Unconventional Phase Transitions in Constrained Single Polymer

Chain

L. I. Klushin∗

Department of Physics, American University of Beirut,

P.O. Box 11-0236, Beirut 1107 2020, Lebanon

A. M. Skvortsov†

Chemical-Pharmaceutical Academy, Prof. Popova 14, 197022 St. Petersburg, Russia

1

AbstractPhase transitions were recognized among the most fascinating phenomena in physics. Exactly

solved models are especially important in the theory of phase transitions. A number of exactly

solved models of phase transitions in a single polymer chain are discussed in this review. These are

three models demonstrating the second order phase transitions with some unusual features: two-

dimensional model of β-structure formation, the model of coil-globule transition, and adsorption a

polymer chain grafted on the solid surface. We discuss also models with first order phase transitions

in a single macromolecule which admit not only exact analytical solutions for the partition function

with explicit finite finite-size effects but also the non-equilibrium free energy as a function of the

order parameter (Landau function) in closed analytical form. One of them is a model of mechan-

ical desorption of a macromolecule which demonstrates an unusual first order phase transitions

with phase coexistence within a single chain. Features of first and second order transitions become

mixed here due to phase coexistence which is not accompanied by additional interfacial free energy.

Apart from that, there exist several single-chain models belonging to the same class (adsorption of

a polymer chain tethered near solid surface or liquid-liquid interface, and escape transition upon

compressing a polymer beween small pistons) that represent examples of a highly unconventional

first order phase transition with several inter-related unusual features: no simultaneous phase coex-

istence, and hence, no phase boundary; non-concave thermodynamic potential and non-equivalence

of conjugate ensembles. Analysis of complex zeroes of partition functions upon approaching the

thermodynamic limit is presented for models with and without phase coexistentce.

∗Electronic address: leo@aub.edu.lb†Electronic address: astarling@yandex.ru

2

I. CONTENTS

1. Introduction

(a) Specifics of phase transitions in a single macromolecule

(b) Exactly solved models of the phase transitions

(c) Main features of the first and second order phase transitions

2. Second order phase transitions in a single chain

(a) Zwanzig-Lauritzen model of 2d crystallization

(b) Coil-globule transition

i. General considerations

ii. Mean field theories of the coil-globule transition

iii. Comparison with the Landau theory of phase transitions

iv. Comparison with the Landau theory of phase transitions

v. Experiments

vi. Computer simulations

vii. Coil-globule transition in higher dimensions

viii. Collapse vs. freezing

(c) Adsorption a single polymer chain grafted on a solid surface

i. Adsorption of ideal lattice chain

ii. Adsorption of macromolecule with excluded volume interactions

A. Scaling ansatz

B. Crossover exponent φ

iii. Continuum model of adsorption of an ideal chain

A. Exact Green’s function

B. Partition function as realization of crossover ansatz

C. Comparison with the Landau theory

D. Mapping of continuum and lattice parameters

3

E. Analytical partition function for finite lattice chains

iv. Adsorption on curved surfaces

3. First order phase transitions with phase coexistence within a single chain

(a) Mechanical desorption of a macromolecule in f -ensemble

i. The Green’s function and the partition function

A. Adsorption-force symmetry and the phase diagram

B. Partition function around the transition line f = c

ii. Unconventional features of the first-order transition with phase coexistence

iii. Landau free energy

(b) Mechanical desorption of a macromolecule in z-ensemble

i. Analogy with gas-liquid transition in (N,P, T ) and (N, V, T ) ensembles

ii. The partition function in the z-ensemble

iii. Evidence of phase coexistence

iv. Local order parameter

v. Comparison of two ensembles

4. First order phase transitions without phase coexistence within a single chain

(a) Polymer chain tethered near an adsorbing solid surface

i. Model and partition function

ii. Landau function

iii. Stretching force

iv. Microcanonical ensemble and the entropy gap

(b) Polymer chain end-tethered near an absorbing penetrable interface (near step

potential)

i. Chain attached near a liquid-liquid interface

ii. Chain attached at a liquid-liquid interface (rolling transition)

(c) Escape transition

4

i. Escape transition in 3d geometry

A. Partition function for ideal chain in D-ensemble

B. Compression curve (average force vs separation)

C. Conjugate ensemble (average separation vs force)

D. Nenequvalence of ensemble

E. Order parameter and Landau function

F. Escape of chain with excluded volume

ii. Escape transition in 2d space

iii. Another escape transition setup: dragging a polymer chain in tube

5. Complex zeroes of partition functions for a single-chain.

(a) Complex zeroes in transition with phase coexistence (mechanical desorption of

ideal chain)

(b) Complex zeroes in transition without phase coexistence (adsorption of ideal chain

tethered near solid surface)

6. Conclusions

7. Acknowledgments

8. References

A. Specifics of phase transitions in a single macromolecule.

Phase transitions were recognized among the most fascinating phenomena in physics since

the days of van der Waals, Boltzmann and Gibbs. Historically, the question of whether sta-

tistical mechanics is strictly applicable to phase transitions was much debated [1]. Onsager’s

solution of the 2d Ising model showed [2] that the full description of the phase transition is

indeed contained in the exact partition function. A lot of progress has been made since then

in understanding both physical and mathematical aspects of the phase transitions phenom-

ena. However, as the phase transition concepts are applied to broader areas new insights

and new problems arise. The aim of this review is to discuss phase transitions in a single

macromolecule and to demonstrate their unconventional features.

5

Phase transitions in macromolecular systems can be fundamentally divided into two types.

The first type occurs in the condensed bulk matter, involves a macroscopically large number

of molecules, and is basically similar to that in ordinary fluids or solids. Crystallization,

segregation of incompatible polymer liquids, and liquid-crystalline ordering of polymers can

serve as examples. The number of molecules in a typical experiment is of the order of the

Avogadro number and finite-size effects are completely negligible unless specific geometry

with the large surface-to-volume ratio is examined. The size of a single macromolecule plays

only minor roles in the nature of these transitions at least as far as equilibrium aspects are

concerned.

The second type of phase transitions belongs exclusively to the realm of polymers since

it is realized at the level of a single macromolecule and does not have any analogies in the

physics of low molecular mass systems. Note, that even a single polymer coil in dilute solu-

tion is characterized by anomalously large size fluctuations analogous to critical phenomena.

The best-known examples of phase transitions in a single chain are the transition from a

loose coil to a compact globule, as well as adsorption-desoption transition of a polymer at a

solid or liquid interface.

The notion of a phase transition in a single chain requires a conceptually clear understand-

ing. The number of the repeat units in a single macromolecule, N , is not really macroscopic,

usually 102 − 104, (for DNA 109 − 1010), while the traditional concept of a phase transition

requires taking the thermodynamic limit, N → ∞ . It turns out, however, that the frame-

work of the phase transition theory including the study of finite-size effects (which are the

rule rather than exception for the single-molecule transitions) proves to be very fruitful. The

approach to the thermodynamic limit for single chains is in principle accessible to experi-

mental studies since many polymers can be synthesized with their molecular weight being

varied over several decades without changing the intra-chain interactions. In many cases

these finite-size effects are completely different from those in bulk matter: for a droplet,

the finite-size effects are dominated by the surface contribution to the free energy, while a

polymer taking a typical coil conformation does not have a well-defined surface at all.

The interest in single-chain phase transitions has been spurred by recent advances in

experimental manipulations techniques. Nowadays, the existing potential includes various

powerful experimental methods such as AFM, magnetic levitation, photon microscopy, op-

tical tweezers, etc.

6

Manipulations of single chain have become available since 1986 [3] and the most impres-

sive progress was achieved in the physics of nanoscale objects. It possible now to measure

the elasticity of single chain [4–6], estimate the intensity of the receptor-ligand bond [7], to

study the physical characteristics of fundamental biological objects (DNA, proteins, molec-

ular motors) [8–13], and to determine the energy dissipation in the case of friction when a

single molecule moves directionally on a solid surface [14]. The forces required for unfolding

of DNA were measured [15, 16], the strength of the specific interaction between antigen-

antibody molecules was estimated [17], the conformational transitions in polysaccharides

were observed [18] as well the transitions during reversible protein folding [19]. Force-strain

curves for individual polymer chains have been studied extensively [5, 6, 20–22] , for several

synthetic polymers including both polyelectrolytes and uncharged polymers, biopolymers

(polymerase, proteoglucans, and xanthane), etc. Evident achievements in this direction

have triggered the development of a new science coined the “nanomechanics of polymers”. A

typical AFM experiment involves measuring the average force while controlling the tip-to-

sample separation. There is another AFM operation mode in which the force is fixed while

the separation is adjusted and measured. These two modes correspond to two different sta-

tistical ensembles. In principle, a combination of the two types of measurements would allow

to address the question of ensemble equivalence experimentally. There are two aspects in

the problem of relationship between different ensembles. First, although standard theorems

of statistical mechanics state their equivalence in the thermodynamic limit this does not

apply to finite-size systems. Second, single chain phase transitions include unique examples

of ensemble non-equivalence even when the thermodynamic limit is taken, as disscussed in

detail below.

Theoretical description of single-chain phase transitions may include the notion of phase

coexistence within one macromolecule, which we demonstrate in Section IV. On the other

hand, first order phase transitions in a single chain may also include unconventional examples

when no phase coexistence is possible in principle, see Section V.

B. Exactly solved models of phase transitions

Exactly solved models revealing details of the phenomena without making a priori con-

jectures or approximations are especially important in the theory of phase transitions. The

7

thesaurus of exact solutions is rather modest [23? , 24]. First, there are two-dimensional lat-

tice models (Ising, Potts, eight-vertex, three-spin and segnetoelectric). Two somewhat more

artificial models have also played a significant role: the one introduced by Kac [25, 26] with

infinitely weak, infinitely long-range interaction, and the spherical model corresponding to

the infinite number of spin components. The models mentioned admit exact solution in the

thermodynamic limit N →∞, where N is the number of particles in the system; finite-size

corrections are available for the Ising model. Only some of these models are solved in the

presence of the external field.

A number of models from polymer physics can be added. Surprisingly, even most simpli-

fied models exhibit non-trivial behavior which could be absent in their low-molecular mass

counterparts. Exact solutions describing second-order phase transitions are known for the

model of two-dimensional β-structure formation [27] and for some closely related models of

directed polymer adsorption [28–30]. Another inherently related set consists of the ideal

model of DNA melting [31] (see [32] for a review), and the model of adsorption of ideal

lattice chain on planar and cylindrical surfaces [33–39]. For all the models mentioned exact

solutions are obtained in the grand canonical formalism, and finite-size effects are analyzed

numerically or by scaling considerations.

There is yet another class of single macromolecule models which admit exact solutions

for the partition function for finite number of segments, N , and demonstrate true phase

transitions in the N → ∞ limit. These are ideal continuum chains interacting with some

external potential and described by the Edwards Hamiltonian

H(r(s))/T =

Nbˆ

0

ds

[3

2b2

(dr

ds

)2

+ U(r(s))

](1)

where b is the segment length, the segments are specified by the contour length distance s

counted from one end, and is the vector specifying the position of the corresponding segment

in space. The partition function for a chain with two ends fixed is

G(r0, rN , N) =

ˆD(r(s)) exp [−H(r(s))] δ(r(0)− r0)δ(r(Nb)− rN) (2)

where integration is over all paths r(s) connecting the two end positions .

It is demonstrated by direct verification that the partition function satisfies the equation

8

∂G(r0, r, s)

∂s=Nb2

6

∂2G(r0, r, s)

∂2r− U(r)G(r0, r, s) (3)

Eqs (1) and (2) establish the well-known analogy between the statistical mechanics of

ideal polymers and the path-integral formulation of quantum mechanics of a particle in

potential U(r) [40]. The wavefunction, ψ(r, t), for a particle of mass m in a potential U(r),

is given by the path integral

ψ(r, t) =

ˆD(r(t)) exp [iS/~] δ(r(t)− r), (4)

where

S =

ˆdt

[m

2

(dr

dt

)2

− U(r(t))

](5)

It is equivalent to the time-dependent Schrödinger equation

∂ψ(r, t)

∂t= − ~2

2m

∂2ψ(r, t)

∂2r+ U(r)ψ(r, t) (6)

A mapping between the quantum-mechanical problem and Gaussian chain in an external

field requires the associations: t ↔ s,ψ(r, t) ↔ G(r0, r, s),r(t) ↔ r(s),m ↔ 3/(Nb2),~ ↔

i,S ↔ H(r(s)),U(r(t))↔ −U(r).

Exactly solved models based on the continuum ideal chain description include examples

of first-order and continuous phase transitions with some very unconventional features that

are discussed below in Sections I, II, and III.

It is worth noting that the mathematical techniques employed in solving different classes of

models are quite distinct. Classical solutions for 2d lattice models are based on the formalism

relating the partition function to the eigenvalues of a transfer matrix. The ideal lattice

chain models are solved by finding the grand canonical partition function and analyzing its

singularities. The grand canonical formalism hinges on the superposition principle expressing

the ideal nature of the model whereby different parts of the chain are non-interacting except

for the connectivity condition. Finally, for the continuum chain model the path-integral

representation of the partition function leads to an equivalent formulation in terms of a

Green’s function of a partial differential equation (analogue of the Schrödinger propagator).

Again, the superposition principle plays a fundamental role.

9

The exactly solvable continuum models with analytical description of finite-size effects can

be pushed beyond the equilibrium partition function to obtain exact solutions for the Landau

free energy as a function of the order parameter[41]. This description covers configurations

that are far away from equilibrium and gives the full free energy landscape defining the first-

order transition kinetics including the possible metastable states. Metastable states as well

as the barriers separating them from stable equilibrium states are described analytically

for finite N . This is in contrast to non-polymeric models, where the Landau free energy

is normally postulated on the basis of symmetry considerations, while its exact analytical

calculation is not possible.

In conclusion we point out yet another fascinating feature of some exactly solved polymer

models. A very distinct approach to describe phase transitions was introduced by Yang and

Lee [42, 43], and is based on representing the partition function in terms of its zeros in the

complex plane of fugacity (or temperature, as suggested later by Fisher [1]. The theory is

mathematically rigorous which brings about a hefty price to be paid: the problem of finding

the actual distribution of zeros for the partition function of even a simple model turns out

to be formidable. Yang and Lee proved a circle theorem, which states that for ferromagnetic

Ising models and for the attractive lattice gas models, the zeros are located on a circle.

Later the theorem was further extended to other ferromagnetic lattice systems. However,

the densities of zeros are exactly known only for the simplest models (ferromagnetic Ising

models in 1d [43], on the Bethe lattice [44], on fractal lattices [45], and for an ideal Bose-gas

in an external potential [46]).

A number of exact results concerning temperature zeros have been obtained for the Ising

model with spin 1/2 [47–55] and with higher spin [56–59].

There exists a phenomenological approach relating certain characteristics of the transition

(amplitudes and critical indices) to the parameters of the distribution of zeros assumed to

be known [60], as well as scaling predictions for this distribution [51]. In practice, for a given

model the zeros are calculated numerically for small samples and then some extrapolations

are employed [61, 62].

There are at least two polymer models [63, 64] that admit analytical solution for the

distribution of complex zeros of the partition function in the cases when first-order and

continuous phase transitions are involved. Exact analytical expressions appearing in the

thermodynamic limit substantiate the scaling considerations existing in the framework of

10

the Yang-Lee-Fisher approach. We discuss this in last part of this review.

C. Main features of the first and second order phase transitions.

The conceptual framework of theory of phase transitions cannot be applied automati-

cally to a single macromolecule without some re-thinking and re-interpretation. We start

be recalling some basic facts about the standard analysis of phase transitions. The earliest

formal classification goes back to Ehrenfest [65]. If the first derivatives of the thermody-

namic potential (entropy, volume per particle, etc.) experience a jump with the change in

a control parameter the transition is first order. If these are changing continuously and the

singular behavior (jumps or divergences) appears only in higher derivatives the transition is

continuous or second order (in the contemporary language these terms are used as equiva-

lents irrespective of which particular higher derivative exhibits a singularity). Importantly,

the above classification applies in the N → ∞ limit, and only to specific statistical ensem-

bles with two fixed intensive parameters, such as the Gibbs NPT ensemble or its magnetic

NHT counterpart, or the grand canonical µV T ensemble. Here, the equilibrium state of

the system as a whole is a homogeneous phase before and after the transition point; phase

segregated states can exist only at the first-order transition points, and their relative phase

composition is not independently controlled.

On the other hand, the Helmholtz NV T ensemble is perfectly suited to study well defined

phase coexistence as we learn from textbook sections on van der Waals fluid[65]. However,

naïve application of the Ehrenfest criteria to ordinary gas-liquid condensation in the NV T

ensemble turns out to be confusing and counterproductive. Identification of the appropriate

ensemble is an important aspect that may be far from obvious in the context of polymer

models as shown by examples discussed below.

The basis for conceptual understanding of the essential background of phase transitions

was provided by the theory due to Landau [41, 66]. The central idea is to identify an order

parameter and then to analyze the non-equilibrium free energy as a function of the order

parameter. Historical examples of superfluidity and superconductivity show that defining a

suitable order parameter may be a non-trivial task by itself.

Landau’s original suggestion was to expand the free energy in powers of the order pa-

rameter, s, assuming analyticity at s = 0. Although it was recognized early enough that

11

F

s

a

T > Tc

T = Tc

T < Tc

F

s

h = 0h > 0

h < 0b

Figure 1: Nonequilibrium Landau free energy Φ as a function of the order parameter s for an

Ising-type model at second-order (a), and the first-order (b) phase transitions. In the first case the

transition is driven by a decrease in temperature T , while in the second case - by changing the

external field h coupled to the order parameter.

the original version is equivalent to a mean-field approximation (and therefore generally

incorrect) the Landau-Ginzburg formulation which incorporates interacting fluctuations of

the local order parameter field is the cornerstone of the critical phenomena theory. For the

sake of qualitative discussion the simple original version is quite sufficient. Two possible

scenarios are illustrated in Figure 1a,b showing the change in the shape of the Landau free

energy as a function of the order parameter, Φ(s).

In the first scenario Φ(s) = A (T − Tc) s2 + Bs4, always has only one minimum corre-

sponding to the true equilibrium state; in the high temperature range T > Tc the minimum

stays at s = 0 and represents the stable disordered phase, while for T < Tc the minimum

gradually moves away signifying a continuous onset of spontaneous ordering. Here, tem-

perature is taken as a particular example of a control parameter: other control parameters

will be discussed below in specific models. In fact, there may be two (or many) equiva-

lent minima representing the symmetry of the original Hamiltonian but only one is relevant

once the symmetry is spontaneously broken. Importantly, in the vicinity of T = Tc the

minimum is anomalously shallow which implies large fluctuations in the order parameter

and correspondingly, anomalous susceptibility to a conjugate field. Thus three features are

intrinsically linked together in the scenario of a second order (continuous) phase transition:

the order parameter starts growing at the transition point without a jump; the order pa-

rameter distribution always remains unimodal; and the fluctuations in the order parameter

12

display anomalous growth in the vicinity of the transition point.

Figure 1b shows the other scenario corresponding to a generic first-order transition. Here,

at T < Tc , the Landau free energy Φ(s) = −hs−A (T − Tc) s2+Bs4 has two minima meaning

that the order parameter distribution is bimodal. The deeper minimum corresponds to the

true equilibrium whereas the other minimum describes a metastable state. The equilibrium

transition temperature is defined by the condition that both minima are of equal depth. At

this point, the two different states are equally probable which would normally imply phase

coexistence. Upon crossing the transition point, the equilibrium state switches from one

minimum to another; this is accompanied by a jump in the order parameter and in the

first derivatives of the equilibrium thermodynamic potential. Again several distinct features

are packaged together in this scenario: a jump in the order parameter accompanied by a

δ-peak in the order parameter fluctuations and in the heat capacity (thus giving rise to

latent heat); bimodal distribution of the order parameter and metastability effects but no

anomalous pre-transitional fluctuation growth; phase coexistence at the transition point. A

more detailed picture incorporating spatial distribution of the order parameter leads to the

notions of an interface, nucleation, etc [66].

Note, that when the Landau free energy applies to the system as a whole, scenarios shown

in Figs1a,b are relevant only in the NPT or similar ensemble where macroscopic phase

segregation is restricted to the transition point only. If the order parameter and the density

of the Landau free energy are understood as local quantities, the choice of ensemble is largely

irrelevant since even in the NV T ensemble the system is always approximately homogeneous

locally. Traditionally, a useful criterion for identifying the nature of the transition when

studying finite systems is the shape of the distribution in the order parameter (or energy):

bimodal in the case of first-order transitions or unimodal in second order transitions.

Some very unconventional behavior in a class of single-macromolecule models demon-

strating first order transitions will be discussed in detail in the present review. Classically,

phase coexistence is a necessary attribute of first-order phase transitions. This always goes

together with metastability effects in the vicinity of the equilibrium transition point and a

nucleation mechanism for the appearance of new phase. The models to be discussed below

defy this common wisdom. In one case, phase coexistence can be properly identified but

nucleation mechanism is absent and no metastable states are to be found; in the other case,

metastable states do exist but phase coexistence is precluded by the very nature of phases.

13

aT*

n*

b

0 5 10 15 20 25

0.100

0.050

0.030

0.150

0.070

¶E*

¶T* Γ

T*

c

0 2 4 6 8 10

0.5

1.0

1.5

2.0

Figure 2: (a) Zwanzig-Lauritzen model of 2d cristallization; (b) the average number of bends per

monomer as a function of reduced temperature T∗ = kT/ε; (c) square-root singularity of the heat

capacity vs. T∗. (According to [27]).

We show examples when the convexity of equilibrium thermodynamic potentials is violated

leading to nonequivalence of different statistical ensembles in the thermodynamic limit and

in one instance, to an equilibrium negative compressibility.

II. SECOND-ORDER PHASE TRANSITIONS OF SINGLE-CHAIN.

A. Zwanzig-Lauritzen model of 2d crystallization.

One of the earliest exactly solved models demonstrating a non-trivial critical behavior in

a single macromolecule was introduced by R. Zwanzig and J.Lauritzen [27, 67]. The model

is relevant for crystallization, β-sheet formation and surface roughening phenomena. The

chain on a 2d plane is folded into a sequence of straight segments (not necessarily of the

same length) densely stacked in one direction, see Figure 2 a.

There are two parameters in this model: the attraction energy per unit length ε and the

bending energy u . The energy is due to short-range attraction between adjacent stacked

segments and for each pair of segments is proportional to their overlap length. Each bend

carries additional energy due to deformation and the surface effects. The grand canonical

partition function as a function of the chemical potential µ is evaluated exactly

Ξ(µ) = −1 +2

σ

Jν(σν)

Jν−1(σν)(7)

where Jν(x) is the Bessel function, ν = εµ−ε , σ = 2

√e−βµ

βε, and β = 1/(kT ).

If bends are energetically disadvantageous (u > 0), the crystal is a perfect (infinite)

square at T = 0, and the number of bends per monomer n∗ is zero in the L→∞ limit. The

14

number of bends per monomer serves as an order parameter and can be used to distinguish

between the low-temperature crystalline phase and the high-temperature phase which is

entropy dominated and is characterized by finite order parameter (the number of bends

being proportionate to the chain length), see Figure 1a. Since the model does not allow

random coil configurations, the high-T phase can be loosely described as a crystal destroyed

by extreme roughening. Above the critical temperature the average order parameter grows

as n∗ = A(T − Tc)1/2.

Although the critical index for the order parameter growth is 1/2 coinciding with the

mean-field value, other characteristics deviate from mean-field predictions. In particular,

the heat capacity displays a square-root singularity rather than a finite jump, Figure 1c.

The exact partition function of the Zwanzig-Lauritzen model was used to extend the

theory and include 3d coil configurations [68]. This changes the nature of the transition and

makes it first order with bimodal distribution in energy, and infinite heat capacity. Finite-

size effects for this system was analyzed numerically [69]. Later the theory was subsequently

extended to include the possibility of α-spiral formation and selective adsorption in a chain

with α-helix, β-sheet and random coil sequences [70, 71]. This theory combined with MC

simulations of finite chains with N = 20 was used to explain experimental data on structural

changes in beta-coil transition and alpha-beta-coil transitions in polypeptides induced by

adding low molecular mass detergents and to the surfactant-induce transitions in globular

proteins [72]. In all of these theories and calculations, interaction between different structural

elements of the chain as well as excluded volume interactions within coiled sub-chains was

ignored.

B. Coil-globule transition.

Interest in the coil-globule transition and in the features of globular conformations in

homopolymers arouse in connection with the problem of denaturation of globular proteins,

protein folding, and DNA packing. Fundamentally, the coil-globule transition is an extension

of the gas-liquid condensation for particles connected into a linear chain. Chain connectivity

brings two important distinctions as compared to the standard gas-liquid transition: first,

the collapse of an isolated chain takes place at zero external (osmotic) pressure; second, the

translational entropy is replace by configuration entropy due to internal degrees of freedom

15

in the macromolecule.

1.General considerations. The standard description of the collapse transition is a tricrit-

ical point related to a field theory for an n-component vector field with ϕ4−ϕ6 interaction,

in the limit n → ∞ [73]. One might then expect that above the upper critical dimension

(d = 3) some type of self-consistent mean-field theory would give a full description of the

transition [74].

For finite polymer length N the transition is rounded. In the language of the finite-size

scaling analysis this means that in the behavior of various thermodynamic quantities in the

vicinity of the Θ-point follows certain scaling laws. In particular, the average monomer

density in the chain ρ = Na3/ 〈R〉3 (where 〈R〉 is the gyration radius of the chain at a given

temperature) is given by

ρav(N, τ) = ρav(N, 0)Φ(τNφ), (8)

where τ = (T − Θ)/Θ is the relative deviation from the Θ-point, ρav(N, 0) ∼ N−1/2 is the

density at the Θ-point. The crossover exponent φ describes how the model rescales as the

tricritical point is approached from different directions [75], and Φ(z) is the crossover scaling

function. Some properties of the scaling function can be deduced on general grounds. At

large negative values of the argument the scaling function must be a power law Φ(z) ∼ |z|x

with the index x determined by the condition that the density of a well-formed globule is

independent ofN , which gives x = 1/(2φ) . For large and positive values of the argument, the

scaling function describes the approach to self-avoiding limit with 〈R〉 ∼ N ν and ρav ∼ N1−3ν

, where ν = 0.5878 is the Flory index for the gyration radius of a self-avoiding chain in 3D.

It follows that Φ(z) ∼ zy , with y = (3/2− 3ν)/φ. Altogether,

Φ(z) ∼

|z|1/(2φ) z � 1

1 z = 0

z3(1−2ν)/(2φ) z � 1

(9)

and the average density is

16

ρav(N, τ) ∼

|τ | τ � 1

N−1/2 τ = 0

τ 3(1−2ν)N1−3ν τ � 1

(10)

Mean-field type theories give a linear growth of density with |τ | which implies x = 1 and

φ = 1/2 (see below). This is all consistent with the general conclusions of the field-theoretical

approach for d=3 since the space dimensionality is just equal to the upper critical dimension

(though renormalization-group arguments suggest logarithmic corrections).

2.Mean-field theories of the coil-globule transition. We discuss briefly the existing

mean-field theories of the coil-globule transition. The interaction free energy can be written

as a virial expansion series, and the configuration entropy is evaluated separately using the

ideal chain statistics. In the simpler version of the theory due to Birshtein and Pryamitsyn

(BP) [76], one deals only with the global monomer density ρ = Na3/R3 defined by the

gyration radius of the chain. Here R is a parameter characterizing the state of the chain and

may be quite different from the equilibrium average value. The entropy is also expressed in

terms of the gyration radius R. Equilibrium size of the chain is found by minimizing the

total free energy as a function of R following the idea proposed by Flory [77] to describe

swelling under good solvent conditions .

The interaction free energy is written keeping the two dominant terms in the virial ex-

pansion:

Fint(R,N)

kT=R3

a3

(A2ρ

2 + A3ρ3)

(11)

The virial coefficients appearing in this expressions should be understood as renormalized

coefficients [78, 79] in order to account for the distinction between local and global densities.

In the standard approach A2(T )is taken as a linear function of temperature in the vicinity

of the Θ-point, A2 = bτ , and A3(T ) - as a constant: A3 = c. The entropy of an ideal

chain with a given radius of gyration, S(R), was studied by Fixman [80]; combining the full

function S(R) with the interaction free energy (11) provides a unified description of swelling

for τ ≥ 0 and compactification for τ ≤ 0. A simple analytical form

S(R)/k = 5 ln(R0/R)− 9R20/4R

2 (12)

17

can be used to describe compact configurations with R < R0 where R = (N/6)1/2a is the

equilibrium average gyration radius of an ideal walk.

Minimization of the total free energy F (R,N) = Fint−TS(R,N) with respect to R leads

to the following equation for the expansion factor α = R/R0:

α3 − α = BN1/2τ + C

(1

α3− 1

), τ < 0, (13)

where B and C differ from b and c, respectively, only by numerical coefficients. In the

thermodynamic limit N → ∞ this equation leads to a simple picture of the transition,

consistent with the crossover scaling form discussed above. Below the Θ-point, (τ < 0, α�

1) Eq.13 gives the swelling coefficient as α ∼(

CBN1/2|τ |

)1/3

. At the Θ-point itself, τ = 0 and

α = 1. Above the Θ-point (τ > 0, α � 1)the configurational entropy has to be modified

appropriately, giving the classical Flory picture of chain swelling with α ∼(BN1/2τ

)1/5. If

one considers the average density as the order parameter, its variation with temperature can

be summarized as:

ρ =

τ−3/5N−4/5 τ > 0

N−1/2 τ = 0

bc|τ | τ < 0

(14)

coinciding with the scaling prediction of Eq.10 with the Flory value of the index ν = 3/5.

The equilibrium free energy in the globular state counted from that at the Θ-point is F/kT =

−b2Nτ 2/(4c). This implies that the transition is second order with a finite jump in the heat

capacity, as one expects in the mean-field approach.

3.Comparison with the Landau theory of phase transitions. The phenomenological

theory of phase transitions by Landau [41] is based on consideration of the non-equilibrium

free energy Φ as a function of the order parameter, s. For Ising-class systems with a scalar

order parameter and mirror-reflection symmetry, Φ is expanded in powers of s in the vicinity

of the second-order transition point, only the terms with even powers being present due to

symmetry considerations:

Φ(s) = Φ0 + A(T − Tc)s2 +Bs4 (15)

Analysis shows that the entropic contribution to the total free energy is negligible in the

well-formed globular state [76]. Then the virial expansion (15), re-written as

18

Φ(ρ,N)

NkT= bτρ+ cρ2 (16)

can be interpreted as the non-equilibrium Landau free energy as a function of the order

parameter. Since the order parameter ρ is non-negative by its physical meaning, a complete

analogy with the original Landau theory is achieved by the mapping ρ = s2. At or above

the Θ-point the minimum is located at ρ = 0; as the temperature drops below the Θ-point,

the minimum shifts linearly with τ . This is a classical picture of a second-order transition

within the tenets of the Landau theory. For finite N the transition is smoothed over the

temperature range δτ ∼ N−1/2 (this can be detected, e.g.. in the shape of the heat capacity

jump), consistent with the crossover exponent value φ = 1/2.

A more sophisticated version of the mean-field theory due to Lifshits, Grosberg, and

Khokhlov (LGKh) [78, 79] is based on the density-functional approach and historically ap-

peared earlier. The total free energy is written as functional of the local density profile;

minimization yields the equilibrium density profile, and all the other characteristics are ex-

pressed in terms of it. Both versions (BP and LGKh) give the same results for all global

chain characteristics and the fundamental features of the transition. We have concentrated

on the simpler BP version since it allows a straightforward analysis in the framework of the

Landau theory and is easily extended to arbitrary space dimension.

Interestingly, the two versions differ in their starting points as far as the transition itself

is concerned. The LGKh version is based on the ground-state dominance approximation

which implies that the globule is already well-formed. The transition is therefore approached

from the globular side of the phase diagram, and the entropic contribution gets a natural

interpretation as the surface free energy. The starting point of the BP approach is a weakly

perturbed ideal coil at the Θ-point, and the entropy term just accounts for compactification

with respect to the ideal coil size.

4.Experiments. Most experimental studies of the coil-globule transition have been carried

out using various solutions of homopolymers in organic solvents: polystyrene in cyclohexane

[81–87] and poly(methyl-metacrylate) in appropriate theta-solvents [88–93]. Monitoring an

individual chain in the process of coil-to-globule collapse and observing fully collapsed ther-

modynamically stable single chain globules is extremely difficult since, on one hand, van der

Waals interaction driving the transition in organic solvents is very weak and on the other,

multi-chain precipitant is formed even at low polymer concentrations. Experimental difficul-

19

ties have been overcome only recently. Fairly monodisperse very high molar mass polymer

samples (up to N ∼ 104 − 105 segments) of temperature responsive poly(N-isopropyl acril-

amide) and poly(N,N-diethylacrilamide) in aqueous solutions were investigated at extremely

low concentration in a wide temperature range near the phase transition and reversible coil-

to-globule transition was demonstrated [94–103].

Theoretical analysis of the collapse of chains of finite length is based on expression (8)

which turns out to be quite useful for interpreting experimental data. In particular, it sug-

gests that the data points presented in the scaling coordinates (α−α3)/(α3−1) vs. |τ |N1/2

should collapse onto a straight line, its slope and intercept giving the coefficients B and (-C),

respectively. Experimental data for coil-to-globule transition in organic solvents show good

agreement with the predictions of the mean-field theory for the expansion factor, see Fig-

ure 3a. Temperature-responsive polymers in aqueous solutions demonstrate a considerably

sharper collapse, Figure 3b.

Thus the coil-globule transition seems to be established as a universal phenomenon at least

for a broad class of typical flexible polymers, including those that demonstrate anomalous

temperature behavior with the collapse induced by increase in temperature.

5. Computer simulations. The first computer simulation of the coil-globule transition

of a flexible model chain on a cubic lattice was undertaken over 40 years ago [105]. Later

on this transition was studied extensively using both Monte Carlo [109–111] and Molecular

Dynamics [112–114] techniques. Earlier computer simulations were restricted to relatively

short chains with N ∼ 102 − 103 where profiles with a well-formed core of constant density

and a diffuse boundary could not be obtained. However, very recent MC simulations with

N up to 104 [115] demonstrate density profiles with compact core of constant density that

are qualitatively very similar to those predicted by the functional-density theory [116–119],

see Figure 4. Systematic quantitative comparisons are not available yet.

6. Coil-globule transition in higher dimensions. Although the overall picture of

the collapse transition for flexible polymers as presented above seems to be quite clear and

satisfying some intriguing questions are left unanswered. Field-theoretical approach to the

collapse phenomenon is far from being fully consistent and clearly understood. Confusion is

revealed most convincingly when the transition is studied in 4-dimentional space. Standard

theoretical arguments suggest that finite-size scaling with the mean-field exponent φ = 1/2

should be exact since one is above the upper critical dimension. However, other approaches

20

æææ

æ

æ

æ

æ

æ

à

à

à

à

à

à

ì

ì

ì

ìì

ì

ì

ì ì

ΤN1�2

1-Α-3a

Α3-Α

1-Α-31 2

3

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

ææ

ææ

ææææææææææææææææææ

ààà

à

ààààà

àà

ààààààààààà

N = 108 107 106

105à PDEAM

æ PNIPAM

0.96 0.98 1.00 1.020.0

0.5

1.0

1.5

Θ�T

ΑS

Figure 3: (a) Experimental data for PS in cyclohexsane (1) [104] , PS in dioctyl phthalate (2)

[81], and results of MC simulations for lattice chain (3) [105, 106] in the coordinates suggested by

the BP theory [76]. (b) Temperature dependence of the expansion factor αs = 〈Rg〉T / 〈Rg〉Θ for

poly(N,N-diethylacrylamide) (PDEAM) with M = 1.7 · 107 g/mol in water [103] and for poly(N-

isopropylacrylamide) (PNIPAM) with M = 1.0 · 107g/mol in water [96, 97, 107]. The Flory Θ-

temperatures are 28.5oC for PDEAM , and 30.5oC for PNIPAM. Solid lines represent calculations

on the basis of yet another version of the mean-field theory [108] with several values of the number

of monomers, N . (The data are taken from [76], and [103])

lead to completely different and contradicting values of the crossover exponent (φ = 0 [120]

or φ = 1 [121]. MC simulations by PERM kinetic growth algorithm with N up to 16384

give clear evidence that close to the Θ-point the crossover scaling is not applicable at all.

It is destroyed by pre-transitional behavior with clear features of a first-order transition:

bimodal distribution of energy and a sharp drop in the chain size [122] .

Quite unexpectedly, this unusual behavior was first suggested on the basis of the mean-

21

Figure 4: Segment density distribution as a function of the distance from the center of mass of the

lattice model chains for various chain lengths N (indicated in the Figure) and energy interaction

parameters between units χ = 0.37 (a) and χ = 0.45 (b), both in the poor solvent regime. For this

model, χ = 0.28 corresponds to Θ-conditions. (c) Swelling parameter α = R/R0 as a function of

τN1/2: simulation data for different N fall on a universal master curve. (According to [115])22

field theory [123] appropriately generalized for arbitrary space dimension d . This can be

easily done within the approach presented in this section taking the Landau free energy as

a function of the average monomer density. The virial expansion terms (16) remain the

same; the entropy of compact conformations still has the form of Eq.12 with the dominant

term S(R) ∼ N/R2 although numerical pre-factors will change. The simplest modification

that adequately describes the (nearly) unperturbed ideal coil at τ = 0 consists of adding the

expansion branch of the entropy so that the ideal Gaussian size is recovered at b = c = 0[124].

The resultant Landau free energy as a function of the global density is given by

F (ρ,N)

NkT= bτρ+ cρ2 + γ

(ρ2/dN−2/d + ρ−2/dN−2+2/d

)(17)

where γ is a numerical coefficient related to d. For d > 3 and large N the Landau free energy

(12) displays all the features that lead to the collapse scenario articulated by Owczarek

and Prellberg [121]. This includes two competing minima at a certain temperature shift

|τ ∗| ∼ N−1/(d−1) causing a sharp drop in the chain size from N1/2 to N1/(d−1). Upon further

decrease in temperature the chain compactifies smoothly and the density grows linearly with

τ . Within this scenario the higher is the space dimensionality, the worse is the failure of the

finite-size scaling ansatz, Eq. (14). The case of 3d is marginal: bimodality can occur only

when the third virial coefficient, c, is numerically very small, and even then the drop does

not sharpen with the increase in N .

It seems that the difficulties in the systematic field-theoretical treatment of the Θ-point

stem exactly from the dramatic asymmetry between the two directions of approach to the

Θ-point which appears in high-dimensional spaces. For τ > 0 the Landau free energy always

has one minimum, while for τ < 0 bimodality appears and undermines the very basis of the

renormalization-group approach.

7. Collapse vs. freezing. Another intriguing aspect of the single polymer chain collapse

was found recently in MC simulations of the bond-fluctuation model [125]. Upon lowering the

temperature, the coil-globule transition is followed by freezing and formation of an ordered

crystalline state which proceeds as a discontinuous first-order transition. This by itself

would not be surprising. However, extrapolation to N → ∞ has lead to a conclusion that

the abrupt freezing transition happens at the Θ-point thus preempting the much smoother

coil-globule transition.

This was in contrast to a big body of data from earlier simulations based on simple cubic

23

lattice walks that display only one collapse transition. It is well known, however, that the

lattice gas on the simple cubic lattice is equivalent to the Ising model and there is only

one condensation transition in this case. Bond-fluctuation model turns out to be closer to

off-lattice models where two distinct transitions (gas-liquid and liquid-solid) are found for

simple fluids.

A much broader point of view has been formulated very recently by Taylor et al. [126,

127]. Off-lattice simulations has shown that the relative position of the two transitions

depend on the range of attraction between the repeat units of the chain. For potentials with

hard repulsive core and a square-well attractive part the relevant parameter is the ratio

λ = r1/r0 where r0 is the radius of the repulsive core, and r1 is the outer radius of the

attractive well. It was shown that a smooth change in λ leads to shifts in both transition

temperatures, and for λ sufficiently close to 1 the coil compactifies directly to an ordered

solid state without an intermediate liquid-like globule (see Figure 5 ). Gas-solid transition

bypassing the liquid state is not specific to the collapse of a single flexible macromolecule

and has been known in the context of phase transitions in colloidal systems. For colloids

(in contrast to atoms and molecules) the ratio of the attraction/repulsion distance can be

varied in a broad range. Phase diagrams of colloidal suspensions include all three phases

(gas, liquid and solid) for large values of λ but the range of existence of the liquid phase

shrinks and eventually disappears with the decrease in λ.

One may speculate that real polymers composed of very bulky repeat units connected

by short flexible links will show abrupt collapse from a random coil directly into an ordered

solid-like state. In particular, the findings of Taylor et al open up intriguing venues of

research on the role of the building block size in the specifics of proteins folding.

C. Adsorption a single polymer chain grafted on the solid surface.

The study of polymer adsorption from a solution onto a solid surface has a long his-

tory and is important for many applications such as adhesion, coating of surfaces, wetting,

adsorption chromatography, etc. A fundamental issue is the adsorption-desorption second-

order transition of an isolated long polymer chain grafted on a planar surface. There is

an important analogy with the melting transition of a double-stranded DNA because the

conformational features in these two cases are similar, consisting of an alternation of trains

24

Figure 5: Temperature-range interaction T−λ phase diagram for the N=128 chain with square-well

potential between units with depth −ε. The filled circles locate the continuous coil-globule collapse

transition and the filled squares locate the discontinuous globule-crystal or coil-crystal freezing

transition. The connecting dashed and solid lines are simply interpolations between these points

and uncertainties are smaller than the symbol size. The simulation snapshots show representative

chain conformations coexisting with different energy -E/ε=404 and 89 at the freezing transition for

interaction ranges λ=1.05 and with−E/ε=379 and 299 at λ=1.15 , as indicated. For short-range

interactions λ <1.05 the chain freezes directly from an expanded coil state while for longer range

interactions λ >1.06 the chain freezes from the collapsed globule state.(From the work [127])

(or native helix pieces) and single (or double) loops.

1. Adsorption of infinite ideal lattice chain

Historically the adsorption of a flexible polymer chain on a solid surface was described by

ideal random walks on regular lattices [33, 128, 129]. The interaction of each step with the

surface is described by a square-well potential with a depth ε and a width b. We take ε and

b as energy and length units and use the dimensionless interaction parameter χ = βε. Exact

partition function was calculated by Rubin in the N →∞ limit [33, 128, 129]. In particular,

for a chain on a six-choice cubic lattice the free energy per segment (in the adsorbed state,

this coincides with the chemical potential) is given by

25

βµ (χ) =

ln[

6(eχ−1)eχ

(√4 + 1

eχ−1− 2)]

χ ≥ χc

−N−1 ln[(e−χ − e−χc)

√6πN

]χ ≤ χc

(18)

The critical value χc below which a chain does not adsorb is χc = ln(6/5) . Close to

the critical point βµads(χ) ≈ (χ − χc)2. The quadratic dependence implies a second-order

adsorption/desorption transition with a finite jump in the heat capacity. The adsorption

order parameter is defined as the fraction of segments, θ, in contact with the surface. The

average order parameter is given by 〈θ〉 = ∂(βµads)/∂χ. Its initial growth is linear in the

deviation from the critical adsorption point: 〈θ〉 ∼ χ− χc.

2. Adsorption of macromolecule with excluded volume interactions.

Scaling ansatz. Great efforts were put into elucidating the excluded volume effects in

the adsorption transition. Since the adsorption point in the grand canonical formulation is

of a tricritical nature [130], the crossover scaling picture presented in the previous section

is expected to hold. For the canonical partition function in the near vicinity of the critical

point this corresponds to the following scaling ansatz

QN(χ) = QcΨ(τNφ) (19)

where QN(χ) is the canonical partition function, Qcis the partition function at the critical

adsorption point, τ = χ/χc−1 is the deviation from the critical point, and φ is the crossover

index. Analysis of the scaling function Ψ behavior similar to that presented in the previous

section gives predictions about its asymptotic behavior:

ln Ψ(x) ∼

x1/φ x > 0

0 x = 0

− ln(−x) x < 0

(20)

The average bound fraction (order parameter) is found from〈θ〉 = 1N∂ ln Ψ∂χ

:

26

〈θ〉 ∼

τ−1+1/φ τ > 0

Nφ−1 τ = 0

(Nτ)−1 τ < 0

(21)

Note that in contrast to the previous section, the choice of τ is such that positive τ

corresponds to adsorbed (condensed) phase; this convention facilitates the comparison to the

exactly solved continuum ideal chain model. The mean-field value φ = 1/2 is exact for ideal

Gaussian chain. For real interacting chains the index φ is evaluated from renormalization-

group calculations or from the numerical simulations data.

Crossover exponent φ. It is clear that the crossover exponent defines not only the near-

critical behavior but also the weak adsorption of asymptotically long chains. De Gennes

[131] originally proposed that φ = 1 − ν ≈ 0.41, where ν = 0.5878 is the Flory exponent.

This was based on the assumption that the density profile of a critically adsorbed chain is

essentially the profile of an unperturbed coil cut in the middle by a fictitious plane, as is

the case in ideal chain adsorption where, indeed, φ = 1 − ν = 1/2. It was soon recognized

that the relation is not valid for adsorption of real chains at a solid substrate, although

it was suggested that it may still hold for polymer adsorption at liquid-liquid interface

[132]. From a theoretical point of view, the mean-field value of φ = 1/2 applies to higher-

dimensional spaces with d ≥ 4 where excluded volume effects are negligible. On the other

hand, conformal-invariance approach [133] in d = 2 also gives the exact value of φ = 1/2

rather unexpectedly. It was also found that φ = 1/2 applies rigorously to adsorption of lattice

animals (randomly branched self-avoiding polymers) [134, 135]. This prompted an exciting

suggestion that φ = 1/2 is a superuniversal index. However, field-theoretical calculations

for d = 3 based on two different methods (ε-expansion with Pade-Borel summation [136],

and massive theory in fixed 3d space [136, 137] seem to contradict the superuniversality

hypothesis giving the estimates of φ = 0.483 and φ = 0.52 , respectively. In any case,

theoretical considerations strongly suggest that φ must be at least quite close to 1/2.

Numerical studies using different Monte-Carlo algorithms were employed to attack the

problem of adsorption (see [138] with references). A traditional way of evaluating φ from

numerical data involves plotting various scaled chain characteristics vs. the finite-size scaling

parameter τNφ in hope of obtaining the data collapse on two universal curves corresponding

27

to adsorbed and desorbed regimes. In this method, χc and φmust be adjusted simultaneously

until the best data collapse is achieved. Earlier studies produced estimates φ = 0.588± 0.03

[36] and φ = 0.53± 0.007 [139]. It was shown that a slight change in the estimated χc leads

to large effect on the estimate for φ: parameter pairs (χc = 1.01, φ = 0.59) or (χc = 0.98,

φ = 0.50) give almost the same overall quality of the data collapse [140]. A more refined

analysis was suggested by van Rensburg and Rechnitzer [141] who studied adsorption of

SAWs in 2d and 3d obtaining extremely accurate data for N up to 300. They demonstrated

that analysis of the specific heat data is fraught with difficulty. A more robust procedure

is based on studying the energy ratios En+1/En for walks of different lengths. The values

reported were φ = 0.501 ± 0.015 (in accordance with the exact theoretical value 1/2) and

χc = 0.565 ± 0.010 for the square lattice and φ = 0.5005 ± 0.0036 and χc = 0.288 ± 0.020

for the cubic lattice.

All the works mentioned above started with the scaling crossover ansatz (19). Explicit

corrections to scaling were introduced by Grassberger [142] in his analysis of the exten-

sive numerical data obtained by a chain growth Monte-Carlo algorithm (pruned-enriched

Rosenbluth method for self-avoiding walks on simple cubic lattice with adsorbing surface)

for N up to 8000. It was shown that corrections to scaling persist even for very large

N beyond 1000, masking the true value of φ which was estimated to be somewhat below

1/2: φ = 0.484 ± 0.002 ; χc = 0.28567 ± 0.0001. An earlier estimate obtained by Hegger

and Grassberger [130] (φ = 0.496 ± 0.004 ; χc = 0.286 ± 0.003) and consistent with super

universality hypothesis seems to be just an apparent value for N ≤ 2000.

It is worth mentioning some results for off-lattice models. MC simulation of an off-lattice

model with N up to 512 demonstrated that the heat capacity extrapolated to N → ∞

experiences a finite jump which is consistent with the mean-field value of φ = 0.5 [143]. MC

simulations of a bead-spring model reported φ = 0.5± 0.02 [144].

One can summarize that the exponent φ is quite close to φ = 0.5 although its exact

value is still being debated. Very importantly, this means that the thermodynamics of real

self-avoiding polymers is to a large extent described by ideal chain models. Quantitatively,

excluded volume effects lead to a shift in the transition point, a change in the numerical

values of the entropy of a strongly adsorbed state etc., but these effects are of the same

nature as the difference between various lattice and off-lattice ideal chain models. This is

in contrast to geometrical characteristics (the size of the chain in the lateral and normal

28

directions) which are governed by the Flory index ν for 2d and 3d cases, respectively, and

therefore quite different for ideal and real chains. There is, therefore, a strong motivation to

consider adsorption of ideal chains in more detail and in particular, finite-size effects. For

lattice model, finite-size effects can be evaluated only by numerical calculations. The only

exactly solved model that includes N explicitly is the adsorption of a continuum Gaussian

chain model.

3. Continuum model of adsorption of an ideal chain.

Exact Green’s function. Chain configurations in the continuum model are described

as Brownian trajectories in the presence of a planar surface situated at z = 0. The total

statistical weight, G0(z, n), of all configurations for a walk of n steps (of unit rms step length)

starting at the surface and ending at position z satisfies the Edwards diffusion equation [145]:

∂G0

∂n=

1

6

∂2G0

∂z2(22)

The initial condition for a grafted chain is G0(0, z) = δ(z). Interaction with the surface is

introduced through a boundary condition G−10 ∂G0/∂z|z=0 = −c , equivalent to a pseudopo-

tential.

Subscript 0 refers to the absence of an external force and will be useful for the later

discussion. The critical point corresponds to c = 0, while the adsorption region to c > 0;

in the latter case, c−1 is the average thickness of the adsorbed layer. For strong adsorption

(large positive c) this thickness becomes smaller than the step length, which is unrealistic

but inherent to the continuum model neglecting the finite monomer size.

The solution of (22) is well known [35, 36] and may be written as:

G0(z,N, c) =(R√π)−1

e−z2/4R2 [

1 + cR√π Y (z/2R− cR)

](23)

Here R =√N/6 is the rms gyration radius and Y (x) = exp(x2)erfc(x), where erfc(x) is

the complimentary error function. Integrating G0 over the position z of the free end gives

the partition function:

Q0(N, c) = Y (−cR) (24)

29

The average number of adsorbed segments in the continuum model is 〈m〉 = ∂ lnQ0/∂c.

Partition function as realization of crossover ansatz. The asymptotes of the partition

function are:

Q0(N, c) = Y (−x) ≈

2ex

2x� 1

1− 2x/√π |x| � 1

(√πx)−1 −x� 1

(25)

Comparing these results with the general crossover scaling ansatz, we see that the argu-

ment x = c√N/6 , of the partition function in the continuum model, is nothing but the

crossover scaling parameter τN1/2 for Gaussian chain. Thus the scaling function Ψ(τN1/2

)is obtained explicitly as Y (−cR). This is a unique example of the scaling crossover func-

tion known exactly. All the strong and weak points of the continuum model stem from

this fact. In particular, the free energy per monomer in the adsorption region (cR � 1)

is βµads = −N−1 lnY (−cR) ≈ c2/6 and the average fraction of surface contacts 〈θ〉 = c/3

grows linearly without any saturation. Saturation effects in the strong adsorption regime

necessarily involve corrections to the crossover scaling.

Comparison with the Landau theory. For a Gaussian chain adsorbing onto a planar

surface, the Landau free energy will be a function of the fraction of adsorbed segments

θ = m/N . The partition function for a given fraction of contacts P (θ) was obtained in [146]

and has the form

P (θ) =

(πN

6

)−1/2

exp

(−Ncθ − 3

2Nθ2

)(26)

This leads to an exact expression for the Landau free energy

Φ(θ) = −N−1 lnP = Φ0 − cθ +3

2θ2 (27)

Note that the Landau free energy as a function of the adsorption order parameter is valid

not only in the thermodynamic limit, but for finite chains as well and is in complete analogy

with the coil-globule transition discussed in the previous section, see Eq. (15).

Mapping of continuum and lattice parameters. The continuum model is exactly solv-

able and allows analytical solutions for any chain length but it has a few inherent drawbacks.

For large values of c the fraction of adsorbed segments shows unlimited growth. There are

30

no explicit temperature effects because the boundary condition corresponds to an effective

pseudopotential. A lattice model in which a chain is described as a walk on a regular lattice

does not suffer from these deficiencies and takes into account many features of real poly-

mers and sorbents except for the interaction of monomers distant along the chain contour.

However exact analytical solutions for the partition function of lattice models exist only in

the infinite chain limit. There is a simple approach [147, 148] that allows to incorporate

finite-size effects into lattice models and to obtain analytical expressions for the partition

function Q(N, T ) .

The basic idea is to enforce the exact correspondence between the asymptotic free energy

expressions in the lattice and continuum models, µcont(c) = µads(χ). This gives the following

mapping between the interaction parameters in the continuum model and the chain on cubic

lattice:

c(χ) =

√

6 ln(

eχ

eχ−1

)− ln

(√4 + 1

eχ−1− 2)

χ > χcr

6 (e−χcr − e−χ) χ < χcr

(28)

Analytical partition function for finite lattice chains. The obtained mapping allows

one to write the partition function of a finite lattice chain in terms of the lattice interaction

parameter χ

Q0(N, c) = Y (−√N/6c(χ)) (29)

This leads to closed form analytical expressions for the equilibrium average characteristics

of finite chains as functions of temperature. In the strong adsorption limit the correct lattice

behavior is guaranteed automatically, while in the vicinity of the critical point the scaling

crossover ansatz is recovered. The two branches (28) match at the critical adsorption point

together with their first derivatives. Higher derivatives are discontinuous, which results

in a small jump of order N−1in the heat capacity at the critical point . To avoid of these

discontinuities a more complicated N -dependent mapping can be used [148]. This procedure

can be considered as an implicit introduction of sub-leading corrections to non-analytic

scaling function at the critical point. Figure 6 shows how the average bound fraction < θ >,

and the heat capacity C for finite lattice chains in the absence of a force depend on χ/χcr.

The solid curves give analytical results for N = 100, 300, and for the N →∞ limit.

31

<Θ>

aΧ

Χcr

12 3

1 2 3 4

0.2

0.4

0.6

0.8

Χ

Χcrb

1

2

3

C

1 2 3 4

0.05

0.10

0.15

0.20

0.25

Figure 6: The average bound fraction < θ >(a), and heat capacity C (b), for chains on a 6-

choice lattice in the absence of an external force as a function of χ/χc, for N = 100, 300, and

the N → ∞ limit (curves 1, 2, and 3, correspondingly). The symbols are the exact numerical

results.The critical adsorption point for chain on six-choice cubic lattice is χc = ln (6/5) w 0.182.

A small discontinuity of order N−1appears in the analytical curves for heat capacity at the critical

point which is an artifact of the mapping.

Figure 6 demonstrate clearly how the second-order adsorption transition develops when

the chain length is varied. In the thermodynamic limit the heat capacity shows a finite jump

at the transition, which is in contrast to the diverging behavior seen in most second-order

transitions for low-molar-mass systems[149]. A finite heat-capacity jump is predicted by the

Landau mean-field theory [41], and adsorption of a polymer chain is one of the few examples

(along with the transition to a superconducting state [150]) which is perfectly consistent

with this framework.

32

4. Adsorption on curved surfaces

Ideal chain. Exact solutions for the grand canonical partition function were obtained

for adsorption of an ideal lattice chain onto a thin thread [34], and on a cylinder of finite

radius R [37]. It turns out that the free energy has an essential singularity at the critical

point of adsorption so that all the derivatives of the order parameter vanish at this point.

In particular, the order parameter in the adsorbed phase vanishes upon approaching the

critical temperature as

〈θ〉 ∼ exp(−b/τ)

where b is a model-dependent numerical constant. According to the Ehrenfest classification

this indicates a phase transition of “infinite order”. The most general result based on a

continuum Gaussian model was presented by Eisenriegler et al[151] for adsorption on a

“generalized cylinder”. This object is embedded in the space of dimension d, has an infinitely

extened axis of dimension d − d1 and a curved surface of constant curvature radius R in

the co-dimension d1. An ordinary cylinder obtains for d = 3, d1 = 2, a normal spherical

object - for d = d1 = 3. A closed-form grand canonical partition function is expressed in

terms of the modified Bessel functions Kα and Kα+1 with α = d1/2−1. Due to separation of

degrees of freedom for a Gaussian chain, the value of the embedding space dimension , d, is

irrelevant. It follows that the scaling ansatz 19 generally holds with the crossover exponent

given by

φ = |d1 − 2|/2 (30)

for 1 ≤ d1 < 4 and d1 6= 2. For d1 = 4 the order parameter at the critical point scales as

〈θ〉 ∼ N/ lnN , while for d1 > 4 one obtains φ = 1 implying a first-order transition. It is

clear that the cylinder case is also special, and formally results in φ = 0. The true finite-size

scaling at the critical point is, however, logarithmic: 〈θ〉 ∼ lnN/N .

It was also proved[152][153] that the adsorption of a Gaussian chain onto a rigid rod

is strictly equivalent to the unzipping transition of two flexible Gaussian chains A and B

if monomer s of chain A can only interact with monomer s of chain B reflecting the key-

lock principle of complementary pairs in natural DNA. This can be shown by a standard

transformsation familiar from the two-body problem reduction in classical mechanics: r(s) =

rA(s)−rB(s), rCM(s) = 12

(rA(s) + rB(s)) and recognizing that the center of mass coordinates

rCM(s) and the relative coordinates r(s) are decoupled due to the special properties of

33

Gaussian chains.

Chain with excluded volume interactions. General scaling results for adsorption

of a self-avoiding walk on “generalized cylinders” were obtained by Hanke [153] by using

renormalization group arguments in conjunction with available results for quantum field

theories with curved boundaries. An estimate for the crossover exponent in the case of a

thin rigid rod in 3-dimensional space was obtained by assuming that φis a smooth function

in the (d, d1)plane. The crossover exponent is known on two lines in this plane, namely

the Gaussian line d = 4 where Eq.30applies and the so-called “marginal” line at which the

unperturbed self-avoiding walk intersects with the generalized cylinder at a manifold of zero

dimension, and as a result, one expects φ = 0. Linear interpolation gives the estimated

value φ ' 1/6 at the point (d = 3, d1 = 2) representing the rigid rod.

III. FIRST ORDER PHASE TRANSITIONS WITH PHASE COEXISTENCE

WITHIN A SINGLE CHAIN.

A. General remarks. In this section we present a unique model which allows exact

analytical investigation of first-order transitions with finite-size effects: a polymer chain

end-grafted to a solid adsorbing substrate with a normal force applied to the free end. The

transition is referred to as mechanical desorption and may be described in two different

statistical ensembles [154]. In the force ensemble the external end-force is fixed while the

fluctuating position z of the free end. The force ensemble with only one extensive variable

N conforms to the traditional analysis of phase transitions. The first-order transition in the

force ensemble has some unusual features: order parameter fluctuations and the heat capac-

ity demonstrate an anomalous pre-transitional growth according to a power law, metastable

states are completely absent, and instead of a bimodal curve, the distribution of the order

parameter has a flat region which becomes more pronounced with increasing chain length.

We demonstrate also that the origin of this anomaly lies in phase segregated configurations

providing the dominant contribution in the transition region. Coexistence of the adsorbed

and the stretched parts within the same macromolecule is quite peculiar due to the absence

of an excess surface energy at the boundary (this boundary consists of one segment only).

Near the transition point, strong fluctuations arise since all the conformations with different

phase composition have approximately the same free energy. To demonstrate clearly the

34

intra-chain coexistence of adsorbed and stretched phases we analyze mechanical desorption

in the conjugate z-ensemble and discuss an analogy with the gas-liquid transition.

A. Mechanical desorption of a macromolecule in f-ensemble.

The first analytical theory for mechanical desorption of a grafted polymer chain from a

solid surface by external end-force was constructed a long time ago [155]. It was based on an

ideal lattice model and treated only the infinite chain length limit. Later, a theory for finite

chains within a continuum Gaussian model was proposed [63, 146, 156] and subsequently

used to interpret AFM experiments [157]. The response of an adsorbed polymer chain

to a pulling force with a lateral component was also analyzed [14]. Temperature effects

and the reentrant nature of the phase diagram in the force-temperature variables at fixed

adsorption energy were discussed in [158, 159]; Monte-Carlo simulations were performed

for mechanical desorption of an off-lattice chain with excluded volume interactions [143,

160, 161]. The authors also proposed a theory incorporating excluded volume effects in the

limit of asymptotically long chains. Importantly, the unusual features of the mechanical

desorption mentioned above are completely unaffected by excluded volume interactions.

The Green’s function and the partition function. This is a generalization of the

standard adsorption-desorption problem discussed in Section II. A constant force is applied

to the free end of the chain in the direction normal to the surface. Within the continuum

model of ideal Gaussian chain the short-range interaction with the surface is described by the

adsorption parameter c (the pseudopotential amplitude). The Green’s function G0(z,N, c)

of the macromolecule with one end fixed at height z is given by Eq.(23). The presence of the

external end-force f changes the statistical weight of every configuration by the Boltzmann

factor ezf , where the temperature T is taken as unity and incorporated into f . Positive f

corresponds to stretching the chain while negative force means pressing the free chain end

down to the surface. The force-modified Green’s function becomes:

G(z,N, c, f) = G0(z,N, c)ezf =(R√π)−1

ezf−z2/4R2 [

1 + cR√π Y (z/2R− cR)

](31)

Integrating G over the position of the free end gives the partition function :

35

Q(c, f) =cY (−cR)− fY (−fR)

c− f(32)

The average height of the free end is 〈z〉 = ∂ lnQ/∂f . The average number of adsorbed

segments is 〈m〉 = ∂ lnQ/∂c.

Adsorption-force symmetry and the phase diagram. The partition function of an

adsorbing chain with the external field is remarkably symmetric with respect to an inter-

change of the adsorption parameter c and the external force f . Hence all the moments of the

distribution functions for the number of adsorbed segments P (m|c, f) and for the height of

the chain end P (z|c, f), as well as the functions themselves, coincide under the interchange

of the conjugated parameters c↔ f [156] . In particular, a restricted partition function for

configurations with a given number of adsorbed segments, m, follows from Eq.(31)

P (m|c, f) =(πR2

)−1/2ecm−(m/2R)2

[1 + fR

√π Y (m/2R− fR)

](33)

The fraction of adsorbed segments θ = m/N is the standard choice for the order parameter

in the adsorption problem [63]. A second order parameter conjugated to the end-force can

be defined as the stretching degree ζ = z/N . In the thermodynamic limit N →∞ one can

speak of a definite phase state for the macromolecule.

Possible states of the chain for various values of the adsorption interaction parameter c

and the force f are expressed by the phase diagram, Figure 7. Phase diagram in terms of

the parameters f and c is remarkably symmetric.The solid line c = f for positive f and c in

the phase diagram is the first-order transition line between adsorbed and stretched states.

The vertical dashed line at c = 0, f < 0 and the horizontal dashed line at f = 0, c < 0

correspond to second-order transitions involving single - loop conformation. The point at

u = 0, c = 0 is the bicritical point.Typical conformations of a chain in stretched phase , and

in adsorbed phase are shown on Figure 7.

It is clear from the diagram that the classical problem of adsorption with no force applied

corresponds to moving along the horizontal dashed line of second order transitions through

the bicritical point. An equivalent dual situation is realized when the adsorption parameter is

fixed at c = 0 and the force applied is changed from pressing down to stretching. Adsorption

of a chain with the end pressed down (constant f < 0) is equivalent to stretching of a

chain anchored to an inert or repulsing surface, and is also a second order phase transition.

36

ISOTROPIC

PHASE

ADSORBED

PHASE

Second order transitions

Second

order

transitions

First

order

transitions

c

f

Bicritical

point

STRETCHED PHASE

Figure 7: Phase diagram in terms of the parameters f and c. The solid line for positive f and c

in the phase diagram is the first-order transition line between adsorbed and stretched states. The

vertical dashed line at c = 0, f < 0 and the horizontal dashed line at f = 0, c < 0 correspond

to second-order transitions involving single - loop conformation. The point at f = 0, c = 0 is the

bicritical point.Typical conformations of a chain are shown in stretched phase (a), and in adsorbed

phase (b).

Adsorption of a chain under constant stretching force f > 0 applied is equivalent to tearing

off an adsorbed chain under the condition of c = const.

Partition function around the transition line f = c. The coexistence line can be

crossed by changing the force f , the adsorption parameter c, or both. Experimentally, a

natural way to induce desorption is to increase the force keeping the adsorption parameter

constant. The detachment point is then at ftr = c. For small deviations (f − ftr) and rela-

tively large total free energy of a pure phase at the transition line, F ∗ = −Nc2/6 = −Nf 2/6

, |F ∗| � 1 , the partition function in the vicinity of the transition can be approximated as

Q = 2cec

2R − fef2R

c− f= Qtr e

t sinh t

t(34)

where t = τ |F ∗|, τ = (f/c) − 1 is relative deviation from the transition point, and

Qtr = −4F ∗ exp(−F ∗) is the partition function exactly at the coexistence line [159]. In

this approximation, F ∗ is to be treated as a constant and all the dependence on the control

parameters f and c resides in the universal parameter t which is negative on the adsorption

37

side and positive on the stretched side of the line.

It follows from (34) that the change of the reduced order parameter 〈ζ〉 /ζtr in the vicinity

of the transition is described by a universal function of a single parameter t:

〈ζ〉ζtr

= 1− t−1 + coth t, (35)

where ζtr = ftr/6.

Unconventional features of the first-order transition. As discussed in the Introduc-

tion, the standard picture of a first-order transition includes a bimodal distribution of the

order parameter with a corresponding two minima in the Landau free energy. Two dis-

tinct competing minima imply also that there is no pre-transitional growth of fluctuations.

Contrary to that, fluctuations in both order parameters in mechanical desorption display

power-law growth upon approaching the transition point. The approximate form of the par-

tition function, Eq.(34) predicts that the reduced mean-square fluctuations for both order

parameters are described by one and the same universal function of the composite parameter

t:

〈δζ2〉〈δζ2〉tr

=〈δθ2〉〈δθ2〉tr

= 3(sinh−2 t+ t−2) ≈

3t−2 |t| � 1

1− t2/5 |t| � 1, (36)

where t = (f/c−1)Mc2/6 and 〈δζ2〉 = 〈δθ2〉 = 13(ctr/6)2 [159]. The mean-square fluctuations

of the adsorption order parameter are directly related to the heat capacity which is commonly

discussed in more conventional phase transitions. The heat capacity per segment is C/N =

(βε)N 〈δθ2〉 where ε is the energy per contact. A way to express the c parameter in terms of

βε was discussed above: this is a smooth function not affecting the anomalous growth near

the transition point. It follows from (34) that the heat capacity demonstrates a power-law

growth with index α = 2 . This is in contrast to the δ-peak singularity in the heat capacity

typical for “normal” first-order transitions. Figure 8 displays the reduced heat capacity per

segment (βε)−2C/N calculated from the full partition function for several chain lengths

N =500, 750, 1000 at fixed adsorption strength c = 1 as a function of force f .

Anomalous fluctuation growth is consistent with a unimodal distribution function of the

order parameter that broadens upon approaching the transition. Figure 9 displays the

distribution of the stretching order parameter near and exactly at the transition point.

38

N<∆Θ 2 >

f1

3

2

0.85 0.90 0.95 1.00 1.05 1.10 1.15

2000

4000

6000

8000

10 000

Figure 8: Fluctuations of the adsorption order parameter N⟨δθ2⟩vs. force f for lattice ideal

chains at fixed adsorption parameter with ftr = c = 1 for several chain lengths N = 500, 750,

1000 (curves 1, 2, and 3, respectively). The dashed line corresponds to the power-law asymptotics

C ∼ (f − ftr)−2.

P(z)

z

f =0.99 c

f =1.01 cf = c

0 100 200 300 4000.0

0.1

0.2

0.3

0.4

Figure 9: Distribution of the free-end position of a lattice ideal chain with length N = 1000 at

external end-force around the transition point ftr = c = 1

Unimodality in the order parameter distribution and pre-transitional fluctuation growth

are normally associated with continuous second-order transitions.

Landau free energy. The most important insight about the nature of the transition comes

from the detailed analysis of the Landau free energy as a function of the order parameter.

For simplicity we consider only the stretching order parameter, ζ = z/N . Due to the

symmetry discussed above the same results apply to the fraction of contacts, θ The Landau

free energy per segment is Φ(ζ) = N−1 lnG(z, c, f). In the thermodynamic limit N → ∞,

the function Φ(ζ) consists of two branches:

39

Ζ

F

f<c

f=c

f<c

0.2 0.4 0.6 0.8 1.0

-0.2

-0.1

0.0

0.1

0.2

0.3

Figure 10: Landau free energy per segment Φ as a function of the stretching order parameter

ζ = z/N at mechanical desorption of a Gaussian chain under external end-force. Three cases are

shown: the value of the force smaller then adsorption interaction parameter c (low curve); the force

is equal to c (middle curve); and the force is stronger then adsorption interaction (upper curve). In

two last cases the dependence Φ (ζ) is nonanalytic.

Φ(ζ) =

(c− f)ζ ζ ≤ c/3

(c− f)ζ + 32(ζ − c/3)2 ζ ≥ c/3

(37)

The function Φ(ζ) is explicitly non-analytic since the second derivative is discontinuous

at the junction point. It is clear from Figure 10 that the Landau free energy has always only

one minimum so that no bimodality in the order parameter distribution is ever present. At

the transition point itself, part of the Φ(ζ) curve is strictly flat, which brings about a finite

jump in the average order parameter. On the other hand, the gradual flattening of the Φ(ζ)

curve in the vicinity of the transition leads also to anomalous pre-transitional fluctuation

growth. The origin of this discrepancy with the classical picture is due to configurations

that consist of two coexisting phases within one macromolecule but without any interfacial

contribution to the total free energy of phase-segregated state. In order to see this more

clearly we consider the polymer chain desorption in the conjugated ensemble, where the

position of the free chain end is fixed at height z above the plane while the reaction force

acting on the end monomer is fluctuating.

40

B. Mechanical desorption of a macromolecule in the z-ensemble.

In the previous sub-section we considered desorption of a single end-grafted chain by

applying an external force to its free end. We demonstrated that there is abrupt transition at

the critical value of the force with some very unusual features. To undestand more clearly the

nature of the transition we discuss the situation when the free end of the end-grafted chain

is fixed at some distance z above the adsorbing solid surface, and consider the equilibruim

characteristics of the chain as functions of z. We show below a certan critical distance the

chain can be viewed as comprised by two coexisting phases, namely, the adsorbed and the

stretched parts. Above the critical distance, the adsorbed phase disappears and the chain

exists in a pure stretched phase. It is important that in a rather broad range of z values the

two phases coexist and their local properties do not change.

Analogy with gas-liquid transition in (N,P, T ) and (N, V, T ) ensembles. It was

mentioned in the Introduction that the standard classification and the qualitative picture

of phase transitions commonly taken for granted applies only to statistical ensembles with

only one extensive variable, such as the Gibbs (N,P, T ) ensemble for fluids. Gas-liquid

condensation in the canonical (N, V, T ) ensemble displays some very unorthodox features if

common-wisdom criteria are applied without proper caution. We specifically analyze some

details of gas-liquid transition in the two conjugate ensembles using the simplest textbook

model of van der Waals fluid. We feel that this discussion is very instructive as it pro-

vides a simple analogy and enriches intuition for understanding single-macromolecule phase

transitions.

Gas-liquid transition is a classical example of a first order phase transition [27, 31, 67].

In the Gibbs ensemble, the phase state is uniquely defined by the two intensive parameters

P and T , independently of the sample size. At fixed P and T the Gibbs free energy is

proportional to the number of particles, N . In the P − T phase diagram the coexistence of

liquid and gas phases corresponds to a certain line Pcoex(T ), which terminates at the critical

point. Upon crossing the coexistence line volume, entropy, average energy, etc. change

abruptly. Each point on the coexistence line corresponds to many phase-segregated states

with different ratios of relative phase volumes. In order to study more closely the states with

phase coexistence canonical ensemble is particularly useful. At a fixed temperature and a

fixed number of particles, volume changes may also induce the liquid-gas phase transition,

41

Figure 11: Thermodynamic analogy between gas-liquid system and an adsorbed polymer chain at

several positions of the end. Coexistence of two phases under piston and inside a single shain are

shown in two situations. The difference in interfacial effects is dicussed in the text.

T=0.8Tc

a

V

Vc

P

Pc

2 4 6 8

0.2

0.4

0.6

0.8

1.0

1.2V

Vc

U

NTc

b

2 4 6 8

-3

-2

-1

1

Figure 12: (a) Reduced isothermal pressure P/Pc (a), and the average potential energy per particle

(b) as a function of reduced volume V/Vc are shown for the van der Waals fluid at fixed temperature

T/Tc = 0.8; coexistence (binodal) curve is shown by the dashed line.

but this transition looks quite different due a wide interval of phase coexistence. Following

an isotherm along the V axis one passes from a homogeneous liquid state at low V into the

region where liquid and saturated vapor coexist in equilibrium (Figures 11b, 11c).

Upon further increase in the volume the liquid phase disappears (Figure 11d) and then

the vapor becomes unsaturated (Figure 11e). For a real gas, the isotherm or the P (V )

dependence at a fixed temperature shows a horizontal step at P = Pcoex, which corresponds

to the coexistence range of the two phases Figure 12 a.

The average potential energy per particle is shown in Figure 12 b as a function of the vol-

ume (kinetic energy of 3/2kT is not included to make a closer comparison with the polymer

42

system where kinetic energies are typically omitted). Note that the average potential energy

per particle in a particular phase depends on the local density and has to be calculated

separately for the two phases. The global average is then found by weighting according to

the relative sizes of the phases.

In the region of phase coexistence, the fraction of particles belonging to the liquid phase

changes linearly as a function of volume. This leads to a linear portion in the range of

phase coexistence as shown in Figure 12 b. It is clear from Figure 12 that in the (N, V, T )

ensemble, the phase transition is not abrupt but extends over the wide interval of changes in

volume V . Jump-wise changes typical of the first order phase transitions are observed only

in the P ensemble, where the region of phase coexistence collapses into a single point. The

V (P ) isotherm is obtained via rotation (Figure 12 a) and shows an abrupt volume change

at P = Pcoex. At this pressure, abrupt changes in energy and entropy likewise occur and the

fraction of molecules existing in the condensed phase changes from unity to zero.

The canonical ensemble allows yet another setting for inducing the gas-liquid transition:

one can change the temperature of the sample under fixed volume conditions. Cooling an

initially homogeneous fluid one would observe the onset of phase segregation below certain

temperature, and phase coexistence will persist for all lower temperatures. Figure 13 a shows

the change in the average potential energy per particle with temperature at two values of

the volume V/Vc = 1; 3, where Vc is the critical volume.

Again, instead of a jump only a discontinuity in the slope is observed. Without proper

awareness of the ensemble involved one may be wrongly tempted to interpret this as a sign

of a second-order transition. However, this is just a hallmark of the underlying extended

phase coexistence. It is worth noting that in this setup the coexistence line Pcoex(T ) is

never crossed (but only followed) . The process of cooling a fluid at fixed volume can be

conveniently illustrated in the (T, n) diagram, Figure 13 b, n = V/Vc is the reduced average

density (n = 1 at dense packing). The binodal curve in the (T, n) diagram gives the boundary

of the region unstable with respect to phase segregation and consists of two branches joining

at the critical point. The branches describe the local densities of the coexisting liquid and

gas phases as functions of temperature. The path followed by the sample upon cooling at

fixed average density is indicated by arrows: for an initially homogeneous fluid the path is

vertical; upon reaching the binodal line the sample splits into two phases with different local

densities.

43

T

Tc

U

NTc

V = Vc

V = 3 Vca

0.6 0.7 0.8 0.9 1.1

-2.5

-2.0

-1.5

-1.0

-0.5

T

Tc

n

b

0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

Figure 13: (a) The average potential energy per particle (in units of Tc) of the van der Waals fluid

vs. reduced temperature T/Tc at fixed volume V = Vc and V = 3Vc. (b) Paths followed by the

fluid upon cooling at fixed average density n: moving down the temperature axis, an initial state

of uniform density separates into two phases with different local densities described by the two

branches of the binodal line; dotted lines represent cooling at fixed critical volume (V = Vc), solid

lines - at V = 3Vc.

Interestingly, the shape of the 〈ε〉 vs. T curve for the van der Waals model remains

qualitatively the same even if the system actually passes through the critical point (see the

curve with V = Vc). In sharp contrast to the Gibbs ensemble, the temperature dependence

of the average energy〈ε〉 vs. T measured in the (N, V, T ) ensemble is completely insensitive

to the distance from the critical point, although other characteristics (e.g. compressibility)

would signal the onset of critical phenomena.

An advantage of the (N, V, T ) ensemble in studying phase coexistence is that if one

analyses, e.g., the local density profile instead of quantities averaged over the whole sample,

the inhomogeneity of the phase-segregated sample becomes obvious.

The partition function of the desorbing chain in the z-ensemble. The partition

function in the z-ensemble coincides with the Green’s function of the original continuum

model of adsorption:

Q(z,N, c) = G0(z,N, c) =(R√π)−1

e−z2/4R2 [

1 + cR√π Y (z/2R− cR)

](38)

The average reaction force, 〈f〉 = ∂ lnQ/∂z, at the end-monomer at point z and the

44

z

a

<f>

c =0.6

c =1

0 100 200 300 400 500

0.2

0.4

0.6

0.8

1.0

1.2

1.4

< E >

N

z

c =1

c =0.6

b

100 200 300 400 500

-0.4

-0.3

-0.2

-0.1

Figure 14: The average force (a), and the everage energy (b) of the ideal chain with N = 1000

vs. the distance z of the nongrafted end above the surface at two different values of the adsorption

interaction parameter c=0.6, and c = 1.

average bound fraction 〈θ〉 = ∂ lnQ/∂c for asymptotically long chains are simplified to:

〈f〉 =

c, z/N ≤ c/3

3z/N, z/N ≥ c/3(39)

〈θ〉 =m

N=

c3− z/N, z/N ≤ c/3

0, z/N ≥ c/3(40)

One can relate the bound fraction to the average potential energy of interaction with the

adsorbing surface per monomer, 〈E/N〉 = −ε 〈θ〉 , to be directly compared to Figure 14b

Evidence of phase coexistence. The fact that in the same range of z the average

reaction force is constant, similar to the constant pressure portion of the P (V ) isotherm, is

a strong indication that phase coexistence is involved. In the same interval of z values the

average energy per particle increases linearly in the same way it changes with V in the phase

segregated fluid. Both features are consequences of the change in the phase composition with

increasing z, see the cartoon of Figure 11 b.

In the adsorbed part of the chain the probability for a segment to be in contact with

the plane is not affected by the end coordinate z and remains equal to c/3; in the torn

off tail, this probability is zero. Weighting these two contributions with the relative phase

size results in the curve shown in Figure 14 b. Beyond ζ = z/N = c/3 the adsorbed part

disappears and the chain consists of a single stretched phase. Simultaneously, the average

force starts growing with further stretching. The free energy of the polymer in the range

ζ ≤ c/3 can be understood on the basis of the standard discussion of phase coexistence.

45

The total free energy is a sum of two contributions:

F (N, c, z) = Fads(N − n, c) + Fstr(n, z), (41)

where the first term Fads(N − n, c) = −(N − n)c2/6 describes the adsorbed part containing

(N − n) segments, and the second term Fstr(n, z) = 3z2/(2n) stands for the free energy of

the stretched tail of length n with no contacts with the plane. Both terms are written in the

dominant asymptotic form although finite-size corrections are available. The equilibrium

distribution of segments between the phases is determined by the condition that the two

chemical potentials be equal: µads = µstr with µads = −c2/6 and µstr = −(3/2)(z/n)2.

This imposes the local stretching parameter in the tail in terms of the adsorption strength,

〈ζtail〉 = c/3 irrespective of the value of z as long as phase coexistence persists. The condition

of mechanical equilibrium (the equality of the normal forces) gives 〈ftail〉 = c coinciding with

the equation of the coexistence line in the f -ensemble. Using the coexistence conditions one

obtains the total free energy of the phase-segregated state as

F (N, c, z) = −Nc2/6 + cz , 0 ≤ z ≤ Nc/3, (42)

the same as the asymptotic form obtained from the full partition function (Eq.32). This

form of the free energy in the z-ensemble gives rise to the linear branch of the Landau free

energy in the f -ensemble discussed in the previous section:

Φ = N−1(F (N, c, z)− fz) = −c2/6 + (c− f)ζ (43)

Note that the form of the total free energy (43) does not include any interfacial contribu-

tions since the boundary between the adsorbed and the stretched parts in the flexible chain

is just one segment. This is directly linked to the linear structure of the polymer chain and

is eventually responsible for the absence of bimodality and of metastable states.

Local order parameter. One can still feel very much confused when the curve of the

bound fraction 〈θ〉 vs. the end coordinate z, Figure 15, is recognized as the change in the

average order parameter with the control parameter of the ensemble. It goes against the

accepted conventions to associate this smooth curve with a first order transition.

The way out of the dilemma is to realize that the order parameters 〈ζ〉 and 〈θ〉 as

introduced in the context of the f -ensemble are defined for the chain as a whole and do

46

<Θ>

z

c =1

N = 1000 a

0 100 200 300 400 500

0.1

0.2

0.3

æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ

æ

æ

æ

àààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààà

à

à

à

à

à

à

à

à

à

àààààààààà

ìììììììììììììììììììììììììììììììììì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ììììììììì

òòòòò

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

òòòòòòòòò

ô

ô

ô

ôôôôôôôô

<Θ>loc

k

z=5z=100

z=200

z=300

z=400

b

0 200 400 600 800 10000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Figure 15: The average order parameter (average bound fraction) vs the end coordinate z of the

long chain at the fix (a), and the average local order parameter as a function of local place k

along the chain at several z (b). The value of chain length N = 1000, and the value of adsorption

interaction parameter c = 1 are the same for both cases.

not allow to distinguish between the two coexisting phases. However, it is possible to define

local order parameters 〈θ〉loc = d〈m(k)〉dk

and 〈ζ〉loc = d〈z(k)〉dk

where z(k) is the z-coordinate of

k-th segment and m(k) is the number of contacts in the sub-chain from the zeroth (grafted)

segment to the k-th segment. The average number of contacts in the sub-chain (0, k) depends

on the position zk of the k-th segment and can be obtained from the Green’s function

G(zk, k, c) given by Eq.(23). Then, the result 〈m〉 (k, zk) has to be averaged over zk with

the weights W (zk) = G(zk, k, c)G(zk, z, N − k, c) where the second term is the Green’s

function of the (N − k)-segment sub-chain with the two ends fixed at zk and z. For the

continuum Gaussian model this Green’s function is well-known and is given by Eq. (23).

Hence calculating the local order parameter profiles is a straightforward exercise. Figure 15

a,b compares the behavior of the global order parameter 〈θ〉 as a function of z for chain of

length N = 1000 at fixed adsorption parameter c = 1 with the series of local order parameter

profiles 〈θ〉loc (k) for the same system. The profiles are calculated at several values of z = 5,

100, 200, 300, 400.

It is clear that with the increase in z the boundary between the adsorbed and the stretched

phase moves closer to the grafted end but the local properties of the adsorbed part of the

chain do not change. Thus the gradual decrease in the global order parameter shown in

Figure 15 a does not contradict the fact that the transition is first order and reflects the

specifics of the z-ensemble allowing extended phase coexistence.

A major difference between the gas-liquid transition and the mechanical desorption is

47

related to the interfacial effects. It is the interfacial free energy that leads to metastable states

and the bimodal distribution of the order parameter in the gas-liquid transition. Tearing

off an adsorbed flexible macromolecule by one end is a unique example of phase coexistence

without interfacial effects. The boundary between the two phases is essentially nominal and

consists of one segment. As a result, in the force ensemble close to the detachment point

fluctuations involving vastly different conformations with different phase composition (from

purely adsorbed to completely torn off chain) without any barrier separating them. This is

the ultimate origin of large pre-transitional fluctuations and of unimodal order parameter

distribution. The argument based on phase coexistence without a physical interface is valid

not only for ideal chains but for chains with excluded volume interactions as well. That

is why the nature of mechanical desorption transition is qualitatively unaffected by these

interactions.

Comparison of two ensembles. Although mechanical desorption looks quite differently

in the f - and z-ensembles, the fundamental theorem of statistical mechanics stating that all

ensembles are equivalent in the thermodynamic limit, holds. Indeed, ensemble equivalence

means that the equations of state are the same irrespective of which set of parameters is

independently controlled and which are considered as a fluctuating response to be measured

or evaluated in the framework of statistical mechanics. For the system under consideration,

〈z〉 vs f represents the force-extension curve (equation of state) in the f - ensemble, while

〈f〉 vs z corresponds to z-ensemble. Ensemble equivalence would mean that one function is

the inverse of the other. Figure 16 displays the force-extension curves obtained in the two

ensembles for a finite but fairly long chain (N = 1000). It is clear that there is still some

difference between the curves but both tend to the simple limiting shape described by

〈zf〉Na

=

0 βf < c

βfa3

βf > c(44)

A more subtle consequence of the ensemble equivalence is that other thermodynamic

averages (e.g. the bound fraction〈θ〉 expressed in terms of z or 〈z〉 should also coincide.

This is indeed correct in the proper N →∞ limit [154].

Ensemble equivalence does not extend to the description of fluctuations. The fact that

fluctuations in the canonical and grand canonical ensembles are different is well recognized

[65]. The difference in the character of fluctuations for mechanical desorption in f - and

48

z/N ,<z>/N

f

<f>

z-ensemble

f-ensemble

0.1 0.2 0.3 0.4 0.5 0.6

0.6

0.8

1.2

1.4

1.6

1.8

Figure 16: Force-extension curves at mechanical desorption of a Gaussian chain in the f -ensemble

(〈z〉 vs f , dashed line) and in the z- ensemble ( 〈f〉 vs z , solid line) at fixed adsorption parameter

c = 1; N = 1000.

z-ensembles is quite dramatic as illustrated below.

The change in the distribution function for the number of contacts is shown for the case

of c = 1 and N = 500 in Figure 17. In the f -ensemble (Figure 17 a), the distribution is

anomalously broad exactly at the transition point f = 1 and much narrower both above

and below the transition (f = 0.9; 1.1). In the z-ensemble (Figure 17b), the values of z are

chosen to coincide with the average chain extension values〈z〉 of Figure 17a. In contrast to

the f -ensemble the width of the distribution does not change appreciably with the change

in z.

We conclude Section III by re-stating the main qualitative results. First, the notion

of phase coexistence withing a single macromoecule provides a close analogy between the

phenomenon of mechanical desorption and standard gas-liquid transition. It is exactly this

analogy that gives a natural explanation of the apparently different pictures observed in the

f- and z-ensembles. Whereas in the force ensemble coexistence is restricted to a line in the

phase diagram (or to a single transition point when one of the control parameters is fixed)

in the conjugate z-ensemble it can be observed in a wide range of change in z, similar to the

gas-liquid coexistence in the NVT ensemble. These ideas are not yet commonly accepted, see

for example the discussion in [160, 161]where the transition in both ensembles was referred to

as “dichotomic” without any phase coexistence. Unconventional behavior of all the common

thermodynamic functions in the z-ensemble can be quite confusing if one does not realize

that this is just a result of re-distribution of matter between the two coexisting phases: the

49

Figure 17: Distribution of the number of contacts during for mechanical desorption in the f -

ensemble (a) and in z- ensemble (b) at c = 1 and N = 500. The curves correspond to the three

values z = 1, 85, and 200 in the z-ensemble and to the same three values of 〈z〉 in the f -ensemble

at f = 0.9, 1.0, and 1.1, respectively.(From [154])

same effect is observed in the van der Waals fluid in the Helmholtz ensemble. If the order

parameter is defined as a local quantity this becomes unambiguously clear. Although the

basic analogy is qualitatively correct, mechanical desorption still possesses some very unusual

specific features that include pre-transitional divergence of fluctuations and the absence of

bimodal distributions. These are due to the absence of the interfacial free energy at the

boundary between the two phases.

IV. FIRST ORDER PHASE TRANSITIONS WITHOUT PHASE COEXISTENCE

WITHIN A SINGLE CHAIN.

In Section IV a we consider the conformational properties of an ideal polymer chain

tethered by one end at distance z0 near a solid adsorbing surface. The short-ranged interac-

tion between the segment and surface are describes by a pseudo-potential with adsorption

interaction parameter c. The exact expression for the partition function and the Landau

free energy are available for this model. A first-order conformational transition occurs from

homogeneous coil state to a nonhomogeneous state composed of a strongly stretched stem

and a pancake that collects the remaining adsorbed segments. We discuss the impossibility

of simultaneous coexistence between two states (phases) and the related abnormal behavior

50

Figure 18: Illustration of the systems of interest. A liquid-solid interface is a plane . An isolated

Gaussian chain is fixed by one end at a position z0 as indicated by the black dot, the other end is free.

When the grafting point is sufficiently from the solid surface the chain may take an unperturbed coil

conformation or an inhomogeneous flower-like conformation with a stretched stem and the adsorbed

crown (pancake) .(From [162])

in the microcanonical ensemble.

In Section IV b we consider a similar situation where the chain is end-tethered at some

distance from a penetrable interface (instead of a solid adsorbing surface). The interface is

modeled as an external potential of a Heaviside step-function form. Special attention will

be paid to exactly solvable model with an unusual phase transition: the rolling transition of

a polymer chain tethered exactly at the penetrable interface.

Escape transition of a polymer chain compressed between pistons is considered in Section

IV c.

A. Polymer chain tethered near an adsorbing solid surface.

Model and partition function. An isolated chain near end-fixed at some distance

from an adsorbing surface as schematically presented in Figure 18.

Experimentally, such a situation can be realized with the help of an atomic force micro-

scope (AFM). The chain would be chemically attached by one end to the AFM tip. One

has to ensure that the chain does not adsorb onto the tip which can be achieved by using a

separate probe particle glued to the tip and used as a grafting substrate. This chain is then

brought near the adsorbing surface. When the tip is still far from the adsorbing surface the

chain will be in a mushroom conformation. However, the free energy of a mushroom con-

formation is approximately the same as that of an unperturbed coil, and the simple model

51

illustrated in Figure adequately represents the experimental situation. With the AFM appa-

ratus it is possible to measure the force on the chain as soon as it takes the partially adsorbed

flower conformation. In effect, the flower bridges the gap between the tip or probe, and the

adsorbing surface. The force which is easily picked up by the AFM apparatus is expected

to be independent of the separation and is only a function of the adsorption parameter that

characterizes the affinity of the chain for the adsorbing surface. In AFM experiments it is

possible to control the time of contact or the time of close proximity of the chain to the

adsorbing surface. Therefore, one should expect to observe the hysteresis effects discussed

bellow in full glory.

The exact partition function Q(z0, N, c) of an ideal polymer chain was obtained in [162]

and has a form

Q(z0, N, c) = Qcoil(z0, N) +Qfl(z0, N, c) = erf(z0/2R) + e−z2/4R2

Y (z0/2R− cR) (45)

The two terms correspond to two distinct states of the macromolecule. The first one

describes the state of a weakly perturbed ideal coil, while the second describes an inhomo-

geneous conformation consisting of an adsorbed part and a stretched stem connecting the

grafting point and the adsorbing plane

The intersection of the two branches of the free energies gives the line of first-order

transitions. For z0 � R and c� 1/R the transition line is defined by

( z0

Na

)tr

=ac

6(46)

Along the transition line the average number of segments in the stem is N/2, i.e. exactly

one half of the chain. The remaining half constitutes the adsorbed pancake.

Landau function. It was suggested [163, 164] that the chain stretching can be used as

the order parameter. For the coil states, this parameter refers to the chain as a whole,

s = (z − z0)/Na . The maximum value of the order parameter in the coil state is achieved

when the free end is touching the surface - at z = 0. For the flower states, only the stem is

stretched, and the order parameter is defined as s = z0/na where n is the number segments

in stem. The following simple analytical expressions for the two branches of the Landau free

energy were obtained in thermodynamic limit:

52

s

F

c = 0.6 0.9

1.2

1.5a

-0.2 0.2 0.4 0.6 0.8 1.0

-0.05

0.05

0.10

0.15

s

F z0 /N = 0.4

0.3

0.2

0.15b

-0.2 0.2 0.4 0.6 0.8 1.0

-0.1

0.1

0.2

0.3

Figure 19: The Landau free energy Φ as a function of the order parameter s for Gaussian chain

at fixed reduced distance z0/N = 0.2 from the adsorbing solid surface at several values of the

adsorption parameter c (a), and for fixed adsorption parameter c = 1.2 and several values of the

reduced distance z0/N (b). N = 1000 in both cases.(According the work [64]).

Φ(s, c) =

32s2, s ≤ s0

32ss0 − c2

6

(1− s0

s

), s ≥ s0

s0 = z0/Na (47)

The two branches of the Landau function match each other at s = L/Na.

The Landau free energy Φ(s) calculated according to (Ref:eq:53) is presented in Figure

19 as a function of the order parameter. A set of typical curves Φ(s) for several values of the

adsorption parameter c at a fixed reduced distance z0/N = 0.2 (s0 = 0.2) is shown in Figure

19 a . The coil state branch and the point where two branches meet do not change with the

adsorption parameter c, while the flower state branch is clearly affected by it. With affinity

for the surface, the minimum in the flower branch becomes more pronounced.

In Figure 19 b the other control parameter, z0, is varied, and the adsorption parameter

is kept constant. Again, the Landau function of the coil state remains the same, but the

matching point is shifting with z0.

The binodal condition, when the two minima are equally deep, leads to equation (46)

found from the analysis of the partition function. The equation of the spinodal line has a

form(z0Na

)sp

= ac3. The barrier height counted from the coil state is ∆coil = 3z0/2N , and

corresponds to the free energy of stretching the chain Fstr. The barrier height with respect

to the flower minimum is given by the combination of the free energy of stretching and that

of adsorption

53

∆fl =

(√3z2

0/2Na2 − ca

√N/6

)=(√

Fstr −√Fads

)2

(48)

Along the spinodal line the average number of segments in stem is equal to N , which means

that the pancake just has not developed. We conclude that metastable flower state has more

than N/2 segments in stem.

Stretching force. The average stretching force acting at tethering point is given by relation:

f = Q−1e−3z20/2Na2

cY (z0/2R− cR) (49)

and in the thermodynamic limit of N →∞ and finite z0N

has a simple behavior:

f =

0, c < ccr

c, c > ccr

(50)

In the vicinity of the transition point the force the force decreases linearly with z0:

f ≈ ftr −Nf 2

tr

2

( z0

Na− ca

6

)(51)

The dependence of the force on the position of the fixed end for several values of the

adsorption parameter c is displayed in Figure 20 a. As the chain end is moved away from

the adsorbing surface, the force remains constant until we approach closely the binodal

distance.

At larger distances, the chain is effectively torn away from the adsorbing surface and

the force is practically zero. In the coordinates f vs z0/N used in the phase diagram and

Figure 20 a this results in a sharp drop of magnitude ∆f = c. The sharpness of the drop is

proportional to N(ftr)2 ∼ Nc. The description in terms of the Landau function allows us to

introduce the force associated with metastable states and demonstrate the hysteresis effect

on Figure 20b,c.

The influence of the excluded volume effects on the transition point and spinodal equation

was discussed in [164, 165] for a very similar situation of the escape transition, see subsection

C below. Due to the model equivalence established in [162] this applies equally to the chain

attached near an adsorbing surface considered here.The binodal and spinodal lines come

close to each other than in the case of ideal chains. The region where metastable flowers

54

Figure 20: (a) The force f needed to keep the tethered chain with the end at z0 versus the reduced

distance z0/N . The sign of the force is in reality negative because the chain is attracted to the

surface. The number of segments is N = 1000 and the c parameter is varied as indicated. (b) The

force versus the adsorption parameter c for various values of the chain length N as indicated and

for z0/N = 0.2. The dashed lines with the arrows represent the hysteresis effects. (c) Example of

a reduced force as a function of the reduced distance z0/N is presented for c = 0.6 and N = 1000,

with special attention to the hysteresis effect indicated by the dashed lines with the arrows.(From

[162])55

exist is relatively smaller, since the adsorbed part at the coexistence line comprises only

2N/5 segments.

Microcanonical ensemble and the entropy gap.

A very unusual feature of the first order transition described above is that, at the tran-

sition point, the two states cannot coexist simultaneously due to their very nature. Both

of them, the coil and the “flower” (stem + adsorbed pancake) are essentially defined for a

macromolecule as a whole, but not for constituent sub-chains. It is therefore impossible

to construct the mixed state in which both phases coexist in arbitrary proportions. This

peculiarity is closely linked to some abnormal thermodynamic properties of the same system

considered now under conditions of the microcanonical ensemble. The role of the appropri-

ate thermodynamic potential would be played by the entropy considered as a function of

energy and number of segments, S(E,N). In the ideal chain model the energy is only due

to the contacts made by chain segments with the adsorbing surface, E = −cm, where m is

the number of contacts. The entropy can be found from the restricted free energy calculated

under condition that the number of contacts is fixed. The corresponding partition function

is represented as a sum [63]

Q(N, c,m, z0) =

N̂

m

Q(k, c,m, 0)G(N − k, 0, z0)dk (52)

where the first term in the integrand represents the partition function of the adsorbed

pancake consisting of k segments with both ends in contact with the surface, and the second

term is the standard Green’s function of the stem with two ends at z = 0 and z = z0,

correspondingly. The restricted partition function Q(k, c,m, 0) is found from Eq. 33 by

taking the negative (pressing down) force. Evaluation of the integral by steepest descent

method yields the following asymptotic result for the restricted free energy F (c,N,m) =

−Ncm + 3/2m2 and S(c,N,m, s0) = −Ncm + 3/2m2 for the entropy as a function of the

number of contacts. All of the above considerations do not apply to a special case of zero

energy (no contacts with the surface). Asymptotically, the entropy of such a state is just

zero since a Gaussian coil serves as a reference state. Figure 21a displays the entropy per

segment as a function of the number of contacts, m, for a chain of length N = 500 at two

different grafting distances. The special state with m = 0 is indicated by a big dot. A very

unusual feature of the entropy as a function of energy is that it is not just non-analytic

56

but discontinuous. One can say that the coil state with no contacts is separated from all

the other states by an entropy gap. The magnitude of the gap increases with the reduced

grafting distance z0Na

. We stress that the standard jump in the entropy associated with

a first order transition in the canonical ensemble (where the entropy is a first derivative

of the thermodynamic potential) should not be confused with the discontinuous behavior

shown in Figure21since the microcanonical entropy is the thermodynamic potential by itself.

Physically the origin of the entropy gap is linked to the polymeric nature of the system and

to the effect of grafting: in order to form the first contact with the surface, the chain has to

deform anf lose configurational entropy.

The second unconventional feature of the microcanonical entropy with far-reaching con-

sequences is that S(E) is not convex everywhere. Figure 21b shows S(E) together with

its convex hull. Concavity of the microcanonical entropy will lead to inequivalence of the

microcanonical and canonical ensembles. The states corresponding to the entropy branch

lying below the convex hull are not thermodynamically stable in the canonical ensemble

(they are only metastable). In the canonical ensemble the system would undergo a jump

between the two states indicated by dots connected by a dashed segment.

Ensemble inequivalence will be disussed in more detail below in the subsection dealing

with the escape transition. There the pair of conjugate ensembles (analogs of NVT and

NPT ensembles) is more closely related to experimentally accessible situations; however, the

underlying physical reason for this inequivalence lies in the existence of two configuration

subsets separated by an entropy gap.

B. Polymer chain end-tethered near a penetrable interface (near step potential)

A liquid-liquid interface is modeled as a Heaviside step function of the external dimen-

sionless (reduced by kT ) potential u(z): it assumes a value u > 0 for positive z and zero

otherwise. An isolated Gaussian chain is fixed with one of its ends at a position z0. When

the grafting point is at any negative z0or at positive z0 far enough from the interface the

chain is approximately in a Gaussian conformation indicated by the big spheres. When the

grafting end is situated in the high potential region (z0 > 0)and close enough to the interface

a flower may form consisting of a stretched stem and a coiled crown, as presented in Figure

22. Comparing Figures 18 and 22one can clearly see the similarity between the two models.

57

ææ

ææææææææææææææææææææææææææææææææææææææææææææææææææ

àààààààààààààààààààààààààààààààààààààààààààààààààà

m

S

z0 = 250

z0 = 150

a

0 10 20 30 40 50 60

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

æ

S

Ε

z0

N= 0.3, c = 1

b

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Figure 21: (a) The entropy per segment S as a function of the number of contacts, m, for a chain of

length N = 500 at two different grafting distances z0above the solid surface at contact interaction

parameter c = 1.The values of z0 are shown on Fugure. The special state with m = 0 is indicated

by a big dot. (b) The microcanonical entropyS(E) at the fixed values z0/N = 0.3, and c = 1. The

states lying below the convex hull (below dash line) are metastable.

This similarity can be put on a firm mathematical basis [162] by comparing the Landau free

energy given by Eq. 53 and the corresponding Landau free energy of the chain grafted near

the step potential [166]:

Φ(s, u) =

32s2 + u, s ≤ s0

32ss0 + u s0

s, s ≥ s0

(53)

It follows immediately that all the results for the binodal and spinodal lines of the previous

subsection apply with the change ca =√

6u , namely(z0Na

)tr

=√

u6and

(z0Na

)sp

=√

2u3.

The phase diagram for lattice chain with finite extensibility is presented in Figure 23 while

the average fraction of contacts for the lattice chain with N = 200 vs the reduced grafting

distance z0/N for several values for the external potential, u, demonstrating the first order

transition is displayed in Figure 24.

1. Rolling transition: Chain attached at a liquid-liquid interface.

We consider now the special situation when a polymer chain attached directly at a liquid-

liquid penetrable interface at z0 = 0. In this case the coil is sitting with most of its segments

in the region where the potential u is most favorable. It does not leave the interface as it is

grafted with one of its ends to the boundary .

58

Figure 22: A liquid-liquid interface is modeled as a Heaviside step function of the external potential

u(z): it assumes a value u for positive z and zero otherwise. An isolated Gaussian chain is fixed

with one of its ends at a position z0 as indicated by the black dot. When the grafting point is at any

negative z or at a very positive z the chain is approximately in a Gaussian conformation indicated

by the big spheres. When the grafting end is at z0 > 0 and near the interface, a flower may form

which consists of a stem from of stretching blobs and a crown as perturbed coil; the dashed spheres

are drawn to help understand the flower structure.(From [166])

Figure 23: Phase diagram for chain end-tethered at distance z0 near step-potential u. The nor-

malized distance z0/N is plotted against the square root of the potential field. The binodal line is

shown as solid and the spinodal is displayed as a dotted line. Regions of stability and metastability

of the coil as well as the flower conformations are indicated. In the thermodynamic limit the second

spinodal line coincides with the abscissa and not indicated.(From [166]

59

Figure 24: The average fraction of contacts numerically evaluated for the lattice model withN = 200

plotted against the reduced grafting distance z0/N for several values for the external potential felt

by each segment of the polymer chain, u, as indicated.(From [166])

If in addition the molecular nature of the components that are responsible for the presence

of the interface is ignored, and replaced by an artificial external potential field felt by the

polymer units, one arrives at a model which is in fact quite general. In the following it is not

necessary to specify whether the step in the external potential has an entropic origin e.g.,

when the interface is the boundary between two polymer gels which differ only in polymer

density and not in chemical composition or is enthalpic in nature e.g., in a liquid/liquid

interface.

The exact partition function for this model was obained in [167, 168], and has a form:

Q(u,N, ) = e−uN/2 I0(uN/2), (54)

where I0(x) is the modified Bessel function. The free energy, which is counted from that of

an unrestricted Gaussian chain taken as a reference point, is given by

F (u) = − lnQ(u) ≈

uN2

[1− uN/8 + (uN/8)3] , uN � 1

12

[ln(πuN)− 1/(2uN)− 1/(2uN)2] , uN � 1(55)

From Eq. (55) is may seem odd that the free energy only grows logarithmically with N

for large uN . However, one should keep in mind that Eq. (55) is only the correction of the

free energy on the reference state of Gaussian coil which is proportional with N . The free

energy is symmetric, F (uN) = F (−uN).

60

Figure 25: (a) Average fraction of segments 〈s〉 in the negative half-space, (b) the fluctuations in

the fraction of segments in the negative half-space, as a function of the external potential u for

Gaussian chain attached at a liquid-liquid penetrable interphase . The degree of polymerization is

indicated.(From [167]).

The free energy is continuous at u = 0, but the derivative with respect to u is not. According

to the classification of Ehrenfest this discontinuity indicates that the system passes a first-

order phase transition at u = 0 - see Figure 25.

The thermodynamic quantities of the system: the full free energy, entropy, energy are

considered in [167].The behavior of the Landau free energy at rolling transition was discussed

in [168]. From the analysis of the free energy it was determined that the transition is of

the first order type. The energy, however, remains continuous at the transition, but its first

derivative jumps. This is indicative of a second-order phase transition. Finally, the entropy

in the system, and its derivatives, remain continuous at u = 0. From the exact analytical

equation for the Landau function it is found that there are no metastable states associated

with the transition . In addition the structure of the coil does not change dramatically during

this process (the end-point distributions remain a function with a single maximum).Roughly

speaking, when the step potential changes its sign, the polymer coil rolls from one region to

the other. These features show that the system discussed above is rather special.

C. Escape transition.

In this section we consider a phase transition in a single macromolecule that has received

much attention recently, namely the escape transition observed for an end-tethered chain

compressed between two pistons of circular cross section - Figure 26. At weak compressions,

the chain is deformed uniformly forming a relatively thick pancake. In this “coiled state” the

61

Figure 26: Typical snapshots showing a configuration of an imprisoned (a) and an escaped (b)

polymer chain of length N = 200, grafted under a cylindrical piston of radius L = 40 at hight

D = 12(a), and D = 4(b) over the substrate. A repulsive Lennard-Jones potential , smeared on all

surface , makes them impenetrable for the chain.The piston edge is rounded so as to preserve the

continuety of the force when a bead moves around it.(From [164]).

resistance force due to the compressed chain increases monotonously as the distance between

two pistons, D, decreases. Beyond certain critical compression, the chain conformation

changes abruptly to an inhomogeneous “escaped state” consisting of a “stem” (a stretched

string extending from the anchoring site to the piston border) and a “tail” outside of the

confining pistons. The resistance force decreases abruptly, indicating a first-order transition.

There is a very close relation between the escape transition and the conformational phase

transition of a polymer chain grafted at some distance from a solid adsorbed surface con-

sidered in the previous section. In both cases one state (the coil or the compressed coil)

is homogeneous. The other phase is inhomogeneous and consists of a stretched stem and

the tail (adsorbed or escaped). There are several reasons to discuss the escape transition in

some detail.

First, the setting of the escape transition is close to what is currently experimentally

feasible. One can envision a force spectroscopy experiment, wherein a single polymer chain

of contour length of ∼ 100− 200 nm and of gyration radius Rg ∼ 10− 15 nm is tethered to

an inert, nonadsorbing surface and compressed by the probe tip with the radius of curvature

of ∼ 20 − 50 nm. In this example it is relevant that the radius of curvature is larger then

the radius of gyration. Currently, it is difficult to make a tip in the form of a cylindrical

piston with a flat surface; however, the model of a macromolecule compressed by a piston

62

includes all physical attributes of a real system.

Second, the escape transition allows considerable modifications in geometry without af-

fecting qualitatively the underlying physics. From a theoretical point of view, an intriguing

question is whether the transition persists for chains with excluded volume interactions in 2d

space where the compressed state is not coiled but already stretched into a quasi-1d string

[165]. A similar situation occurs when a 3d chain is trapped inside a tube of diameter D .

This setting is very relevant to biophysical applications including DNA fragments confined

inside membrane channels . For both cases analytical theory can be created which is in good

agreement with the simulation results.

Last, but not least, the escape transition can be considered theoretically and (at least

in principle) realized experimentally in two conjugate ensembles. It was demonstrated

[154, 164, 169] that in the D-ensemble where the distance D between two pistons serves

as an independent parameter the force-separation curve has a loop with a negative com-

pressibility region. In contrast to that, in the conjugate f -ensemble the force-separation

curve is monotonic with a flat portion resembling the normal gas-liquid coexistence . We

analyze both situations and demonstrate that the escape transition is a unique example of

a highly unconventional first order transition with negative compressibility region and non-

equivalence of conjugate ensembles which persists even in the thermodynamic limit. We

address the general question of nonequivalence of statistical ensembles and the underlying

physical reasons leading to this nonequivalence in the case of a single polymer chain.

Escape transition in 3d geometry. In this section we present an analytical theory of the

escape transition for Gaussian chains in two conjugate D- and f - ensembles, demonstrate

qualitative differences in the behavior of thermodynamic properties of the system in these

two ensembles and discuss the origin of this nonequivalence. We analyze the origin of the

negative compressibility region as obtained from the equilibrium canonical partition function.

We also visualize the metastable and the coexisting states in escape transition on the base of

Landau approach. In the conclusion we discuss the escape transition in chains with excluded

volume interactions in good solvent.

Escape transition in 3d dimension has been studied by scaling arguments [170, 171], via

numerical methods [157, 172], and by computer simulations for good solvents and theta

solvent conditions [173–175] .

Numerical methods have been used to study the effect of the curvature of pistons

63

[176, 177], adsorption of a chain on piston walls [178], and compression of a star shaped

macromolecule [179] and a molecule based on a diblock copolymer [180]. A rigorous ana-

lytical theory addressing equilibrium and kinetic aspects of an ideal chain compressed by

pistons has been constructed in [162, 163]. The effects of metastability and negative com-

pressibility and the behavior of a chain in the conjugated ensembles have been described

in [169, 181, 182]. The theory taking into account the excluded volume between monomer

units has been compared with the Monte Carlo calculations and presented in [165]. We start

from the main result of the rigorous analytical theory of escape transition for a Gaussian

chain .

Partition function for ideal chain in D-ensemble. The full partition function in D-

ensemble was obtained in [163] and has a form

Q(D,N) = Q1 +Q2 = erf

(L

2R

)exp

[−(πR2

D

)]+ exp

(−πLD

)[1− erf

(L

2R− πR

D

)](56)

The first term takes into account all coil configurations, when the chain is confined in the

inter-piston volume; the second describes escaped configurations. Two asymptotic branches

of the free energy F (D,L,N) = − lnQ(D,L,N) are:

F (D,L,N)

N≈

π2

6D2 , D > Dtr

πLDN

, D < Dtr

(57)

The phase transition point occurs at the compression distance Dtr which depends on the

ratio of the contour length of the chain N to the piston radius L:

Dtr =πN

6L, (58)

At the transition point the tail escaping outside the inter-piston volume comprises exactly

1/2 of the chain.

As discussed later, both branches of the free energy can be continued into the metastable

regions. One of them terminates at the spinodal point D = D∗, the other persists down to

arbitrary small D (which is a specific feature of the phantom Gaussian chain). Figure 27

shows the Helmholtz free energy F as a function of the separation distance D for L = 90;

64

0

50

100

150

200

3 4 5 6 7 8

D

F

Figure 27: Free energy of a Gaussian chain at escape transition as a function of the piston separation,

D, for a cylindrical piston’s radiusL = 90, and chain length N = 800. The equilibrium branches

are shown by solid lines and the metastable branches by dotted lines. At the transition point, the

free energy has a discontinuity in the slope. The region where F (D) is not concave is limited by

two triangles.(According to [169])

N = 800. The equilibrium branches are shown by solid lines and the metastable branches

by dotted lines.

The free energy curve F (D) has a very unconventional feature: is not concave within

the range of 33/25 < D/Dtr < 32/23 . This contradicts a general statement of statistical

mechanics on the concavity of thermodynamic potentials. In the standard discussion of

statistical mechanics, a non-concave region must be replaced with the help of the double-

tangent construction that corresponds to phase coexistence and lowers the total free energy.

However, in the present case phase coexistence is impossible due to the very nature of the

phases, as discussed later. Here we just stress that the free energy displayed in Figure 27 is

obtained from a complete analytical partition function that includes all possible configura-

tions. Indeed, in the previous section concerning mechanical desorption we were able to see

that phase-segregated states can be naturally included in the exact partition function.

Compression curve (average force vs. separation). The average compression force

〈f〉 can be obtained by differentiating the free energy F (D,L,N) with respect to D. The

two asymptotic branches are given by:

65

0

10

20

30

40

50

0 2 4 6 8 10

<f>

D

1

2

3

Figure 28: Average compression force < f > for Gaussian chain compressed under piston

with radius L as a function of compression distance D for (L;N) = 40; 400(1), 80; 800(2), and

160; 1600(3).(According to [169])

β 〈f〉 ≈

πLD2 , D < Dtr

π2Na2

3D3 , D > Dtr

(59)

With progressive squeezing, the average force jumps down from 72L3/πN2a4 in the impris-

oned state to one half of this at the escaped state.

In Figure 28 the average compression force < f > vs. compression distance D is given

for three sets of parameters (L;N) = (40, 400); (80; 800), and (160; 1600) at fixed ratio

L/N = 0.1. At fixed ratio L/N , the transition becomes sharper with increasing chain

length while the region of negative compressibility narrows down and eventually shrinks to

a point as the limit N → ∞ is approached. We stress that negative compressibility is a

strictly equilibrium result that follows from the exact partition function and is not related

to unstable states.

Conjugate ensemble (separation vs. force). The force ensemble wherein the external

compression force plays the role of the independent variable is conjugate to the D-ensemble.

The partition function

66

Qf =

∞̂

0

QD exp(−fD) dD (60)

in the f -ensemble is the Laplace transform of the partition function QD. The transition

point in the f -ensemble is ftr = 16π

(43

)4 L3

N2 . The Gibbs free energy is globally concave as a

function of the compression force f . The average compression distance 〈D〉 = −∂ lnQf/∂f

has two asymptotic branches

〈D〉 ≈

(π2N3f

)1/3

, f < ftr(πLf

)1/2

, f > ftr

(61)

and experiences a downward jump at f = ftr .The region of negative compressibility is

absent.

Non-equivalence of ensembles. To compare the above two ensembles we present on

Figure29 reduced average compression force 〈f〉 / 〈f〉tr vs reduced distance D/Dtr as well

reduced compression force f/ftr vs. reduced average distance 〈D〉 / 〈D〉tr . In all cases, the

ration of piston radius to chain contour length, L/na, is kept constant, while the number

of chain units, N , and the piston radius, L, are varied. For small system with N = 100,

D = 15 (Figure 29 a) the phase transition is very wide and in the force ensemble it is

hardly identifiable. At N = 1600, D = 240 (Figure 29 b) the phase transition becomes well

pronounced in both ensembles and demonstrate the different behavior. In D-ensemble this

transition is accompanied by abrupt change f vs 〈D〉, while 〈f〉 vs D has the appearance of

a van der Waals loop. In contrast to a liquid-gas transition, all points of this loop correspond

to the equilibrium states of the system. Therefore the negative compressibility close to the

transition point is not related to any metastable effects. As follows from Figure 29, the

difference between the force-strain curves in two conjugated ensembles do not decrease with

N but become even more pronounced. This scenario disagrees with well-known theorem of

statistical mechanics that predict that pressure is a monotonic function of volume [154, 164].

To understand more clearly the situation of nonequivalence of two ensembles we note that

the curve of the free energy from compression distance F (D) is convex exactly in the region

where nonequivalence occurs - see Figure 29 and Figure 27. A general theorem of statistical

mechanics states that a convex region of the free energy signals that a homogeneous state

is unstable with respect to phase segregation. The convex region is then replaced by the

67

0,6

0,8

1

1,2

1,4

0,7 0,8 0,9 1 1,1 1,2 1,3 1,4

af / f*< f >/ f*

N=100L=15

<D>/D*D/D* 0,6

0,8

1

1,2

1,4

0,7 0,8 0,9 1 1,1 1,2 1,3 1,4

вf / f*< f >/ f*

< D>/D*D/D*

N=1600L=240

Figure 29: Average separation < D > vs. force f (solid lines) and average force < f >vs. separation

D (dash lines) for Gaussian chain compressed under piston with radius L in two conjugate ensembles

forN = 100;L = 15 (a) andN = 1600;L = 240 (b). Nonequivalence of the two conjugate ensembles

becomes more pronounced as system size is increased. (According to [154])

standard double-tangent construction that lowers the total free energy and represents phase-

segregated states. However, in the escape transition simultaneous coexistence of the two

phases is impossible due to the nature of the states. Both of them, the compressed coil and

the partially escaped “flower” (stem + tail) are essentially defined for a macromolecule as a

whole, but not for constituent sub-chains. It is therefore impossible to construct the mixed

state, in which both phases coexist in arbitrary proportions. This is the main difference with

the typical low-molecular mass systems and with the macromolecule undergoing mechanical

desorption, see Section III.

In the f -ensemble the Laplace transform of Eq. (60) automatically creates the convex

envelope of the F (D) function which is effectively equivalent to the double-tangent construc-

tion: hence the flat portion of the f vs.〈D〉 curve in Figure 29. We stress that the results

discussed are not artifacts of the simple Gaussian model. Their origin is quite general and

is eventually linked to the “entropy gap” discussed in Section 4.1. Indeed, Monte-Carlo

[174, 175] and Molecular Dynamics [164] simulations of the escape transition in a chain with

excluded volume interactions demonstrated qualitatively the same negative compressibility

loop in the D-ensemble. Anomalous behavior is also independent of other details such as the

exact position of the grafting point, the geometry and the precise alignment of the piston

68

(within certain limit).

One might ask whether examples of ensemble non-equivalence are known in other ar-

eas of statistical physics. Indeed, self-gravitating systems also fall outside the ensemble-

equivalence class and the difference between microcanonical and canonical pictures of finite

self-gravitating clusters has been discussed recently [183]. Experimental verification of this

effect is, however, at present hardly within reach. On the other hand, experimental study

of the escape transition is feasible and would be highly warranted.

Ensemble equivalence is rigorously established for a broad class of systems composed of

indistinguishable particles with the interaction vanishing at infinity. Existence of a well-

defined thermodynamic limit with a finite energy per particle is also required. All known

examples in the physics of condensed low molecular weight matter seem to fall into this

class. The last condition is not satisfied in self-gravitating systems. A single polymer chain,

considered as a statistical system, allows taking a proper thermodynamic limit in which

the total free energy is extensive. However, the first condition concerning the nature of

interactions is violated. The topological connectivity of linear chain is provided by bonding

interactions that do not satisfy the conditions stipulated above. This does not mean that

phase transitions in the single chain should always violate the ensemble equivalence; rather

that these are promising candidates for research in the present direction. The details of

intrachain interactions seem to be irrelevant since the ensemble non-equivalence is present

even for ideal Gaussian chain. On the other hand, we note that end-grafting is essential in

setting the stage for the escape transition. At the fundamental level, grafting removes the

translational degrees of freedom as well as the more subtle translational invariance along the

chain contour.

Order parameter and Landau function. It was suggested in [162, 164] that the param-

eter that characterizes the chain stretching can serve as the order parameter of the escape

transition. For the imprisoned coil states, this parameter refers to the chain as a whole,

s = r/N where r is the end-to-end distance. For the partially escaped states, only the

stem is stretched, and the order parameter is defined as s = L/n, where n is the number

of segments in the stem. The stem is the sub-chain composed of n imprisoned segments,

starting from the fixed chain end and ending with the first segment that reaches the edge of

the pistons. The two definitions of the order parameter match smoothly for the border-line

configuration in which the stem includes all the segments with r = L and n = N . The

69

following simple analytical expressions for the two branches of the Landau free energy were

obtained in the asymptotic limit :

Φ(s,D) ≈

32s2 + 1

6

(πD

)2, s ≤ L

N

3L2Ns+ L

6N

(πD

)2s−1, s ≥ L

N

(62)

Comparing this result to Equations 47 and 53 it becomes obvious that the three models

discussed in this Section (chain tethered near an adsorbing surface, near a step potential, and

compressed between pistons) are equivalent in the thermodynamic limit with the following

mapping between parameters: z0 ↔ L; and ca ↔√

6u ↔ πaD

[162]. The shape of the

Landau free energy as a function of the order parameter is typical for standard first-order

transitions with two minima separated by a barrier located at s = L/N , similar to the curves

of Figure 19. This means that as far as the behavior of the average order parameter and

its fluctuations is concerned, the escape transition is perfectly normal. In the vicinity of

the transition metastable states are possible and can be easily analyzed from the shape of

the Landau free energy. In particular, the spinodal point where the escaped state becomes

unstable is given by the simple relation: Dsp/Dtr = 2.

We stress again a close analogy between the escape transition and the adsorption tran-

sition for a polymer chain end-fixed at a distance from the solid surface or a liquid-liquid

interface. Although these are two distinct phenomena that are quite different at first glance

their fundamental similarity is revealed in the shape of the Landau free energy. In the

asymptotic limit, there is a simple mapping between the three partition functions.

The new escaped (or partially adsorbed) phase emerges via a single nucleus when the

free end of the chain reaches the piston edge (or adsorbing surface/interface) and forms a

seed crown of a few or one segments; its appearance involves a global change in the chain

conformation with a finite jump in the entropy, see Figure 21. The nucleation barrier height

is determined by the entropy gap ∆ = 3L2/2Na2 (or 3z20/2Na

2) only.

Escape transition (as well as the two other eqivalent models discussed in the current

section) is remarkable since not only the exact partition functions with finite-size effects

are known for them but also the exact expressions for the Landau free energy, containing

all the information on the metastable states. It can also serve as a pedagogical example

allowing a simple visual interpretaion of a state at the spinodal point. In classical systems

undergoing first order phase transitions it is not easy to form a simple picture of spinodal

70

states. In contrast to this, spinodal conformation in the escape transition is just a uniformly

stretched stem that contains all the segments and a vanishingly small crown. More detailed

characteristics of the binodal and spinodal states were dicussed in [182].

The classical spinodal decomposition occurs via growth of long-wave fluctuations of the

order parameter. The escape transition, as well as the eqivalent models, is adequately

described in terms of a single global order parameter that allows one to construct a simple

theory for both equilibrium and kinetic aspects of this phenomenon. The nature of the

order parameter also dictates the basic mechanism leading to a decay of an unstable state.

Note that not all single-chain phase transitions are naturally described in terms of a global

order parameter. For instance, in the coil-globule transition the order parameter is the local

monomer density. Correspondingly, kinetics of the coil collapse is governed by growth and

coalescence of multiple nucleation centers, although its initial stage does not necessarily

involve any change in the large-scale coil conformation. For a discussion of local order

parameter, see also mechanical desorption in the z-ensemble, Section 3b.

Escape transition in chain with excluded volume. The theory of escape transition for

flexible polymer chain under good solvent conditions was developed in [164] using Landau

free energy approach based on the order parameter defined above. The approach allows a

straightforward incorporation of finite chain length effects. The main results obtained in

[164] are: 1) the line of first-order transitions is given by equation

Dtr = A(N/L)ν/(1−ν) (63)

where the numerical pre-factorA = 0.834 is linked to the non-universal pre-factors in the con-

finement free energy and the lateral end-to-end distance and was evaluated for the Kremer-

Grest model. 2)The spinodal line where the barrier maintaining a metastable state vanishes

is Dsp/Dtr = (4/3)ν/(1−ν). The second spinodal equation is L/N = 0 in the limit N → ∞.

3)The average order parameter in the transition point has a jump from zero to 4L3Na

, while

the average fraction of imprisoned monomers jumps from unity to 34. Thus at the transition

point the escaped tail always comprises 14of the total monomers irrespective to the L/Na

ratio.

A comprehensive study of both static and dynamic properties using Molecular Dynamic

simulations for Kremer-Grest bead-spring model demonstrated good agreement with quan-

titative theoretical predictions. Therefore we can speculate that all the unusual features of

71

the escape transition (as well as the two other eqivalent models ) qualitatively apply to real

polymer chains.

Escape transition in 2d space. The escape transition for a two-dimensional polymer with

excluded volume interactions was analyzed in [165, 184] using the model of self-avoiding

walks on a square lattice. The inter-piston volume becomes a strip in the 2d-setting. Self-

avoidance of the chain confined inside a strip means that the confined coil conformation is

transformed into a linear string of blobs. Common-sense intuition suggests that with the

progressive compression of the chain the blob size decreases and the position of the free end

moves monotonically towards to the edge of a strip. Upon further compression the extra

monomers are squeezed out of the strip and form a coiled tail outside. No phase transition

occurs in this “toothpaste” scenario.

Both simulation results and the analytical theory indicate however that in the thermody-

namic limit of large N and L but finite L/N , there is a weak first-order transition with small

but perfectly identifiable jump the order parameter and in the average number if imprisoned

units.

Simulations and scaling arguments show that the free energy has two distinct branches

depending on whether the chain is completely confined of part of it forms the escaped tail.

The best fit for the free energy as evaluated in simulations is given by

F (N,L,D) =

Fimp ≈ 1.944(2)ND−4/3 D > Dtr

Fesc ≈ 2.03(3)L/D D < Dtr

(64)

The two braches join at transition point Dtr ≈ 0.889(Na/L)3.There is a discontinuity in

the slope when F is plotted as a function of D. The fraction of segments that are escaping

the strip at the transition point is quite small: (Nimp − Nesc)/N = 1 − 3/25/3, instead of14in 3d case (see previous section).The accurate estimation of these values needs precise

calculations for chain length up to 50000 with the help of the pruned-enriched Rosenbluth

method. Since two distinct phases separated by a high barrier are involved introducing a

special bias is required: otherwise, misleading spurious results are very likely [184].

The physical reasons of a weak first-order escape transition in 2d case as opposed to the

smooth “toothpaste” scenario are quite subtle. Indeed, the form of the free energy given by

Eq. 64 suggest that there is also a discontinuity in the slope when the free energy is consid-

ered as a function of L. The derivative ∂F/∂L has the meaning of the average force applied

72

to the fixed chain end. For the fully confined state this force is zero and the blob picture

is applicable without reservations. However, for conformations that include a tail outside

the strip the force at the grafting point is non-zero. Thus the escaped tail is equivalent to a

constant stretching force and therefore, the confined stem experiences additional stretching,

as illustrated in the cartoon Figure 30. The naïve blob picture in the “toothpaste” scenario

fails to take this additional stretching into account while a more rigorous approach based

on the Landau free energy incorporates it automatically and correctly predicts a jump-wise

transition. The blobs in the stem are deformed due to the tail effect as compared to the

blobs in the fully imprisoned state.

Another escape transition setup: Dragging a polymer chain into a tube. A

situation when a 3d chain is confined in a tube of diameter D is qualitatively similar to the

2d escape. An important qualitative result of the analytical theory of the escape transition

is that the transition can be induced by changing any of the three parameters: the chain

length, N,the tube diameter, D, and the distance between the fixed end and the tube

opening. In the case of a tube, changing D (i.e. squeezing the chain) may not be easy to

arrange experimentally unless the tube is made of a soft compressible material. Changing

N would imply a process in which the chain is grown slowly in a controlled way within

the restricted tube geometry. Experimentally this may be feasible although quite difficult

to realize. Finally, the distance between the end-monomer inside the tube and the tube

opening can be changed. This can be realized by slowly dragging a polymer chain by one

end into a narrow tube e.g., by using optical tweezers. Well calibrated nanochennels were

produced recently in fused silica substrates by lithography methods with the width in the

range of 30 to 400 nm, which were used to study the confinement of single λ-phage DNA

molecules driven electrophoretically into these nanochennels [185]. The persistence length

of DNA under conditions used in these experiments is about 50 nm while its contour length

was about 1000 times larger. This means that except for the case of the narrowest channels

DNA behaved essentially as a long flexible macromolecule on the relevant length scales.

Cartoons of a flexible polymer chain with one end dragged into a nanotube in a quasi-

static process are shown in Figure 30. The chain end inside the tube is characterized by its

coordinate x counted from the tube opening. It experiences a reaction force f that balances

the pull of the undeformed swollen coil outside the tube. At the transition point x∗, the

remaining tail is sucked into the tube abruptly by a uniform shrinking of all blobs in the

73

Figure 30: Schematic drawings of a flexible chain confined partially (above) and fully (below) in a

nanotube.In the first case, there is an ejecting force due to the outside tail, and external force is

needed to keep the chain from ejection; no force appears in second case.

stem, and the reaction force becomes zero . As long as the chain is fully confined in the

tube, no reaction force appears at the fixed chain end .

The properties of a single macromolecule confined in a tube have been studied extensively

for decades, both by analytical theory and by numerical simulations for various models of

flexible and semi-flexible chains [186–191] . For a fully imprisoned homogeneous state there

are scaling predictions concerning various chain characteristics which were tested by MC

simulations. We focus here in the nonhomogeneous state where the confined part of the

chain inside the tube forms a stretched stem and the free tail still in solution forms a coiled

crown. This type of conformations appears in a variety of situations including translocation

through a thick membrane [192, 193].

The results of the analysis are qualitatively similar to those of the 2d escape although

the Flory exponent involved is changed from ν2 = 3/4 to ν ≈ 0.587 . Again, there are two

branches of the free energy:

F (N,L,D) =

Fimp = 5.4ND−1/ν x > xtr

Fesc = 1.46 x/D x < xtr

(65)

where x is the coordinate of the chain end counted from the tube edge (the variable to

be changed in this setup) and the numerical prefactors refer to the walks on simple cubic

lattice. The transition occurs at the right end position

74

Figure 31: (a) MC data for the average fraction of imprisoned units, Nimp/N , plotted against

the reduced end coordinate x/x∗ for relatively long chains or various values of tube diameter D

at fixed number of blobs (nb = 60 ). The solid line shows the predicted linear growth below the

transition point. At the transition point x/x∗ = 1, Nimp/N jumps up from 0.76 to 1.(b) Nimp/N

vs the reduced end coordinate x/x∗ for tube diameter D = 21 and for different number of blobs

nb, displaying the rounding of the transition due to finite-size effects.(From [194])

xtrNa≈ 1.26 (D/a)1−1/ν (66)

At the transition point the outside tail comprises about 24% of all segments which is

much larger that the corresponding jump in the 2d geometry (5.5%) and remarkably close

to the jump in the traditional 3d setting (exactly 1/4).

The theory predicts that upon gradual dragging of the chain end further into the tube the

average fraction of imprisoned units, Nimp/N , plotted against the reduced end coordinate,

x/xtr, increases linearly before the transition point is reached and then jumps up from 0.76

to unity. These predictions are plotted on Figure 31a together with simulation data for four

different values of D. Another set of numerical data is shown on Figure 31b, displaying the

rounding of the transition due to the finite chain length.

The analysis of the Landau free energy allows identifying two spinodal points, see Figure

32 b. If the chain end is moved at a finite speed one would expect a standard hysteresis

loop illustrated in Figure 32 a. In that sense, the transition conforms to what is expected

of a standard first-order transition. However, the abnormality associated with the “entropy

gap” and the impossibility of simultaneous phase coexistence still exists and would show up

75

Figure 32: (a)The hysteresis loop of the average fraction of imprisoned units, Nimp/N , associated

with metastable states of a self-avoiding chain in a nanotube. (b) Landau free energy of the chain in

a nanotube of diameterD = 17 as a function of the order parameter at the transition point x = xtr

and at two spinodal points x(1)sp ≈ 0.76xtr and x(2)

sp ≈ 1.30xtr.(From [194])

as a non-concave region in the free energy, Eq. (65) considered as a function of D.

V. COMPLEX ZEROES OF PARTITION FUNCTIONS FOR SINGLE-CHAIN.

At a phase transition point, thermodynamic functions have a singularity, meaning that

they or their derivatives have a finite or infinite discontinuity. On the other hand the

partition function Q =∑

i exp(−Ei/kT ) is just a sum of exponentials and thus has no

singularities as a function of external parameters. Since the free energy is the logarithm

of the partition function, and the logarithmic function has a singularity at zero argument,

the only possibility left is that the partition function should vanish (or at least, in some

sense, tend to zero) when the system approaches the transition point. It is obvious though,

that the partition function is positive and cannot be zero at any real values of external

parameters. It was also realized very early that a true mathematical singularity of the free

energy can develop only in the thermodynamic limit when the number of particles tends

to infinity. The approach pioneered by Yang and Lee [42, 43] related these singularities to

complex zeroes of the partition function.

Their original papers dealt with the liquid-gas transition induced by the change in fugac-

ity, so these were the analytical properties of the grand partition function in the complex

plane of fugacity that were the object of investigation. Similarly, the temperature-induced

76

transitions should be described in terms of the zero distribution for the canonical partition

function in the complex plane of temperature (or β = 1/kT ). These are commonly called

Fisher zeros in order to distinguish them from the Yang-Lee zeroes in the fugacity plane.

Obviously, the general approach is applicable to phase transitions induced by changing any

other external parameter as well.

Yang and Lee showed that for finite N the partition function can have only complex

conjugated zeroes but no zeroes on the real positive axis. The only possibility for a phase

transition to appear is that as the number of particles increases, the complex zeroes come

closer to the real positive axis, and eventually, in the N → ∞ limit, they pinch upon the

real axis at the transition point. While the Yang-Lee theorem states that the zeroes in the

fugacity plane have to be located on a unit circle, there are no general results known for

Fisher zeroes. Empirical regularities show, however, that they tend to fall on smooth arcs

that cross the real axis at a certain angle. In the thermodynamic limit, the free energy and

its derivatives can be represented as integrals over the continuous distribution of zeroes that

could be characterized by some limiting density function. But the problem of finding the

actual distribution of zeros for the partition function of even a simple model proves to be

formidable. In practice, for a given model the zeros are calculated numerically for small

samples and then some extrapolations are employed [61, 195].

There exists a phenomenological approach relating certain features of a phase transition

(amplitudes and critical indices) to the characteristics of the distribution of zeros assumed

to be known [60], as well as scaling predictions for this distribution [51].

Assuming that the zeroes concentrate on two symmetric support lines which cross the

real axis at the point β = βc making an angle ω with it, Grossmann and Rosenhauer [60]

were able present an extensive classification of phase transitions and to express the main

characteristics of a transition (jump magnitudes, critical indices and amplitudes) through

the parameters of the linear density of zeroes g(y), where y is the coordinate along the

imaginary axis. They assumed that the linear density of zeroes is a power function of the

coordinate y along the imaginary axis: g(y) ∼ y1−α. Using this assumption several cases

can be distinguished. Two of them are summarized below.

(1) If α = 1 and the density of zeroes g(y) tends to a constant at small y, then the zeroes

necessarily approach the real axis at a straight angle ω = π/2 and the energy has a finite

jump ∆E = 2πg(0) upon crossing the β = βc point. This obviously corresponds to a first

77

order transition with a δ-peak singularity in the specific heat.

(2) If α = 0 , the density of zeroes grows linearly with y , and the support lines cross

the real axis at ω = π/4, the energy is continuous but the specific heat has a finite jump

discontinuity, as in a classical mean-field second order transition.

As we discussed in the Introduction there are two different types of the first-order phase

transitions in a single chain: transitions with phase coexistence and transitions without phase

coexistence. It is naturally to put forward the question: is it possible to discriminate these

transitions by analyses of complex zeroes of the partition functions these systems. Below

we present complex zeroes for two exactly solvable models. The first model is an ideal chain

desorbing from solid surface by external end force. It was demonstrated in Section III that

this model undergoes the first order transition with phase coexistence. The second model

is an ideal chain tethered be one end near adsorbing surface. This model is also exactly

solvable and undergoes the first order transition without phase coexistence as was discussed

in Section IV.

a) Complex zeroes in transition with phase coexistence (mechanical desorption of

ideal chain). In the model of Gaussian chain desorbing by external end-force , the governing

parameter c (or f) is analogous to the inverse temperature β, while the conjugated variable,

the number of adsorbed segments (or the free end height) would be the analog of the energy

E. The analysis of the partition function Q(c, f0) in the complex plane of the variable c

was presented in [63]. It was shown that the limiting density of zeros is g(ρ) = ρ/6π. At

the point of crossing the real axis ρ = f0, and g tends to the finite value g0 = f0/6π. This

falls into category 1) according to the above classification, and indeed, gives the first order

transition with a finite jump in the order parameter. The magnitude of the jump was derived

in ref. [60] to be 2πg0 = f0/3, in accordance with the value obtained by direct differentiation

of the partition function.

On the asymptotic wings of the hyperbola the distance from the transition point along

the curve, s, is the same as ρ, and the density of zeros increases linearly with it. Taking s ' ρ

and solving for ρ as a function of k one obtains the position of the k-th zero ρk ' (12πk/N)1/2

When one crosses the line of the first order transitions closer and closer to the bicritical

point the focus distance of the hyperbola, f0, decreases and so do accordingly the density

g0 and the magnitude of the jump in the order parameter. Eventually, at f0 = 0, the

curve degenerates into two straight lines at an angle ω = π/4 with the real axis crossing

78

it at the origin (see Figure 33) . Here, the transition becomes second order: changing the

adsorption parameter c one passes exactly through the bicritical point. Strictly speaking,

the asymptotic representation of the Y function used in Eq. (#) is not valid in this case

any more. However, the limiting picture of the distribution of zeros does apply.

Numerical results shown in Figure 33 suggest that zeroes indeed condense on a support

line at an angle ω = π/4 with the real axis. The limiting density of zeros has a form

g(|c|) = |c| /6π. This case falls into category 2 according to the above classification, and,

indeed, results in a second order transition of a mean-field type with a finite jump in the

order parameter fluctuation squared (see Section IV). The magnitude of the jump is 1/3.

Of course, this coincides with the result obtained directly by differentiating the free energy,

Eq. 24.

The limiting density and the slope of the line of zeros of the partition function upon

crossing the line of second order transitions away from the bicritical point (f0 = const < 0)

are exactly the same: ω = π/4, g(|c|) = |c| /6π. This is consistent with the fact that the

Landau free energies coincide for both cases in the thermodynamic limit (see section IV).

Numerical data for a few first zeros of the exact partition function (27) with N = 100 and

N = 500 for several values of f0 are displayed in Figure 33 and support the analytical results

for the distribution of zeros.

It is worth noting that in contrast to the lattice models where the partition function of a

finite system is a polynomial and therefore has only a finite number of zeros, the model of

the adsorbing chain that we discuss results in a partition function with an infinite number

of zeros for any value of N. The reason for this is that we deal with a continuum model as

opposed to lattice models with finite sets of discrete states.

b) Complex zeroes in transition without phase coexistence ( adsorption of ideal

chain tethered near a solid surface). The exact partition function Q(z0, c) of an ideal

chain tethered at the distance z0 near surface with adsorption interaction parameter was

presented before, Eq (45). The complex zeroes this partition function were analyzed in [64].

For zeros close to the real axis (i.e. with small χ) was obtained: χk = (2k+1)π6Ns2

.The closest

zero is characterized by the polar angle χ0 = (π/6N)(N/z)2. For a fixed value of the ratio

s = z/N and increasing N , χ0 approaches the real axis as N−1 . At distances of order

z ∼ N1/2 the deviation of polar angle from zero is on the order of unity and one certainly

cannot speak of a first order transition.

79

Figure 33: Distribution of zeros of the partition function for ideal polymer chain at mechanical

desorption under external end-force f in the complex plane of the adsorption parameter c for chain

length N = 100 (a), and N = 500 (b). Various values of the end force magnitude, f , indicated in

the plot. The analytical limiting curves are shown by solid lines.(From the work [63])

At the point of crossing the real axis (χ → 0 ; ρ → 6s) , the density of zeros tends

to a constant value g0 = s/2π. According to Grossman and Rosenhauer, this falls into

the category of first order transitions with a finite jump in the energy (for our adsorption

model- in the number of contacts). The magnitude of the jump must be 2πg0 = s . Direct

analysis of the partition function shows that the jump of the number of contacts is, indeed

〈m〉 /N = z/N .

On the asymptotic wings of the hyperbola χ→ π/4, the density of zeros increases linearly

80

Figure 34: The distribution of zeros of the partition function in the complex plane of the adsorption

parameter for ideal polymer chain tethered at the distance z0 near solid surface . The chain length

N = 400 and three values of the tether point z0/N = 0; 0.1; 0.2 are indicated.The lines are drawn

to guide the eye.(From the work [64])

with the distance from the transition point along the curve: g ' ρ/6π. Along these wings

the position of the k-th zero for large enough k, k/N � s2, is ρk ≈ (2πk/N)1/2.

When the distance from the adsorbing plane to the grafting point, z, decreases, so do

accordingly the density g0 and the magnitude of the jump in the order parameter. Eventually,

at z = 0, the curve degenerates into two straight lines at an angle ω = π/4 with the real

axis crossing it at the origin. The density of zeroes turns out to be a linear function of

the distance from the critical value c = 0. Here, the transition becomes second order, as

mentioned before The distribution of the partition function zeroes for chain with both end

fixed can be easily understood now. The transition is again second order with the critical

value of the adsorption parameter being c∗ = 3z/N .

Comparing Figures 33 and 34 one concludes that the distribution of the complex zeroes

does not allow to discriminate between transitions with and without phase coexistence.

CONCLUSIONS

We have considered several polymer models exhibiting phase transitions which admit

exact analytical treatment in the calculation of the partition function. Moreover, almost

all models allow us to obtain exact expression for the Landau free energy as a function of

the order parameter both for first- and second-order transitions. For some models, it is also

possible to study analytically the distribution of complex zeros of the partition function.

This rigorous treatment can be applied not only in the thermodynamic limit, but also to

systems of finite size. The fact that all these solutions are available in closed form is very

81

remarkable; they can serve as good examples for discussing various methods and approaches

in statistical physics and be a useful teaching aid.

Now, there is a natural question: why such a treatment is possible, and what distinguishes

these polymer models from the classical non-polymeric exactly solvable lattice models.

We can point to at least three special features of our polymer models. First, the interac-

tions between individual elements of the systems (the repeat units of the polymer chain) are

taken into account from the outset in writing the initial differential equation for the random

walk chain. By assumption, each unit interacts only with its closest neghbours, while there

are no long-range volume interactions. Second, successful treatment was possible when the

order parameter was not defined locally, but rather characterized the system as a whole.

Because of this, there are no correlation of fluctuations of the order parameter. Third, the

external field is applied to the free end of the chain, and does not affect the loops nor the

adsorbed parts of the molecule.

Almost all the models discussed display some unconventional behavior. Even the well-

studied coil-to-globule transition with its mean-field character becomes very unorthodox in

higher dimentions. For the mechanical desorption model features of first and second order

transitions become mixed due to phase coexistence not accompanied by interfacial free en-

ergy. Models belonging to the escape transition class represent a unique example of a highly

unconventional first order phase transition with several inter-related unusual features: no si-

multaneous phase coexistence, and hence, no phase boundary; non-concave thermodynamic

potential; non-equivalence of conjugate ensembles and negative compressibility region.

All the unconventional features of the phase transitions mentioned above apply not only

to the idealised models that admit closed-form exact solutions. Excluded volume interac-

tions don not change the basic picture which is rooted either in the fact that the interface

between the two phases within a single macromolecule consists of a single segment, or in

the entropy gap between the two classes of configurations. Thus the qualitative conclusions

about the unconventional features of the transitions should apply to real macromolecules.

In particular, the macromolecules capable of escape-class transitions are speculated to be

practically utilized as trigger systems .

Ensemble equivalence is rigorously established for a broad class of systems composed of

indistinguishable particles with the interaction vanishing at infinity. Existence of a well-

defined thermodynamic limit with a finite energy per particle is also required [1? ]. All

82

known examples in the physics of condensed non-polymeric matter seem to fall into this class.

However, a single polymer chain, considered as a statistical system, does not. The topological

connectivity of linear chains is provided by bonding interactions that do not satisfy the

conditions stipulated above. Of course, this does not mean that phase transitions in a

single chain should always violate the ensemble equivalence; rather that these are promising

candidates for research in the present direction. The details of intrachain interactions seem

to be irrelevant since the ensemble non-equivalence is present even for ideal Gaussian chains.

On the other hand, we note that end-grafting is essential in setting the stage for the escape

transition. At the fundamental level, grafting removes the translational degrees of freedom

as well as the more subtle translational invariance along the chain contour. Presumably

it is also important that one of the phases has to be nonhomogeneous and consist of two

mictrophases like stem and and crown in the escaped configuration.

Examples of ensemble non-equivalence are not completely unknown in other areas of sta-

tistical physics. Indeed, non-extensive self-gravitating systems also fall outside the ensemble-

equivalence class and the difference between microcanonical and canonical pictures of finite

clusters with long-range interactions has been discussed recently [183]. Experimental ver-

ification of this effect is, however, at present hardly within reach. On the other hand,

experimental study of the escape transition is feasible even now and would be highly war-

ranted.

Acknowledgments

We are grateful to Prof. Kurt Binder who conceived of the idea of this review and stim-

ulated us throughout this work by his deep insights and thoughtful discussions. Technical

help provided by Dr. A. Polotsky is greatly appreciated. This work is supported by DFG

grant 436 RUS 113/863/0-2 and by the Russian Foundation for Basic Research (RFBR

grants No.09-03-91344-a , and No.08-03-00402-a).

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