi0 = μi

0 − (μ0i−1 + μ1

0) is the free energy of association

between one (i − 1)-mer and one monomer forming one new

i-mer at infinite dilution (Fig. 2). Δμi0 has already been

evaluated as a function of i by a thermodynamic integration

calculation based on the MD calculation in our previous study 7.

In Table 1, the coefficient ak obtained by fitting the polynomial

0

kk

k

a i to Δμi

0 is listed for two regions (1 ≤ i < 66 and 66 ≤

i ≤ 80).

If the system is in equilibrium, ΔG = 0. Then, the mole

fraction, Xieq, in the equilibrium state satisfies,

eq

eq 0 1 11eq

1

expi ii

ii

XX

X

(4)

as easily derived from Eq. (3).

2.3 Activity coefficient of i-mer

Debye-Hückel (DH) theory is used to evaluate the factor

1 1i

i

in Eq. (4). According to DH theory, lnγi∝i2 for

the activity coefficient γi of ions with i charges 27. Using this expression, we obtain the relationship

B1 1

ln 1i

ik T i

(5)

where the coefficient α was determined such that it reproduces

the experimental critical micelle concentration (CMC) 7.

Regarding Xieq, X1

eq, and α as unknown variables, we obtain

from Eq. (4)

eq eq eq 01 1

2

eq1

2

†

, exp 1

exp

ii

kik

ii

kk

X X X k

X

(6)

where Δμk† = Δμk

0 + α(k−1). Further, we obtain a relation

between CCMC and Xieq

eq eqCMC 1

1

,ii

C iX X

(7)

There are several definitions of CMC using Xieq 28. Here, we

adopted the definition that Xieq satisfies the condition of

eq eq eq1 1( , )iX iX X (8)

at CMC 28, where the total number of molecules belonging to

the aggregate of a certain aggregation number i (≠ 1) is the

same as the total number of molecules of the monomer. Then,

we can determine Xieq, X1

eq, and α numerically using Eqs. (6)–

(8). In the previous study 7, α = −4.5 × 10−22 J and X1eq =

0.00015 were obtained, and an aggregation number of i = 57

gave the maximum value of X1eq.

2.4 Free energy of formation of the aggregates

Excess free energy of formation Gi† of the i-mer is the sum of

the free energy of aggregation 0

2

i

k

k

at infinite dilution and

the contribution B1 12

lni

i

ik

k T

of the activity coefficient.

Gi† can be expressed using Δμk

† defined by Eq. (6) as

† †

2

i

i kk

G

(9)

Using this Gi†, Eq. (6) can be rewritten as,

eq

eq†B 1eq

1

exp 1 lnii

XG i k T X

X

(10)

Then, we can obtain the information about the size distribution eq

eq1

iX

X of the i-mer by comparing Gi

† with (i − 1) kBT ln Xieq at

each i.

Fig. 2 Δμi0 and Δμi† as a function of aggregation number i.

The value of Δμi0 obtained from MD calculation is given in Ref. 7. The black solid line

was obtained by fitting the polynomials

0

ka ik

k to the calculated Δμi

0 for two

regions (1 ≤ i < 66 and 66 ≤ i ≤ 80). The red solid line shows Δμi† in Eq. (6),

where the coefficient α was determined to reproduces the experimental CMC.

0 20 40 60 80-1.5

-1

-0.5

0

0.5

i0 ,

i†

(10

-19 J

)

i†

i0

i

Table 1 Fitting coefficients, ak, of Δμi0, where Δμi0 is

approximated as 0

0

kΔμ = a ikik=

.

1 ≤ i ≤ 66 66 ≤ i ≤ 80

a0 4.11 × 10−21 7.11 × 10−21

a1 −5.51 × 10−21 −4.81 × 10−21

a2 3.73 × 10−22 1.21 × 10−22

a3 −1.73 × 10−23 −1.21 × 10−24

a4 4.24 × 10−25 4.00 × 10−27

a5 −5.31 × 10−27 −

a5 3.24 × 10−29 −

a7 −7.65 × 10−32 −


The excess free energy of aggregation, ΔG†i+j, where two

aggregates with sizes i and j associate to form the (i + j)-mer

can be given by

† † † †i j i j i jG G G G (11)

Then, free energy of aggregation, ΔGi+j, may be obtained by

adding ideal free energy of mixing

eq

mixB eq eq

, lni j

i ji j

XG i j k T

X X

(12)

to the excess free energy of aggregation, Eq. (11), as † mix

i j i ji jG G G (13)

This determines whether the aggregation actually goes

forward or not.

3 Results and discussion In this section, free energy of aggregation of an i-mer and

j-mer to form (i + j)-mer is presented. The free energy profile is

also investigated along aggregation pathways, and then,

kinetics in the final stage of the aggregation is discussed.

3.1 Free energy of micelle formation

As shown in Fig. 2, free energy of micelle formation, Δμi0

and Δμi†, rapidly decrease with increasing i for i ≤ 10. This

stability is caused by the increase of coverage of the aggregate

surface by hydrophilic groups with increasing aggregation

number 29. In contrast, the instability for i ≥ 40 is due to the

decrease of the distance between hydrophilic groups 30.

Fig. 3 shows the free energy of formation, G i†, of the i-mer

calculated by Eq. (9). The G i† shows an inverse S-shaped

function with an inflection point near i = 40 where Δμi† in Fig. 2

has a minimum. The shape of this G i† curve is similar to the

reported model functions of the free energy of micelle

formation 2,3; it is convex upward for 1 ≤ i < 40 and convex

downward for 40 ≤ i ≤ 80. Now, X1eq is constant in the range of

0 < X1 eq < 1, such that (i − 1) kBT ln X1

eq is a monotonically

decreasing linear function of i, as shown in Fig. 3. Probability

that the i-mer is found in the system is determined by the

relative magnitude of both G i† and (i − 1) kBT ln X1

eq, as shown

in Eq. (10). Thus, the aggregates with 1 ≤ i < 40 are not found

because of the condition, G i† − (i − 1) kBT ln X1

eq > 0. In

contrast, the aggregate with 40 ≤ i ≤ 80 must be found in the

system because of the condition G i† − (i − 1) kBT ln X1

eq < 0. G i†

in Fig. 3 shows such tendency, giving a correct micelle size

distribution.

G i† − (i−1) kBT ln X1

eq at several concentrations, C, is plotted

in Fig. 3. Here, we assume that G i† does not depend on C for C <

30 CCMC. For C < CCMC, the minimum is found at i = 1,

indicating that the monomer is present most frequently in the

system. At C = CCMC, local minima are observed at i = 1 and

57, and the monomer and the stable micelle, respectively, are

distributed to a similar extent. For C > CCMC, the minimum at i =

57 decreases as C increases. This indicates that the micelle with

i = 57 becomes more stable and is found more frequently in the

system.

In Fig. 3, there is a broad high free energy region around i =

20. The aggregation between the two aggregates with similar

sizes (i ≈ 20) hardly occurs at CCMC because the probability that

the aggregates with i ~ 20 are found in the system is much

smaller than that with i = 1 and 57. This is consistent with the

Aniansson-Wall model 31, where aggregation proceeds through

the sequential reaction in which the aggregate size increases by

addition of monomers.

3.2 Free energy of aggregation of the i-mer and j-mer

Fig. 4 shows a free energy landscape of aggregation, ΔG†i+j,

of an i-mer and j-mer given by Eq. (11). ΔG†i+j is negative for

all i and j where the two aggregates satisfy i + j ≤ 60. The

aggregation under this condition proceeds thermodynamically.

In particular, ΔG†i+j becomes a large negative value when the

two aggregates are of similar size with i ≈ 30. This is because,

as shown in Fig. 3, the aggregates with i ≈ 30 are greatly

unstable, while the aggregates with i ≈ 60 are quite stable.

In contrast, in the aggregation of the large fragments of i ≥

60 (for example, two 60-mers are fused into one to form a

120-mer), ΔG†i+j has a large positive value. In such an

aggregation, the hydrophilic groups in the formed aggregate

Fig. 3 Aggregation number i dependence of G†i − (i − 1) kBT lnX1

eq.

Gi†: black line, G i

† − (I − 1) kBTlnX1 eq at 0.5CMC, CMC, and 30CMC are depicted by the

blue, red and green lines, respectively. The black dashed line represents (I − 1)

kBT ln X1 eq at CMC. Color online.

0 20 40 60 80-4

-3

-2

-1

0

1

Gi†

(10

-18 J

)

i

Gi† - (i-1)kBTlnX1

eq (0.5CMC) Gi

† - (i-1)kBTlnX1eq (CMC)

Gi† - (i-1)kBTlnX1

eq (30CMC) Gi

†

Fig. 4 Free energy landscape, ΔG†i+j, of formation of

an (i + j)-mer from i-mer and j-mer.

No. 10 doi: 10.3866/PKU.WHXB201802271 1167

become close to each other on the aggregate surface 30. It is

quite unstable.

3.3 Free energy profile along possible aggregation

pathways

We consider free energy profile along possible aggregation

pathways from randomly solved structure to complete spherical

micelles. The aggregation pathways are generated randomly

according to the procedure given in Fig. S1. Here, l is the index

of independent aggregation path, and k represents the state of

the system. Then, the free energy profile, G(l, k), was evaluated

using G i† by,

†B

,, ,

,ln i

i ii

nG n G k T

n

l kl k l k

l k

(14)

where ni(l, k) is the number of i-mer in the system at l-th

aggregation path and k-th state. , ,i

i

l k ln n k is the total

number of aggregates in the system. The free energy of mixing,

Gmix(l, k), was evaluated by,

mixB ln

,, ,

,i

i

i

n lG n k T

kl k l

nk

l k (15)

We started with 12 pentamers as an initial state because the

number of aggregation paths from 60 monomers is too large to

calculate. This initial state is reasonable because at the early

stage of aggregation simulation from the randomly dispersed

molecules, the monomers quickly associated to form small

fragments with a several molecules such as pentamers. Totally

39700 aggregation paths were investigated. 800 randomly

chosen profiles, G(l, k), and the contribution of Gmix(l, k) are

plotted in Fig. 5a.

G(l, k) decreases monotonically with increasing k at all l. The

free energy does not increase with the growth of the aggregates.

Further, the absolute value of Gmix(l, k) is several tens of

kJ·mol−1, whereas that of G(l, k) is several hundreds or one

thousand kJ·mol−1. It is more than one order of magnitude larger

than Gmix(l, k). Thus, †,i ii

n Gl k is a dominant factor in

G(l, k).

Fig. 5b shows increment of the free energy ΔG(l, k) = G(l, k) −

G(l, k − 1) along the path. It is noted that they are all negative

values. Although the positive values were found that in several

ΔGmix(l, k) = Gmix(l, k) − Gmix(l, k − 1), the values are in the

order of several kJ/mol, their contributions to ΔG(l, k) being

quite limited. As a result, ΔG(l, k) is in the range of −50 to

−270 kJ·mol−1. Thus, no barrier is found in the free energy

profile from the initial state to the complete spherical micelles.

3.4 Aggregation simulation

Next, free energy change during spontaneous aggregation of

the fragments was investigated based on the aggregation

simulation. The detail of the aggregation simulation is

described in the supporting information.

Snapshots of SDS aggregates in an aggregation simulation

are shown in Fig. 6. From the trajectory of the aggregation

simulation, fragments were identified using the intermolecular

bond between the DS ions. The contact area between the

hydrophobic parts of the two DS ions was used to define the

bond between the DS ions. Voronoi polyhedra were defined

using all atoms except for hydrogen. The surface of the DS ion

was defined using the Voronoi polyhedra. When the contact

area of the two DS ions is greater than a threshold value, the

two DS ions were regarded as bonded. From the contact area

distribution of two DS ions in the stable SDS micelle, the

threshold was set to be 0.05 nm2 23.

Results of the aggregation simulation are shown in Fig. 7.

The largest aggregation number is plotted in Fig. 7a for

independent six MD runs. The aggregates with i = 10–20 were

quickly formed within several tens of nanoseconds. However,

the aggregation did not proceed more. Four largest aggregation

numbers are plotted as a function of t in Fig. 7b for one

trajectory, green one in Fig. 7a. At t = 30 ns, three aggregates

with aggregation numbers of 12, 19, and 29 were formed.

However, their aggregation numbers hardly changed until t =

100 ns.

The time evolution of the maximum aggregation number

presented here is similar to that of aggregation simulation

performed using different potential parameters (CHARMM 36) 23.

Fig. 5 Free energy profile G(l, k) (lower plot of (a)) and Gmix(l, k) (upper plot of (a)), and ΔG(l, k) (lower plot of (b)) and ΔGmix(l, k) (upper plot of (b))

along the aggregation pathways.

Note that the scales on the vertical axes are different between (a) and (b).


This indicated that the results do not strongly depend on the

details of potential parameters. In contrast, as described in the

introduction, the time evolution of the maximum aggregation

number greatly differs depending on whether the hydrophilic

group is ionic, zwitterionic or nonionic 24. The electric charge

of the hydrophilic group has a strong influence on the rate of

aggregation.

The time evolution of the free energy G(l, t) was evaluated

by Eq. (14) for each trajectory. For all l, G(l, t) decreases with

increasing time. As described in Section 3.3, it is

thermodynamically stable to form one aggregate. However, the

aggregation was not actually completed during the 100-ns-long

simulation, and the system did not reach the equilibrium state.

A kinetic mechanism is considered to prevent the fragment

from aggregating each other.

3.5 Aggregation kinetics

The kinetics concerned with the aggregation between the

aggregates is dominated by two factors: (1) collision by

diffusion, and (2) crossing the free energy barrier along the

reaction coordinates of the aggregation.

Here, the frequency of collision between the aggregates

caused by diffusion is evaluated to examine whether the 100 ns

long simulation was sufficient for the aggregates to diffuse and

collide each other. For this purpose, we use the fast passage

time, which is often used to evaluate the reaction time in a

diffusion-controlled reaction where the reaction proceeds by

the contact of two spherical particles in solution 26. First, one

aggregate is fixed in space. The range of motion of another

aggregate is assumed to be inside a sphere of radius R centered

on the fixed aggregate, and the area outside the sphere with

radius Rm (< R), which is the sum of the radii of two aggregates

that are fixed and diffusible. Then, the fast passage time, tFPT, is

given by,

2 2 32

FPT 2

1 5 6 3

15 1

x x x xRt

D x x x

(16)

where x = Rm/R. D is the diffusion coefficient of the two

aggregates, and the sum of the self-diffusion coefficients of the

individual aggregates is commonly used. The self-diffusion

coefficient of the individual aggregate was evaluated from the

Einstein-Stokes law (D = kBT/6πηRm), where the viscosity

coefficient, η, of water at 300 K is 0.000853 Pa·s 32, and Rm is

obtained from the radius of gyration by the MD calculation

(1.34 nm at N = 31 33).

The time required for the contact between two aggregates (N =

31) in the aggregation simulation is estimated to be 20 ns from

tFPT. Therefore, several collisions should have occurred in the

last 50–70 ns. However, the aggregation did not occur for all

independent six simulations. Thus, the aggregation is not a

diffusion-controlled process.

A repulsive interaction coming from free energy barrier must

exist between aggregates. The repulsive force may be produced

by the overlap of the electric double layer formed on the

surface of the aggregates by the negative charge of the

surfactant molecules and the positive charge of the counter

ions. More than 100 ns are necessary to pass the free energy

Fig.6 Snapshot of SDS aggregates in solution at elapsed time (a) t = 0 ns, (b) 7 ns, (c) 50 ns, and (d) 100 ns.

Colors for atoms are the same as in Fig. 1. Water molecules are not depicted for clarity.

Fig. 7 Time evolution of (a) maximum aggregation number, i, of

aggregates in the independent six aggregation simulations, (b) four

largest aggregation numbers for one trajectory, green line in Fig. 7a,

and (c) free energy profile G(l, t) of the system evaluated by Eq. (14) for

each trajectory.

Results of the six independent MD runs are plotted in different colors in (a) and (c).

No. 10 doi: 10.3866/PKU.WHXB201802271 1169

barrier by to form the stable micelles.

4 Conclusions In this study, the free energy landscape, ΔG†

i+j, of aggregation

was investigated using the free energy, Gi†, of micelle formation

obtained by MD calculations. This analysis is based on

chemical species model. The calculated ΔG†i+j was negative for

all aggregations where the sum of the number of DS ions in the

two aggregates was 60 or less. From the viewpoint of chemical

equilibrium, aggregation to the stable micelle is desired.

Further, free energy profile along possible aggregation

pathways was investigated from small aggregates to

thermodynamically stable complete micelles in solution. The

free energy profile G(l, k) of the aggregates was evaluated by

†( ),i ii

n Gl k and B

,,

,ln i

i

i

nn k T

n

l kl k

l k . All aggregation

pathways were obtained from the initial state of 12 pentamers

to the stable micelle with i = 60. The calculated G(l, k) all

monotonically decreases with increasing k. The absolute value

of †( ),i i

i

n Gl k was an order of magnitude greater than that

of B

,,

,ln i

i

i

nn k T

n

l kl k

l k , dominating the aggregation

process.

Independent six MD calculations were performed for the

spontaneous aggregation process of SDS dispersed in water. In

each calculation, 10–30-mers were formed within about 20 ns.

However, the aggregation did not proceed more and was not

completed even after 100 ns simulations, despite the fact that

the SDS micelle with i ≈ 60 is the most thermodynamically

stable. For each run, the free energy decreased sharply with

time for the first 20 ns, but hardly changed after that.

The time required for a collision between aggregates was

evaluated using the fast passage time, tFPT. The calculated tFPT

was about 20 ns for the aggregates with N = 31. Thus, if

aggregation is a diffusion-controlled process, the aggregation

should be completed within the 100 ns simulation. However, in

fact, the aggregation does not proceed. This is due to the free

energy barrier between the aggregates, that is, the repulsive

force acting on them. This may be caused by electrostatic

repulsion produced by the overlap of the electric double layers

formed by the negative charge of the hydrophilic groups and

counter sodium ions on the surface of the aggregates. In order

to aggregate to form stable micelles the aggregates are required

to pass this barrier thermally.

The potential of mean force between aggregates must be

evaluated to obtain the time constant for the aggregation of two

aggregates to form the stable micelle 34. We discuss this free

energy barrier elsewhere 35, based on free energy calculation

combined with molecular dynamics calculation.

Acknowledgment: Calculations were mainly performed at

the Research Center for Computational Science, Okazaki,

Japan, partially at the Information Technology Center of

Nagoya University, and partially at the Institute for Solid State

Physics, the University of Tokyo.

Supporting Information: available free of charge via the

internet at http://www.whxb.pku.edu.cn.

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