Does a surface attached globule phase exist?

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Physica A 318 (2003) 171–178www.elsevier.com/locate/physa

Does a surface attached globule phase exist?P.K. Mishraa, D. Girib;∗, S. Kumara, Y. SinghaaDepartment of Physics, Banaras Hindu University, Varanasi 221005, India

bCenter for Theoretical Studies, I.I.T., Kharagpur 721302, W.B., India

Abstract

A long ,exible neutral polymer chain immersed in a poor solvent and interacting with animpenetrable attractive surface exhibits a phase known as surface attached globule (SAG) inaddition to other adsorbed and desorbed phases. In the thermodynamic limit, the SAG phase hasthe same free energy per monomer as the globular phase, and the transition between them is asurface transition. We have investigated the phase diagrams of such a chain in both two- andthree-dimensions and calculated the distribution of monomers in di1erent domains of the phasediagram.c© 2002 Elsevier Science B.V. All rights reserved.

PACS: 64.60.−i; 68.35.Rh; 5.50.+q

Keywords: Surface attached globule(SAG) phase; Exact enumeration

A long ,exible neutral polymer chain immersed in a poor solvent and interactingwith an impenetrable surface is known to exhibit a very rich phase diagram [1]. Com-petition between the lower internal energy near an attractive wall and higher entropyaway from it results in a transition, where for a strongly attractive surface the polymersticks to the surface and for weak attraction the polymer chain remains desorbed. Onthe other hand, due to the self-attraction in the polymers the possibility of a collapsetransition both in the desorbed and adsorbed states occur. The phase diagram is gen-erally plotted in variables ! = e��s and u = e��u where � is the inverse temperature,�s is the (attractive) energy associated with each monomer lying on the surface and�u represents an attractive interaction energy between pairs if monomer which comeclose to each other and are separated along the chain by more than one unit. The shortrange repulsion between the monomer is taken in a lattice model by self-avoidance [2].

∗ Corresponding author.E-mail address: [email protected] (D. Giri).

0378-4371/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0378 -4371(02)01424 -3

mailto:[email protected]

172 P.K. Mishra et al. / Physica A 318 (2003) 171–178

(b)(a)

(c)

(d)

(e)

Fig. 1. Schematic diagrams of self-avoiding walk conformation at T = 0. Below critical value !c1, (a) thenumber of monomers at the surface are zero and polymer is in the desorbed state; (b) the structure shownin (a) gets adsorbed on the surface at critical value !c1. Average number of pair of neighbouring monomerson the surface 〈Nc〉 scales as Nd−1=d; (c) 〈NC〉 increases with !; (d) At !c2, 〈Nc〉 is maximum; (e) 〈Nc〉decreases sharply with further increase in !.

Considerable attention has been paid to Gnd the full phase diagram of such a chain inboth two dimensions (2D) and three dimensions (3D).In one of the earliest papers on the subject, Bouchaud and Vannimenus [3] derived

the exact phase diagram on a 3D-Sierpinski gasket. The phase diagram consisted ofthe adsorbed expanded (AE), desorbed expanded (DE) and desorbed collapsed (DC)phases. Kumar and Singh [4] have reinvestigated the phase diagram for 3D-Sierpinskigasket and showed that for a certain range of surface interactions, an additional phasehaving the feature of globule attached to the surface exists. Recently Singh et al. [5,6]using extrapolation of exact series expansions calculated the phase diagram for thepolymer chain in both 2D and 3D (Euclidean space). These phase diagrams show theexistence of the surface attached globule (SAG) phase in qualitative agreement withthe one found earlier for the gasket. It is important to note that in the thermodynamiclimit, the SAG phase has the same free energy per monomer as the DC phase, and thetransition between them is a surface transition.As shown earlier [1], it is easy to understand the SAG and its boundary in the limit

T = 0. At T = 0, the polymer conGgurations may be mapped by Hamiltonian walk asshown in Fig. 1(a–c). Its bulk energy is the same as in the DC phase and the surfaceenergy is (�u− �s)Ld−1

‖ +(d−1)�uN=L‖. Where L‖ is size of polymer chain along each

P.K. Mishra et al. / Physica A 318 (2003) 171–178 173

1.0 2.0 3.0 4.0u

1.0

2.0

3.0

4.0

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1.0

2.0

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DE

AESAG

AC

DC

uc3d

uc2d*

*

* c1

ω

ω ω ω

c2

c

AE

DE

c

ω

ω ω

c2

c1

DC

SAG

ω ω

(a) (b)

Fig. 2. (a), (b) These Ggures show the phase diagrams of a surface interacting linear polymer in 3D and2D space, respectively. The transition lines were obtained using method of exact-enumeration.

direction in the plane of surface and L⊥ deGnes chain size perpendicular to surfacerespectively. Minimizing the surface energy with respect to L‖, we obtain

ESAG =−(d− 1)�uN + d�d−1=du (�u − �s)1=dNd−1=d : (1)

While for the DC phase,

EDC =−(d− 1)�uN + d�uNd−1=d : (2)

In the AC phase, we have L⊥ = 1, L‖ = N 1=(d−1), and the free energy at T = 0 is

EAC =−(d− 2)�uN − �sN + (d− 1)�uNd−2=d−1 : (3)

Comparing the energies of these phases, we see that SAG phase has lower free energythan the DC or the AC phases for 06 �s6 �u. Thus the lower and upper boundariesof the SAG phase (lines !c1 and !c2 in Fig. 2) tend to !c1 = 1 and wc2 = u forlarge u.In the case of partially directed polymer in 2D we have found [1] exact phase di-

agram of the SAG phase for T ¿ 0. In this case the polymer has di1erent behaviourdepending on whether it is near the wall perpendicular to the preferred direction(SAG1) or the wall is parallel to the preferred direction (SAG2). The phase boundariesof SAG1 and SAG2 have been found by calculating this orientation dependent surfaceenergy.Since the analytical methods at T ¿ 0 are limited to very few cases such as fractal

lattices or the directed walks on 2D lattices, we have to resort to numerical methodsto calculate the phase diagram in other cases. A lattice model using extrapolation ofexact series expansions (hereinafter referred to as exact enumeration method) has beenfound to give satisfactory results as it take into account the corrections to scaling. Toachieve the same accuracy by the Monte Carlo method, a chain of about two ordermagnitude larger than in the exact enumeration method has to be considered [7].


In this article we report our results which we have found using exact enumerationmethod both for 2D and 3D cases. Let CN (Ns; Nu) be the number of SAWs of N siteshaving Ns monomers on the surface and Nu nearest neighbour monomer pairs. Weanalysed the series for CN (Ns; Nu) up to N = 30 for square lattice and N = 20 for thecubic lattice; thus extending the series by two more terms in both cases.To obtain better estimates of critical points as well as the phase boundaries, we

extrapolate for large N using ratio method [8]. Let

ZN (!; u) =∑

Ns; Nu

CN (Ns; Nu)!NsuNu ; (4)

be the partition function. Then, the reduced free energy per monomer can be writtenas [8]

G(!; u) = limN→∞

1N

log ZN (!; u) : (5)

The phase boundaries may be found from the maxima of @2G(!; u)=@�2s (=@〈Ns〉=@�s)and @2G(!; u)=@�2u (=@〈Nu〉=@�u). However, we have shown [5] that in the limit of largeN the value obtained from the maxima of @2G(!; u)=@�2s approaches to the value foundfrom the G(!; u) vs. ! curve where G(!; u) remains constant up to !c. Therefore, wechoose this method to determine !c and !c1 lines in the phase diagrams.The phase diagram thus obtained is shown in Fig. 2(a) for three dimensions. It

has Gve phases which are denoted as desorbed expanded (DE), desorbed collapsed(DC), adsorbed expanded (AE), adsorbed collapsed (AC) and surface attached globule(SAG). Since in case of two dimensions, surface is a line, therefore, we do not Gndadsorbed collapsed phase [Fig. 2(b)]. The value of uc at !=1 is found to be 1:93±0:02and 1:76 ± 0:02 for 2D and 3D, respectively. The value of !c at u = 1 is found tobe 1:80 ± 0:02 and 1:48 ± 0:02 for 2D and 3D, respectively. The special adsorptionline !c separates the AE phase from that of DE phase. The uc line meets !c linewhich is a multi-critical point (uc; !c) = (1:93; 1:78) for 2D and (1:76; 1:38) for 3D,respectively. It is to be noted that DE to DC phase boundary is vertical both in twoand three dimensions. This line starts bending after !¿!c(uc). This is due to factthat (surface does not a1ect the bulk behaviour) step fugacity remains unchanged tillpolymer get adsorbed on the surface as bulk free energy is not changing. In the abovephase diagrams, we have strong numerical evidence that DE to DC transition is of Grstorder however all other transitions i.e. DE to AE, DC to SAG, SAG to AC and AC toAE transitions are of second kind (continuous type). The value of radius of gyrationexponent � at u= 1 and 1:93 is found to be 0:74± 0:02 and 0:57± 0:02, respectivelyat != 1 for 2D case. The value of cross-over exponent � at u= 1 and 1:93 is foundto be 0:52 ± 0:05 and 0:4 ± 0:05 respectively for 2D case. These values are in verygood agreement with the known results [9,10].In order to show how the monomers distribute when the chain is in the di1erent

regimes of the phase diagram, we calculate the value of each term in Eq. (4) forgiven values of u and ! and plot the results for walks up to 20 steps (3D case only)in Figs. 3–6. Here we have plotted �=�m for four values of ! for di1erent u, where� = CN (Ns; Nu)!NsuNu (�m being the maximum value of �) as a function of Ns andNu. Since the term which contributes most to the sum of Eq. (4) represents the most


05

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ξξ

/

mξ

ξ /

m

ξξ /

m

ξξ

(a) (b)

(c) (d)

Fig. 3. Plot of �=�m as a function of Ns and Nu. Here � = CN (Ns; Nu)!NsuNu and �m is the maximumvalue of � for a given value of ! and u. (a) u = 1:0, ! = 1:0; (b) u = 1:0, ! = 2:0; (c) u = 1:0, ! = 3:0;(d) u = 1:0, ! = 3:5.

05

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(a) (b)

(d)(c)

Fig. 4. Same as Fig. 3, (a) u=1:5, !=1:0; (b) u=1:5, !=2:0; (c) u=1:5, !=3:0; (d) u=1:5, !=3:5.


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probable conGguration of the chain. We can from these Ggures infer the manner inwhich the monomers are distributed in di1erent states. For example, Figs. 3(a–d) and4(a–d) show how the distribution of monomers changes as ! is varied at Gxed u=1:0and u= 1:5 (corresponding to the expanded state), respectively. It is clear from these


0.0 20.0 40.00.0

2.0

4.0

6.0

8.0

<N

c>

u=10.0; N=24u=15.0; N=24u=20.0; N=24u=25.0; N=24

c1

ω

ω

c2

ω

Fig. 7. Diagram shows the variation of number of monomer pairs on the surface 〈Nc〉 with !.

Ggures that Ns and Nu are nearly zero around !=1 (see Figs. 3(a) and 4(a)). On theother hand, for ! = 3:5; Ns is around 20 (see Figs. 3(d) and 4(d)) corresponding tothe adsorbed expanded state of the chain. For u=2:0 and 2.5 (these values correspondto the collapsed state), the distribution of monomers is shown in Figs. 5(a–d) and6(a–d).For != 1, the polymer chain is in the bulk as there is practically no monomer on

the surface (Ns ∼ 0) but Nu is large which indicate that suLcient fraction of monomersoccupy the neighbouring sites. When ! is chosen equal to 2.0 a value which lies inbetween !c1 and !c2 line, as shown in Fig. 2, a fraction of monomers gets attached tothe surface (Ns ∼ 12) indicating the breaking of translational invariance of the globulecorresponding to the SAG phase. With further increase in ! polymer gets adsorbed incollapsed state where Ns ∼ 20 and Nu is around 8.We also calculate number of neighbouring pairs of monomers at the surface i.e.,

〈Nc〉. This can be calculated from the expression

〈Nc〉=∑NcZN (!; u)∑ZN (!; u)

: (6)

The variation of 〈Nc〉 with ! for di1erent value of u = 10; 15; 20; 25 is shown inFig. 7 for two dimensions. At such a high value of u the most likely structure is ofHamiltonian walk as shown in Fig. 1(a). Below certain value of !c1, this structureremains in bulk and 〈Nc〉 and 〈Ns〉 are almost zero. At certain critical value !c1, thisstructure gets stuck on the surface (Fig. 1(b)). At this value of !, the 〈Nc〉 is equalto 〈Ns〉 and goes as Nd−1=d. As ! increases 〈Nc〉 also increases and reaches to itsmaximum at !c2. The corresponding conGguration is shown in Fig. 1(d). With furtherincrease in !, polymer gets adsorbed on the surface with decrease in 〈Nc〉 as shownin Fig. 7.


In this paper we have provided ample of evidences in favour of the existence of SAGphase. The SAG phase which exists in between the boundaries !c1 and !c2 for u¿ucis essentially a globule sticking to the surface in the same way as liquid drop maylie on the surface. In between these boundaries monomer–monomer attraction remainse1ective in holding the monomers in the neighbourhood of each other than the surfacemonomer attraction whose tendency is to spread the chain on the surface as shown inFig. 1(a–e). The bulk free energy remain same up to !c2 suggesting that � is always(d−1)=d in the entire SAG phase. However, amplitude of this term will increase as !increases. The phase boundaries !c1 and !c2 tend to 1 and u, respectively as T goesto zero and is independent of dimension. We once again emphasized that the transitionassociated with !c1 line is surface transition.

We thank Deepak Dhar and R. Rajesh for many helpful discussions. We would alsolike to thank the organizers for giving us opportunity to present this work. Financialassistance from INSA, New Delhi and DST, New Delhi are acknowledged.

References

[1] R. Rajesh, D. Dhar, D. Giri, S. Kumar, Y. Singh, Phys. Rev. E 65 (2002) 056124.[2] P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1979.[3] E. Bouchaud, J. Vannimenus, J. Phys. 50 (1989) 2931.[4] S. Kumar, Y. Singh, Phys. Rev. E 48 (1993) 734;

Y. Singh, S. Kumar, D. Giri, Pramana 53 (1999) 37.[5] Y. Singh, S. Kumar, D. Giri, J. Phys. A 32 (1999) L407.[6] Y. Singh, D. Giri, S. Kumar, J. Phys. A 34 (1999) L67.[7] P. Grassberger, R. Hegger, Phys. Rev. E 51 (1995) 2674;

P. Grassberger, R. Hegger, J. Phys. I (France) 5 (1995) 597.[8] D.S. Gaunt, A.J. Guttmann, in: C. Domb, M.S. Green (Eds.), Phase Transition and Critical Phenomena,

Vol. 3, Academic, London, 1974.[9] D.P. Foster, E. Orlandini, M.C. Tesi, J. Phys. A 25 (1992) L1211;

D.P. Foster, J. Yeomans, Physica A 177 (1991) 443;F. Igloi, Phys. Rev. A 43 (1991) 3194.

[10] T. VrbovPa, S.G. Whittington, J. Phys. A 31 (1998) 3989;T.W. Burkhardt, E. Eisenriegler, I. Guim, Nucl. Phys. B 316 (1989) 559.

Does a surface attached globule phase exist?

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