() - 1406.1246v1.pdf

arXiv:1406.1246v1 [cond-mat.soft] 5 Jun 2014

Theory of Polymer Chains in Poor Solvent: Single-Chain Stru

cture, Solution

Thermodynamics and

Point

Rui Wang and Zhen-Gang Wang

Division of Chemistry and Chemical Engineering,

California Institute of Technology, Pasadena, CA 91125, US

Using the language of the Flory

parameter, we develop a theory that unifies the treatment of t

single-chain structure and the solution thermodynamics of

polymers in poor solvents. The structure

of a globule and its melting thermodynamics is examined usin

g the self-consistent filed theory. Our

results show that the chain conformation involves three sta

tes prior to the globule-to-coil transition:

the

fully-collapsed globule

, the

swollen globule

and the

molten globule

, which are distinguished by

the core density and the interfacial thickness. By examinin

g the chain-length dependence of the

melting of the swollen globule, we find universal scaling beh

avior in the chain properties near the

Θ point. The information of density profile and free energy of

the globule is used in the dilute

solution thermodynamics to study the phase equilibrium of p

olymer solution. Our results show

different scaling behavior of the solubility of polymers in t

he dilute solution compared to the F-H

theory, both in the

dependence and the chain-length dependence. From the persp

ectives of single

chain structure and solution thermodynamics, our results v

erifies the consistency of the Θ point

defined by different criteria in the limit of infinite chain len

gth: the disappearance of the second

viral coefficient, the abrupt change in chain size and the crit

ical point in the phase diagram of the

polymer solution. Our results show

= 0

5 (for the case of equal monomer and solvent volume),

which coincides with the value predicted from the F-H theory

I. INTRODUCTION

Flory-Huggins (F-H) theory is the cornerstone for poly-

mer solution thermodynamics

. The theory assumes

random mixing of ideal chains with solvent molecules,

and describes the polymer-solvent interaction by a phe-

nomenological parameter

. Above the critical

(or

equivalently below the critical temperature), the F-H

theory predicts phase separation of a polymer solution

into a polymer-concentrated phase and polymer-dilute

phase. From the F-H theory, des Cloizeaux and Jannink

found that the equilibrium polymer concentration in

the dilute phase well below the critical point scales as

∼−

(

−

)

. The critical

for polymer chains of

length

is given by

−

, and has the asymptotic

value of

approaches infinity. It is accepted that,

in the limit of infinite chain length, the critical point for

the phase separation of the polymer solution coincides

with the Θ point for the coil-to-globule transition of a

single chain

3–7

. Thus,

in the framework of F-H

theory in the limit of infinite chain length.

The phase diagram calculated by the F-H theory shows

poor agreement with experiments and simulation for the

dilute phase and near the critical point: the F-H theory

predicts a much lower equilibrium polymer concentration

and a lower critical

8–12

. An obvious fact that is not cap-

tured by the F-H theory is the large localized density fluc-

tuation in the dilute solution. The instantaneous picture

of the solution has much higher polymer density where

the chains are located and pure solvents elsewhere, which

is significantly different from the random-mixing picture

envisioned in the F-H theory. For polymers in poor sol-

vents, such localized density fluctuation takes the form

of single-chain globules and multichain clusters

. By

accounting for the large localized fluctuation, we showed

in previous work

that the solubility of the polymer in

the poor solvent is enhanced by several orders of magni-

tude relative to the prediction of the F-H theory – the

logarithm of the solubility scales with

rather than

as predicted from the F-H theory. The critical

the phase diagram for polymers of finite chain length was

also shown to be pushed to higher values than

−

consistent with computer simulation and experiment re-

sults.

Another important fact not reflected in the F-H theory

is the change in chain conformation according to differ-

ent solvent qualities

15–23

. With increasing

, an isolated

polymer chain will undergo the coil-to-globule transition

from the swollen coil (

∼

) in good solvent to

the ideal coil (

∼

) at the Θ condition, and fur-

ther to the collapsed globule (

∼

) as the sol-

vent becomes poorer. The coil-to-globule transition is a

unique property of macromolecules that is different from

the small molecules: even one macromolecule can form

a mesoscopic phase

13,24–26

. It is a fundamental problem

relevant to many interesting phenomena, such as protein

folding

, DNA packing

, gel network collapse

, and

interpolymer complexation

. The coil-to-globule transi-

tion of a flexible polymer is of second-order

25,31

: in the

limit of infinite chain length, there is a well-defined tran-

sition temperature known as the Θ point; whereas for the

chain of finite length, the transition occurs within the Θ

region of the width proportional to

−

. De Gennes

pointed out that the Θ point is a tricritical point, so

mean-field theory can be applied except for logarithmic

corrections

32,33

. For polymer chains with sufficient stiff-

ness, the coil-to-globule transition is predicted to become

first-order

23,34

Given that the F-H theory provides a poor description

of polymers in poor solvent, on consideration of both

the over-simplified physical picture in the theory and the

demonstrated inconsistency with experiments and simu-

lation, it is natural to question the validity of the Θ value

predicted by the F-H theory. The

parameter is a

general language for describing single chains, polymer so-

lutions, blends and block copolymers. The value of

crucially important in polymer science and engineering,

because it serves as the benchmark to categorize differ-

ent polymer-solvent pairs. However, existing theories for

coil-to-globule transition are usually couched in the lan-

guage of either the viral coefficients

34–37

or the expansion

factor

31,38,39

, which makes it difficult to directly connect

the single chain behavior to the solution properties. Fur-

thermore, the second virial coefficient used in the study

of the single-chain behavior in previous work was ob-

tained from the F-H free energy of the homogeneous bulk

phase; the connection between this second virial coeffi-

cient and the effective two-body interaction at the level

of the single-chain Hamiltonian has not be elucidated. In

addition, the consistency of the Θ point defined by the

different criteria concerning different properties – the sec-

ond viral coefficient in dilute polymer solution, the single-

chain conformation, and the critical point in the phase

diagram of polymer solution in the limit of infinite chain

length – has not been theoretically verified. On the other

hand, computer simulation of the Θ point is challenging

due to the limitation of chain length

22,40,41

and numerical

accuracy. On the experimental side, there are ambigui-

ties in the structure of single-chain globule. For example,

by measuring the temperature dependence of the chain

size for an isolated polymer in aqueous solution, Wu and

Wang

17,18

suggested that the chain conformation before

the globule-to-coil transition involves two globule states:

fully-collapsed globule

and a

molten globule

, where, in

their words, quantitative description of the molten glob-

ule is still a challenge. They also found it surprising that

a fully-collapsed globule still contains 66% solvent inside

its volume.

The globule state of polymers has been theoretically

studied extensively by Grosberg and coworkers

13,34–37,42

Using the Lifshitz theory

, these authors have sys-

tematically elucidated the globule structure, globule-to-

coil transition, globule-globule interaction and the phase

equilibrium in dilute solution. The Lifshitz theory as-

sumes ground-state dominance and uses a virial expan-

sion of the local interactions. For both these two as-

sumptions to be satisfied, the globule must on one hand

have a sharp interface and on the other hand have low

monomer volume fraction in the core. The combination

of these two assumptions excludes a significant portion of

the parameter space in the globule state. In addition, the

globule structure in a poor solvent and the melting ther-

modynamics of a globule near the Θ point are studied

by using two different theoretical formulations, the for-

mer by using the Lifshitz theory and the later by using

Flory expansion factor. To complete the knowledge on

the globule state of polymers, it is desirable to develop a

theory that can describe the structural change of a glob-

ule in the full parameter space of the poor solvent, and

can unify the globule structure and the globule melting

in a single theoretical framework. In this regard, we note

that Ref.

showed that the expansion factor calculated

using the F-H free energy has a large discrepancy with

that obtained using the virial expansion. However, the

density profile of the globule and the surface energy or

its connection to solubility were not examined.

In an earlier paper, we developed a new theory in

the language of the F-H

parameter that can unify

the description of the single chain structure and the so-

lution thermodynamics for polymers in poor solvent

There, we focused on the equilibrium cluster distribu-

tion, the solubility limit, and nucleation in the supersat-

urated state. In the current work, we apply the theory

to examine in detail the structure and properties of a

single-chain globule. In our theory, we employ the full

self-consistent field theory (SCFT) and treat the solu-

tion as incompressible, thus allowing us to describe both

a high-density globule and melting of the globule as the Θ

point is approached. From the chain-length dependence

of globule melting characteristics, we investigate the scal-

ing behavior and determine the value of

by extrapo-

lating to the limit of infinite chain length. Because our

theory is capable of describing the phase equilibrium in

dilute polymer solutions, we also determine

from the

solubility of polymers based on the intrinsic connection

between the single chain structure and solution property.

II. THEORY

The system of polymers in poor solvent is a collection

of globules and clusters. We study this structure of large

localized density fluctuation by focusing on a subvolume

of the entire solution that contains only one globule or

one multi-chain cluster

14,43,44

. The density profile and

free energy of the globule and clusters are obtained by

applying self-consistent field theory (SCFT) in the sub-

volume. This information is then used in the framework

of dilute solution thermodynamics to reconstruct the so-

lution behavior of the full system. To distinguish between

the composition within a single globule/cluster and the

overall bulk composition in the solution, we use the nota-

tion

and

, respectively, to denote the volume fraction

of polymer in the globule/cluster and in the solution.

A. Self-consistent Field Theory for an isolated

globule/cluster

We consider a subvolume

consisting of

polymer

chains and

solvent molecules.

= 1 specifies the

single-chain globule. We treat the subvolume as a semi-

canonical ensemble: the number of polymers in the sub-

volume is fixed, whereas the solvent is connected with a

reservoir of pure solvent outside that maintains chemical

potential

. (Here

is defined relative to that of the

pure solvent.) The polymers are assumed to be Gaussian

chains of

Kuhn segments (with Kuhn length

). For

simplicity, the volume of the solvent molecule and the

chain segment are assumed to be the same

. The local

polymer-solvent interaction is described by the Flory

parameter, and the solution is treated as incompressible.

The semi-canonical partition function can be written

as:

Ξ =

∞

∑

βμ

∏

∫

{

}

∏

∫

(ˆ

+ ˆ

−

1) exp

[

−

∫

(

)ˆ

(

)

]

(1)

where

∫

{

}

denotes integration over all chain config-

urations weighted by the Gaussian-chain statistics, and

∫

denotes integration over the solvent degrees of free-

dom. ˆ

(

) and ˆ

(

) are the local instantaneous volume

fraction of solvent and polymer, respectively. The par-

tition function Eq. 1 cannot be exactly evaluated due

to both the energetic interaction and the incompressibil-

ity constraint. We make the self-consistent field (SCF)

approximation, which involves (1) decoupling the inter-

acting system into a noninteracting chains in fluctuat-

ing fields by a identity transformation using functional

integration over the fluctuating fields, (2) replacing the

functional integration over the fluctuating fields by the

saddle point approximation. For systematic derivation

and numerical details we refer readers to the standard

literature

45,46

. The free energy of the system is then:

βF

∫

[

χρ

−

)

−

)]

−

βμ

+ ln (

(2)

is the single-chain partition function in the field

given by

∫

(

, N

), where

(

, τ

) is the chain

propagator determined by the diffusion equation:

[

∂

∂τ

−

∇

]

(

, τ

) = 0

(3)

with initial condition

(

0) = 1.

is the single parti-

cle partition function of solvent in the field

, given by

∫

exp(

−

). The density profile

(

) and the

fields

(

) and

(

) are determined by the following

self-consistent equations:

(

)

−

(

) =

−

(

)]

)

(

) =

∫

dτq

(

, τ

)

(

, N

−

)

−

(

) =

βμ

exp(

−

(

))

)

By solving equations 3 and 4 iteratively

, we obtain

the equilibrium density profile and free energy of a glob-

ule/cluster in the subvolume.

B. Effective Two-body Interaction

If the polymer density in the subvolume is low, it is

convenient to integrate over the solvent degrees of free-

dom, which leads to effective interaction between two

polymer segments. Applying the identity transformation

for ˆ

together with local incompressibility, we recast the

partition function in Eq. 1 into:

Ξ =

∫

Dω

∏

∫

{

}

exp

[

−

∫

[

−

)

−

)] +

βμ

]

(5)

Because of the high volume fraction of the solvent, the

fluctuation of the field

is small, and the functional

integration over

in Eq 5 can be replaced by the saddle

point value, which yields

Ξ =

∏

∫

{

}

exp [

−

βH

]

where

is the Hamiltonian in the form of

βH

∫

[

−

) + (1

−

)(ln(1

−

)

−

1)]

−

βμ

−

)

(6)

Taking the reference such that

is zero in the reser-

voir of pure solvent outside the subvolume, we obtain

βμ

−

1. Eq 6 can then be written in the form of viral

expansion in terms of instantaneous volume fraction of

polymer segments as

βH

∫

[

(

−

1)ˆ

(

−

)

···

]

(7)

from which we identify the effective two-body interac-

tion as

−

. Although this is the identical expression of

the effective two-body interaction commonly used in the

literature, the correspondence between the effective two-

body interaction and the F-H

parameter in the previous

theories

1,3,13

was constructed for the

homogeneous, bulk

polymer solution; to our knowledge, this connection has

not been established at the level of single-chain Hamilto-

nian. Here we have provided an explicit demonstration,

based on a saddle-point approximation which amounts to

neglecting the concentration fluctuation of the solvent,

that the expression is valid at the single-chain level. If

one further includes the short-length-scale concentration

fluctuations, the

parameter should more properly be

interpreted as the effective interaction parameter instead

of the bare interaction parameter

48,49

. Renormalization

group theory

showed that the second viral coefficient

of the polymer solution has corrections from the three-

body interaction. Grosberg and Kuznetsov

pointed out

that this correction vanishes in the limit of infinite chain

length. Therefore, both the effective two-body interac-

tion and the second viral coefficient vanish at

for

infinitely long chain.

C. Phase Equilibrium

The density profile and free energy of the globule and

clusters obtained from SCFT can be included into the

framework of dilute solution thermodynamics to recon-

struct the bulk dilute polymer solution. The free energy

density of the entire solution with volume

, including

the translational entropy of clusters, can be written as

βF/V

∞

∑

[

βF

(ln

−

1)]

(8)

where

and

are respectively the concentration and

volume fraction of the cluster with association number

(called

-cluster henceforth) in the solution.

the free energy of the

-cluster calculated by SCFT (ob-

tained as an excess free energy in the subvolume with

respect to the pure solvent). In Eq 8, we ignore the in-

teraction between different clusters on the assumption

that the solution is sufficiently dilute. The equilibrium

concentration of

-clusters can be obtained by minimiza-

tion of the free energy density in Eq 8 subject to fixed

total polymer concentration

∑

∞

, which results

in the following cluster distribution:

= (

)

exp(

−

∆

)

(9)

where ∆

≡

−

is the free energy of formation

of the

-cluster from m single chain globule.

To study the coexistence between the polymer-poor

phase and the polymer-rich phase, we use Eq 8 with Eq.

9 to account for the large localized density fluctuation

in the polymer-poor solution, whereas the polymer-rich

phase is described by the F-H theory. The phase bound-

ary is determined by the respective equality of the chem-

ical potential of the polymer and the solvent in the two

coexisting phases, which results in

−

∞

∑

(

−

)

+ ln(1

−

) +

χφ

(10a)

βF

+ ln

= ln

−

1 + (1

−

)(1

−

)

χN

−

)

(10b)

where

is the equilibrium volume fraction of polymers

in the polymer-rich phase and

is the equilibrium vol-

ume fraction of the

-cluster in the dilute phase given

by Eq 9. The total volume fraction of polymers in the

dilute phase,

, is given by

∑

∞

III. GLOBULE STRUCTURE

In the poor solvent, a single polymer chain adopts a

compact globular structure due to the unfavorable inter-

action between the polymer segments and the solvent.

The increase in local segment density is countered by ex-

cluded volume (modeled by the incompressibility in our

theory) which prevents the globule from collapsing into

unbounded high density. Figure 1 shows the density pro-

file of a globule for different values of

calculated by

SCFT. Clearly the globule can be divided into a core re-

gion with uniform density (denoted by

) and a surface

region. The globule core resembles a liquid droplet that

contains a number of uncorrelated parts of the chain. By

taking uniform fields

and

in equations 2-4, we ob-

tain the free energy in the globule core to be

βF

[

χρ

−

) + (1

−

) ln(1

−

)] (11)

which is the Flory-Huggins free energy without the trans-

lational entropy of the polymer due to the fixed center of

mass. In the limit of large

, the Laplace pressure due to

curvature can be neglected, and the polymer density of

the globule core can be obtained approximately by bal-

ancing the osmotic pressure inside the globule with that

of the pure solvent outside (

≈

0), which yields

as the

nontrivial root of the following equation

+ ln(1

−

) +

χρ

= 0

(12)

For

close to 0.5, Eq 12 has the symptomatic solution of

≈

−

5). As shown in Figure 1, the core density

becomes lower with the decreasing

, which increases the

size of the globule as a result of swelling. At the same

time, the interfacial region becomes thicker and more dif-

fuse. We see from Figure 2 that the thickness of the in-

terfacial region scales with the Kuhn length

as pointed

out by Grosberg and Khokhlov

and is independent of

the chain stiffness parameter

. Because the de-

pendence on

is relatively straightforward, in most of

the subsequent discussions, we choose a specific

corre-

sponding to

= 4

πb

Based on the density profile, we can define the radius

of gyration (

), the core radius (

) and the surface

diffuseness (

) to characterize the globule size, which are

given as:

(

∫

∞

(

πr

∫

∞

(

πr

)

(13a)

= (3

Nv/

πρ

)]

(13b)

−

(13c)

is the radius of gyration for a uniform sphere with

polymer density

that contains the same total amount

of chain segments as in the globule; in other words,

is the location of the Gibbs dividing surface.

describes the deviation of the globule from the uniform

sphere, thus characterizing the diffuseness of the globule

surface. It should be noted that

is not a linear measure

of the true thickness of the globule surface (denoted by

) within which the polymer density changes from

0.0

0.2

0.4

0.6

0.8

1.0

=0.54

=0.77

(r)

r/(N

1/3

=2.0

FIG. 1. Density profile of a globule for three values of

with

= 10

and

= 4

πb

(

) is the local volume fraction

of polymer with

the radial axis starting from the globule

center.

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

(r)

r/(N

1/3

3/(4

)

(r)

FIG. 2. Effect of the chain stiffness parameter

the density profile of a globule with the same Kuhn length.

= 0

77 and

= 10

. The insert shows the overlap of the

interfacial region when the density profiles are shifted.

to 0. Dobrynin and Rubinstein

pointed out that the

surface thickness of the globule has a length scale of one

thermal blob

∼

b/ρ

. For a globule with a large

uniform core (i.e.,

≪

1), it can be shown that the

surface diffuseness scales as

∼

In this work,

we choose

rather than

to characterize the globule

surface because

is defined based on

, which is an

experimentally measurable quantity.

With the definition of

and

, we can separately

examine the volume contribution and the surface con-

tribution to the globule size. The core density and the

globule size as a function of

for polymer with

= 10

are shown in Figure 3. When

is well above 0

coin-

cides with

while

remains very small and is negligible

compared to

. The globule surface is of the thickness

of one thermal blob (

∼

b/ρ

∼

bρ

−

), and

the entire globule is space-filled by

∼

thermal blobs.

In this regime, the globule can be envisioned as a uniform

sphere: the increase of the globule size as

decreases is

due to the swelling of the globule core (see Figure 3b),

given by

≈

∼

(

Nv/ρ

)

. As

approaches 0

increases rapidly and deviates from

. At the same

time,

increases sharply and becomes comparable to

The thermal blob expands, reaching a size comparable

to the whole globule. The rapid increase of the globule

size in this regime indicates melting of the globule, which

takes place by both the swelling of the globule core and

the widening of the interface.

The

dependence of the globule size and core density

shown in Figure 3 suggests that the single chain con-

formation prior to the globule-to-coil transition involves

three states: the

fully-collapsed globule

, the

swollen glob-

ule

and the

molten globule

, in the order of decreasing

. For large

(

χ >

1), the globule is fully-collapsed,

with core density close to one and a sharp interface:

≈

∼

−

and

h/R

∼

−

. The fully-

collapsed globule can further transform to the ordered

solid-globule as pointed out by Zhou et al.

, which is

out of the scope of this work. In the intermediate regime,

where

−

≪

1 but (

−

)

≫

1, the globule is

swollen. The core density is significantly lower than one,

while the interfacial thickness is still narrow compared to

the core radius:

∼

[

(

−

)

]

−

h/R

∼

[

(

−

)

]

−

. Finally in the Θ regime,

where (

−

)

∼

(1) or less, the globule is molten,

with a very dilute core, comparable to a Gaussian coil,

and an interfacial thickness comparable to the core ra-

dius:

∼

(

−

)

∼

−

∼

(1), and

h/R

∼

(1).

That the globule structure may involve more than one

states has been suggested before. Wu and Wang

17,18

pro-

posed that the globule structure involves two states: the

fully-collapsed globule and the molten globule, based on

the consideration that the ratio between the radius of

gyration and hydrodynamic radius for the fully-collapsed

globule is (3

whereas this ratio for the molten glob-

ule is smaller than (3

(this criterion is equivalent

to whether the globule has a sharp or diffuse surface).

However, the “fully-collapsed” globule observed in their

experiment contains 66% solvent inside the globule vol-

ume, which is more properly considered as the swollen

globule in our definition. If we choose

= 1 as the cri-

terion for separating the swollen globule from the fully-

collapsed globule, then the range of the swollen globule is

roughly between

and

= 1 for large

; this can cor-

respond to a large temperature window in experiments.

The swollen state for the infinitely long chain persists all

the way to the Θ point, as we discuss below.

FIG. 3. a) Radius of gyration (

), core radius (

) and in-

terfacial diffuseness (

) and b) core density(

) of the globule

as a function of

with

= 10

. The schematic inserted in

a) represents the three states in globule structure: the

fully-

collapse globule

, the

swollen globule

and the

molten globule

It is instructive to note an analogy between the swollen

globule and the semidilute polymer solution in good sol-

vent. The existence of both these states is due to the

long chain length. Indeed the molten globule, the swollen

globule and the fully-collapsed globule can be consid-

ered the respective analogues of the dilute, semi dilute,

and concentrated solutions. Just as there is no clear-

cut boundary between the semidilute and concentrated

solutions, the distinction between a swollen globule and

fully collapse globule is not a sharp one. And just as

the concentration range for the dilute regime becomes

narrower as the chain length increases, so the region of

the molten globule shrinks with increasing chain length.

Likewise, with decreasing chain length, the window for

the existence of the swollen globule shrinks, and eventu-

ally disappears for very short chains.

We should emphasize that the density profile of the

globule is calculated using the flexible Gaussian chain

model. If the thickness of the globule surface becomes

comparable to the Kuhn length, which happens for very

large

, the chain segments can become oriented by the

interface. In that case, the flexible Gaussian chain is no

longer a valid description and chain rigidity and local ne-

matic order may have to be accounted for explicitly

However, we find that for

= 1

0 (with the core den-

sity

= 0

7) the interfacial width is

= 5

and is

= 10

for

= 0

77 (with the core density

= 0

52).

The interfacial width decreases to the Kuhn length at

about

= 2

0, corresponding to a core density

= 0

93.

Therefore, there is a significant range of parameter space

in which the Gaussian chain model remains valid.

IV. DETERMINING

FROM THE

SINGLE-CHAIN GLOBULE

The globule structure obtained by our SCFT calcula-

tion is valid under the condition that the surface energy

of the globule is much larger than

; otherwise shape

fluctuation must be taken into account and the assump-

tion of spherical symmetry becomes questionable. This

condition is equivalent to the requirement that the num-

ber of thermal blobs

must be large, i.e.,

≫

To satisfy this requirement, the globule state should be

within the parameter space (

−

)

≫

1, i.e., the

chain should be in the fully-collapsed or swollen globule.

(Equation 17b provides a more quantitative Ginzburg cri-

terion.) This requirement would at first seem to prevent

us from applying the theory to the close vicinity of the Θ

point. However, as the chain length increases, the width

of the Θ region (in

) shrinks and the transition becomes

increasingly sharper

23,31,34

. In other words, the state of

the swollen globule persists closer to the Θ point. In the

limit of infinitely long chain, the swollen globule persists

all the way to the Θ point. Therefore, the chain length

dependence of the melting of the swollen globule can be

used to probe the width of the Θ region and the location

of the Θ point.

0.5

1.0

1.5

2.0

0.00

0.02

0.04

0.06

0.08

0.10

h/R

FIG. 4. The ratio between the surface diffuseness and the core

radius (

h/R

) as a function of

for different chain length.

Figure 4 shows the ratio between the surface thickness

and the core radius of the globule (

h/R

) as a function

for several different chain lengths. As

decreases

close to 0.5,

h/R

exhibits a rapid rise as a result of the

melting of the globule; this rise is sharper for larger

From Figure 4, we can define the onset of the melting

(denoted by

(

)) as the value of

at which

h/R

reaches a threshold

(

) thus represents the onset

of the globule-to-coil transition as well as the boundary of

the Θ region on the globule side. Clearly,

(

) depends

on chain length. Figure 5 shows the linear relation of

(

) versus

−

for three different levels of

; this

linearity demonstrates that the width of the Θ region

is proportional to

−

, in agreement with the results

from the scaling argument

and simulation

0.00

0.01

0.02

0.03

0.50

0.55

0.60

0.65

0.70

=5%

=7%

=9%

(

)

-1/2

FIG. 5. The onset of globule melting

(

) plotted vs

−

for three different

. Points represent the results calculated

by SCFT. Straight lines are the linear fit to the points. Three

lines intersect to a common point as

→ ∞

, which yields

= 0

increases,

(

) for the different choices of the

threshold

become closer, indicating that the transition

becomes sharper for longer polymer chains. Extrapolat-

ing

(

) versus

−

to the limit of

→ ∞

(

)

for different

’s converge to a common point that is inde-

pendent of the choice of

, indicating that the transition

is infinitely sharp as

→ ∞

. From the perspective

of single-chain conformation, Θ point is defined as the

abrupt change in chain size from globule to coil for an

infinitely long chain. The common intersection shown in

Figure 5 yields

= 0

50. This value coincides with the

result of F-H theory and is also consistent with the van-

ishing of the second viral coefficient derived in Sec II. We

comment that the coincidence of

= 0

50 with the pre-

diction of the F-H theory is not a trivial result, nor does

it mean that the F-H theory itself is valid. Indeed, the

F-H theory is a poor description of the dilute solution,

including the solution near the Θ point. However, if we

focus on the core of an isolated globule, it is a homoge-

neous region containing many uncorrelated blobs of the

chain, sharing the same physical picture as the F-H the-

ory. In the limit of the infinite chain length, the swollen

globule state persists all the way to the Θ point where

the effective two-body interaction vanishes.

100

1E-3

0.01

0.1

h/R

(

1/2

FIG. 6. Universal behavior in the scaling region. log-log pl

h/R

vs (

−

)

Using

h/R

can now be plotted as a function of

the scaling variable (

−

)

. Figure 6 shows that

h/R

for different chain lengths collapse onto each when

plotted against (

−

)

, confirming the universal

behavior near the Θ point:

h/R

[

(

−

)

]

(14)

where

is a universal function. In the range

(

−

)

10 (where the globule is in the swollen

state with low core density), the slope of the log-log

plot is

−

4, close to the exponent

−

3 we have ob-

tained in Sec III by using the scaling argument. The uni-

versal behavior of the single chain conformation around

the Θ point, initially predicted by de Gennes

, is usu-

ally difficult to achieve in the globule state in computer

simulation

due to the limitation of chain length. For

short chains, the swollen globule either is not well-defined

or cannot persist to a very dilute core, thus making the

window for observing the scaling regime unclear or too

small. Here, by applying SCFT to long chains (

N >

we are able to more definitively examine this scaling

regime in the globule state.

V. DETERMINING

FROM SOLUTION

THERMODYNAMICS

The single chain structure will of course affect the so-

lution properties. The compaction of chain segments

into a globule significantly reduces the polymer-solvent

contact, which leads to enhancement of the solubility of

polymer in poor solvent by several orders of magnitude

compared to the prediction of the F-H theory as shown

in our earlier paper

. The governing equation for the

phase equilibrium of polymers in poor solvent is given

by Eq 10 in Sec IIC. In Eq 10a, the osmotic pressure for

long polymer chain in the dilute solution is very small

(

∑

∞

/mN

≈

0). Thus, from the similarity between

Eq 10a and Eq 12, we obtain for the volume fraction of

the polymer-rich phase,

≈

(15)

On the other hand, based on our analysis of the globule

structure in Sec III, the free energy of the single chain

globule in Eq 10b can be divided into the volume con-

tribution and the surface contribution as

It can be shown that this definition is consistent with

the definition based on the monomeric osmotic pressure

across the interface used in Ref.

. By substituting

in the form of Eq 11 into Eq 10b, and making use of Eq

12 and Eq 15, we obtain the relation between the equi-

librium volume fraction of polymers in the dilute phase

and the surface energy of the globule

≈−

βF

+ ln

−

γA

+ ln

−

(16)

from which we identify the surface tension

(36

)

(

Nv/ρ

)

is the surface area of the globule.

Noting that the natural dimension of the surface ten-

sion is

b/v

(

is taken to be 1), a dimensionless surface

tension can be defined as

γv

which thus contains an

overall

dependence; this is the origin of the

Eqs. 17a and 17b. Eq 16 indicates that, the excess free

energy of a globule is due to its surface energy, and not

due to (uniform) mixing of polymer segments and sol-

vents assumed in the F-H theory. Equations 15 and 16

reveal the intrinsic connection between the macroscopic

phase behavior of the polymer solution and the micro-

scopic structure and property of the single globule. This

connection also allow us to extract information of single

chain globule from solution thermodynamics.

The linear relation between ln

and

is con-

firmed by numerically solving Eq. 10 as shown in Figure

7. From the slope of ln

versus

, we can extract the

surface tension of a single globule. Figure 8 shows that

for

much larger than

decreases linearly with

γv

= 0

−

07. This linear dependence on

agrees with the result of Weber and Helfand for the in-

terfacial tension of a planar interface

. As

decreases,

deviates from the linear behavior and approaches zero.

The disappearance of the surface tension of the globule

serves as another definition of the Θ point. By extrapo-

lating the numerical data to the limit of

= 0, Figure 8

yields

= 0

50, which coincides with previous results

obtained from other definitions. Figure 8 also shows that,

follows a quadratic form

γv

= 0

52(

−

for

−

≪

1. The quadratic form of

near the Θ point is

in agreement with scaling prediction by Lifshitz et al.

In the framework of F-H theory, des Cloizeaux and

Jannink obtained

(

−

)

exp

[

−

(

−

)

]

By substituting the expression of

and

into Eq 16, our

400

500

600

700

-150

-100

-50

=0.6

=0.7

=0.8

2/3

=0.9

FIG. 7. Linear relation between ln

and

. Points rep-

resent the numerical results based on Eq 10. Straight lines

are the linear fit of the points.

0.5

0.6

0.7

0.8

0.9

1.0

0.00

0.01

0.02

0.03

0.04

0.05

This work

Large

Near the

point

v

2 /3

FIG. 8. Dimensionless surface tension of the globule

γv

a function of

. Extrapolating the numerical data to

= 0

reveals

= 0

is fitted linearly for large

γv

−

07, and fitted by a quadratic function 0

52(

−

near the Θ point.

theory yields the equilibrium volume fraction of polymers

in the dilute phase as

exp

[

−

53)

]

for large χ

(17a)

= 3(

−

5) exp

[

−

(

−

]

for χ

−

≪

(17b)

where we have inserted the dependence on the stiffness

parameter

. We note that an equation similar

to Eq 17b has been given in Ref.

(Eq. 26.8). However,

our Eq. 17b provides the numerical prefactor (1.93) in

the scaling dependence in the exponential, which was not

provided in Ref.

. Eq. 17a is a completely new result.

These equations show very different scaling behavior of

the solubility of polymers in poor solvent predicted by

our theory compared to the F-H theory

, both in the

dependence and in the

dependence.

The exponent in Eq. 17b can be considered as a quan-

titative Ginzburg criterion for the validity of applying the

SCFT, i.e., SCFT is applicable if

(

−

1. Thus increasing the stiffness parameter

allows a

closer approach to the Θ point. This conclusion is con-

sistent with results from computer simulation by Withers

et al.

From Eq 17b, we obtain the slope of the phase bound-

ary near the Θ point as

∂φ

∂χ

[

−

(

−

]

exp

[

−

(

−

]

(18)

Consistent with the validity of the SCFT, Eq 18 is valid

in the swollen globule state, which can persist all the way

to the Θ point for infinitely long chain. In the limit of

→ ∞

∂φ

/∂χ

= 0 if

χ >

5. However,

∂φ

/∂χ

jumps to a finite value at

= 0

5, which suggests that

= 0

5 is the critical point in the phase diagram of a

polymer solution in the limit of infinite chain length

This establishes the Θ point defined from the perspective

of solution thermodynamics.

These results confirm that the Θ point defined as the

abrupt change in chain size from globule to coil and as

the critical point in the phase diagram are consistent with

each other in the limit of infinite chain length. This con-

sistency reveals the intrinsic connection between the sin-

gle chain structure and its solution properties. From the

perspective of solution thermodynamics our results sug-

gest an alternative experimental determination of the Θ

point from measuring the slope of ln

versus

different temperatures. Since this approach is based on

finite polymer concentrations, it may be more easily con-

ducted than the single-chain measurement.

VI. CONCLUSIONS

In this work, we have presented a theory that uni-

fies the study of the single-chain structure and solution

thermodynamics for polymers in poor solvents, using the

language of the Flory-Huggins parameter. Our theory

captures the large localized density fluctuation in dilute

polymer solutions and the change in chain conformation

with the solvent quality that are missing in the F-H the-

ory. The structure of a globule is studied by SCFT, which

affords a more accurate description of the density profile

of the globule and its free energy for finite chain lengths.

By relaxing the assumption of the virial expansion of

the local interaction as assumed in previous theories, our

theory can be applied to globules with relatively high

monomer density, which facilitates the calculation of the

surface tension of the globule and the solubility of poly-

mers in dilute solution for large values of

. On the other

hand, by avoiding the ground state dominance approx-

imation, the SCFT is capable to describe globules with

diffuse interfaces. These advantages allow us to study the

globule structure and the thermodynamics of the globule

melting in a unified theoretical framework. The chain-

length dependence of the globule melting provides the

essential information on the approach to the Θ point.

We briefly summarize the key new results of this work.

First, we show that the chain conformation involves three

globular states prior to the globule-to-coil transition: the

fully-collapsed globule

, the

swollen globule

and the

molten

globule

; this identification clarifies the ambiguity in the

experimental studies of the globule structure. Second, we

provide numerical verification of the universal behavior

as a function of the scaling variable (

−

)

near the

Θ point, which has not been achieved by previous com-

puter simulation in the globule state due to the limitation

of the chain length. Third, we provide new results for the

solubility of the polymers in dilute solution compared to

the results of des Cloizeaux and Jannink based on the F-

H theory. The large

result is completely new while the

result near the Θ point provides the missing numerical

prefactor in previous work. Fourth, we demonstrate the

consistency of the Θ points defined by the different crite-

ria in the limit of infinite chain length: the disappearance

of the second viral coefficient in the dilute polymer solu-

tion, the abrupt change in chain size from globule to coil

and the critical point in the phase diagram of polymer

solution. Fifth, we find

= 0

5 from all three differ-

ent criteria, which

coincides

with the prediction of F-H

theory. Lastly, although the expression itself is known

and widely used in the literature, we provide the explicit

derivation that shows the effective two-body interaction

in the

single-chain Hamiltonian

is given by 0

−

Although

= 0

5 can be directly obtained through

the viral expansion of the local Hamiltonian if the

polymer-solvent interaction is parameterized by the Flory

parameter, the two methods we present in this pa-

per, that is tracking the chain-length dependence of the

melting of the globule and tracking the chain-length de-

pendence of the solubility of polymer solution, are more

general approaches in determining the Θ point. For ex-

ample, the approaches can also be used for heteropoly-

mers. Moreover, the consistency of the Θ point defined

by the different criteria allows us to choose a convenient

way to measure the Θ point in experiment. Our results

suggest that the Θ point can be determined by measuring

the slope of ln

versus

for different temperatures,

which is a finite-concentration instead of a single-chain

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∂φ

/∂χ

→ ∞

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in the swollen globule regime, we are unable to reach the

critical point. Therefore, rather than a divergence, we ob-

tain a discontinuity.