Phase transitions for polymers on fractal lattices

Physnca D 38 ( 1989~ 3 5 1 - 3 5 5 North-Holland. Amsterdam

PHASE TRANSITIONS FOR POLYMERS ON FRACTAL LATriCES J. V.~NNIMENUS Laboratotre de Physique Stattst~que Ecole Normale Supdneure, 24 rue Lhomond. .'523 ! Parts Cedex 05. France and Labora°mre Loms-N~d. CNRS, 38042 Grenoble Cedex, France Models of linear and branched polymers o n finitely ramtfiecl fractal lattices are briefly reviewed. T h e n r foremost interest ts to provnde exactly solvable systems with phase transitions at fintte values of the fugacily or temperature. This gnves insight nnto the behavtour of polymers in inhomogeneous media, their collapse transition and their adsorption at surfaces. Some unexpected results are also found such as the possible existence of essential singularities ,n generaung functnons, or the non-convergence of critical exponents towards their Euclidean limtt.

!. Historical background

This Conference offers a chance 1o reflect on how and why one got involved in the physics o f fractals, and I will give an account of the motivations and general thread of our work, rather than insist on technical details. My first contact with the w o r l d of fractals occurred as earl), as 1974, while a post-doctoral fellow at the IBM laboratory in Yorktown Heights. There, Benoit Mandelbrot shog'ed me the first chapters of his manuscript "'Les Objets Fraclals" [ I ], asking for the reactions of ~ "'typical reader". In fact he needed advice on French literaD' style rather than scientific contents, and my wife was better qualified to make suggestions in that respect - so she, not me, is thanked in the foreword to the book. h was amusing anyway to learn that the coast o f Brittany has infinite length and to discover the concept of self-similarity, but frankly all this looked so far from my research interests - "'serio,-s'" ca!e-!ations on t.he e!eetroni, stru,zlure of metallic surfaces - that ! did not imagine to ~ork some day on tl~ese o~uestions. Iv, hm.dsighl, ! missed a unique oppoi-i.u~tly Io enter a new field as a pioneer, but then very few people around proved more far-sighted! Eight years later, fraetals had become fashionable and provided reD'. useful models for the random meTM

Essays nn honour of Benolt B. Mandelbrol Fractals in Ph,~stcs - ,~. ~haron) andJ. Feder ledttors)

dia we were studying at Ecole Normale. Also, the)" appeared to open a new approach to phase transilions by offenng well-defined realizations of spaces o f non-integral dimensionality [ 2 ]. In particular one could hope to gain a deeper insight into the c-expansion, scaling laws, universality, ... from the solution of new non-trivial models. Many spin models ( lsing. Polls, X Y) have been subsequemly considered m that spirit, but either they have transittons only at T = O [3l, or the)' are not exactly soh'able and one has to resort t o various aporoximations [4-7 i. a iimnalion wlfich lessens their usefalness as Lcstw,g grounds. Polymer models turn out to be the most notable exception to that disappointing situation, in their case a well-defined thermodynamic transition exists as a function of monomer fugacity, and exact results can be obtained on fractal lattices of finite ramification such as the Sierpinski gasket 18,9]. Various critical properties and their relationships can thus be s~udied in detail, and most aspects are found to be qualitatively simniar to their counterparts ot~ reguiar Eu~c~'~dean iamces. ,,nh just d i | T e r e n i ~aiu¢~ o ( t'r~.it,.al ¢,,fi-actals ha~ e ~ieldea severai ....... ,,::.c wa) to new questions and possible generalizatnons. and are the best justnficalion for my lasting interest m that class of s.vstems.

LI. 0167-278% ~9,"$03.50 '~ Elsc~ n~r Sctence pl'.,., shers B ~, I North-Holland Ph.~stcs Publnshmg Dt~ nslon )

352

J i "anmmenus / Poh'mers on fractal latuces

2. Linear polymers The simplest "'polymer'" system on a lattice is the standard self-avoiding walk (SAW), where the links do not interact except for the non-crossing constraint. This models a linear polymer in a good solvent and one expects that the gyration radius o f a n Nmonomer chain behaves for large N as

(R)~N".

(!)

where the average is taken over all configurations with equal weight. We considered this problem on fractals with Rammal and Toulouse [9], as a natural generalization of their work on random walks [ 10]. One first question was to understand on which properties of the lattice the exponent v depends: Is a simple Flops'type formula based on the fraclal dimension D [ l I ],

1'=3/(2+D).

(2)

valid, at least as a first-order approximation? Also, ~e hoped to gain insight in the controversial and deep problem of polymers in random medm [ ! 2-14 ]. SAW mc,0els on fractals had in fact already been studied b.v Dhar [8 ], mti~u,.,gh he ~,medthe ierm "pseudo-latltces'" and his work was not ,cry widel.~ known. Our a.am contribution was to point out ~hat the pro0uct Dv must be an intrinsic quantity, i.e., it is invariant fflhe lattice is crumpled or distorted but its topology is fixed. An "'improved" Flops' expression was proposed, introducing the spectral dimension ~ a s a new ingredient, and it was in reasonable agreement with the kno~n exact cases - but clearly our expression was still veps, crude. Recently that question has been taken up again from a deeper point of view by Bouchaud and Georges [ i 5 ] and by Aharony and Harris [ i 61, ~ he independently obtained the formula

I

4d, - , 7

D 2+2d.-,'t'

(3)

where d, ts the chemica~ or "'spreading" dimension. v, hich describes the average number ofaccesmble sites on the lattice [ 17 ]. Express~on ( 3 ) is based on tea-

sonable assumptions and correctly reproduces the trend among some families of fractals, but in spite of its dependence on three distinct properties it is not yet in very close agreement with all known values for instance it predict.,- v=0.825 and 0.725, respectively for the 2-d and 3-d gaskets, while the exact values are 0.798 and 0.674. Anyhow, it is very useful to test in that way the validity o f arguments proposed to explain the success o f the Flory approximation and of its generalizations to inhomogeneous media. One also sees clearly that universality on fractals is much weaker than on regular lattices [18]: it is possible to determine some properties on which critical exponents must at least depend, but not to list a finite set of properties that uniquely define a universality class.

3. The collapse transition On regular lattices a collapse transition of the polymer may occur when a m o n o m e r - m o n o m e r shortrange attractive interaction is introduced to model the eITect of a bad solvent. At the transilion temperature ( 0 r-c!.',~ ) the polymer behaves on large scales essentially as an Iota; Gaussian chain, with t'o= I / 2 for d>_ 3, and at lower lemt;eralures it forms a compact globule, with vc= I/at. The !dea to look for a similar transition on fraclal lattices is clue to Klein and Seitz [ 19 ], who concluded that none'occurs for finite values of the interaction on the d--2 Sieroinski gasket

(SG). With Dhar, we realized that this negative result might be due to a topological constraint: a SAW can only cross once a given kth-order triangle of the gasket, so self-interactions only occur at the vertices and the)' become rarer and rarer as the size N increases. Thts effect does not occur on the 3-d gasket, and indeed we readily discovered the existence in that case of a new fixed point corresponding to the usual 0 point, with vo=0.5294, in tact the transition was implicitly contained in the recurrence equations written down eight )'ears earher by Dhar! A whole family of fraclals depending on one (m-

J. |'anm menus / Polymers on fractal lattu'es teger) scale factor p, the "'modified rectangular lattices", were then studied along these lines [ 20]. The resulting equations can even be continued formally for p ~ !, enabling us to mimic a "'quasi-Euclidean" system in the sense o f G e f e n et al. [ 2 ]. This trick provides a new MigdaI-Kadanoff type renormalization scheme for the collapse transition problem in two dimensions, which gives an estimate uo-O. 546 + 0.010, in very good agreement with the best numerical determination Uo=0.55+0.01. It also shows how the deep connection between fractal models and realspace renortnalization group equations may be exploited in a quantitative manner.

4. Adsorption In the presence o f an impenetrable surface such as a solid wail, a polymer in dilute solution feels an effective repulsion due to the loss o f configurational entropy with respect to the bulk, so the solution is depleled near the surface. An attractive surface po-. tential is necessary, to balance that effect. Then there exists a transition temperature T~, below which a finite fraction of the monomers remains adsorbed. The transition is analogous to a tricritical pomL and there is a cross-ovL=r region in which the fraction M of adsorbed monomers scales as M~N°~

IT-T~I-'

(4}

The value of the cross-over exponent ~ is 1/2 for an ideal chain and ~ 0.6 for a SAW in d-- 3 [ 2 ! ]. With Bouchaud, we have recently considered that problem ~ hen the polymer is restricted to reside on a fractal lattice and interacts with a fractal surface of dimension d, [22]. By making reasonable physical assumptions and assuming a scaling form for the density profile, as for the Euclidean case, simple bounds on the exponent 0 may be derived: d:lD>O>__ l - { D - d ~ ) ~ , ,

{5)

where t, is the gyration radius exponent in the bulk solution. On the other hand exact results may be obtained

3~.3

Ibr the gaskets, if the natural bounaaD' is chosen as adsorbing surface. We find that an adsorpnon wansition takes place at a finite T~ and close to T~ the polymer behaviour is correctly described by eq. (4), with 0=0.59152 and 0.7481 for the 2-d and 3-d SG, respectively, in agreement with the stringent bounds given in eq. {5). it is also possible to study adsorption in the presence o f attractive self-interactions on the 3-d SG, and one observes a multicritical point, where a collapse and an adsorption transition coexist.

5. Branched pol~iers After linear polymers, it sounds natural to study their branched cousins en fractals, but the generalization is far from straightforward, it took the strong motivation of Knezevic to overcome my pessimism and to enumerate the possible configurations on the SG and get their recurrence relations. Even then, their behaviour was confusing and quite different from the linear polymer case. We had to find ways to extrzct the dominant terms for the asymptotic behavlour be/'ore we could understand what was happening and locate the relevant fixed points: Ihey correspond to finite values lbr specific combinations of the basic variables, which themselves go to zerc or to infinity. Armed with that experience we could also solve the 3-d gasket, where it is necessary to keep track ofabout I 0 ~configurations aad to generate an I I -equation recursion system - about the limit for an "'exact" solution, even by computer! That effort yielded several very i,.teresiing results [231: ( i ) L o o p s are irrelevant on large scales, so branched polymers indeed belo.~g to ~h¢ same ':.~!versality class as lattice animals. t 2 '~ One can calculate e.xac.:iy .-'as wel~ as l:,-~eeiher basic exponent O, ~vhieh appears m the singular pan of the generating function near the critical fugac~t.~

i/u, G(x) ~ (I- ~)'~-'. These

exponents

(6) ha~e

"'reasonable"

values:

354

J. ! "anmmenus / Pol.vmer~on.#actal lauwes

v=0.71655 and 0 = 0 . 5 3 2 8 for the 2-d SG, to be compared toy=0.641 and 0= 1 for the square lattice. The new feature is that the two exponents are independent, while on Euclidean lattices they obey the Parisi-Sourlas relation: ( 0 - 1 ) / ~ , = d - 2 . On fractal lattices one can write by a n a l o g y ( 0 - I ) / v = D - & where J is very close to 2 for some quasi-h.aear lattices, while d = 2.237 for the 2-d SG. For all lattices studied so far, J>_ 2, but there is no interpretation of that finding yet. ( 3 ) A collapse transition exists for the ,~askets, with an exponent vt at the transition temperature extremely close to the value in the compact phase v¢ = I/D. For instance, v,=0.6325, v~=0.6309 for the 2-d SG, and vt=0.5055, v¢= I/2 for the 3-d SG. This is very similar to the situation in d = 2, where the best numerical determination is v, =0.509 + 0.003, akld it suggests the possibility for a deeper expk v,ation in terms of some small parameter. These results confirm that there is a deep similarity with the case of regular lattices, and that man~ features of critical p h e n o m e n a survive the loss oftranslattonal invanance, but further studies revealed more mtrtguing behaviour in other systems.

6. Surprises A first type of unexpected behaviour was discovered for branched polymers on a generalized gasket. the b = 3 member of the GM family introduced by Given and Mandelbrot [24]. As expected intuitively, the exponent v=0.7068 is a little closer to the Euclidean value 0.664 than for the 2-d SG. But the generating function has an essential singularity close to the critical fugacity [2.5 ] G(.r) ~ exp[c! i -,u.~ )-" ]

that possibility in mind, for instance while analyzing numerical data for polymers in random media. Another counter-intuitive effect concerns linear polymers on the G M gaskets. There is no essential singularity in that case, and the critical exponent 7 of the generating function is well defined for all scale factors b. When b~oo, the system becomes more and more similar to an Euclidean triangular lattice, and one would expect the exponents to converge to their d=2 values. This is indeed the case for v, but Elezovic et ai. [26] found numerically that the convergence of 7 was at best very slow and non-monotonic. Through a deep analysis of the scaling properties o f SAW, Dhar was finally able to show [27] that lim y(b--,oo)= 133/32 is completely different from 7(d=2)=43/32. The collapse transition ofself-interacting polymers itself is not always as simple as expectec" .On the 3-d GM gasket a "'topological frustration" effect orevents a liner polymer from filling densely the lattice, but contrarily to the 2-d SG gasket (where no transition occurs at finite T), we discovered a novel "quasic o m p a c t " phase[ 28 ], where the fractal dimension of the polymer is slightly less than the lattice D. Fn- '~. rice trails, i.e. walks which may self-cross at a site but not on a bond. Chang and Shapir [29] found a 0point on the 2-d SG, while none exists for SAW [ 19], thus suggesting that th~ two problems belong to different universality classes. In a similar vein, Maritan pointed out recentb [30] that the random walk and the ideal chain have very different asymptotic behaviour on many fractals, contrarily to the intuition built firm experience with r a n d o m walks on percolation clusters, and that in some cases one also finds essential singularities linked to localization effects. IQt

°

(7)

~ i:h ~...)= In t 3 - ,, ] ~/In ( 3 + ~ 3 ). instead of,,he standard po~er law. eq. ( 6 ). The argument for the power law singularity relies on translation invariance, and it does not apply to inhomogeneous systems, so the behavlour discovered on one par',tcular fractal might %veh be much more general. At least, one should keep

7. Ceaclusio~ in many cases the equations describing polymers on fractal lattices are complex enough to display a rich phase diagram, with sexeral fixed points, and to re',eai uucxpected features. The main challenge raised

J. V a n m m e n u s ,' Polvmer~ on l'ractrJ latn(e.~

by the wealth of exact results so obtained is now to understand in depth when the critical behaviour will be qualitatively similar to the standard one on regular lattices, and when it may be radically different. Also, it would be very useful to develop the conneclions with real-space renormalization [ 20,27 ], in order to achieve a systematic approach applicable to more general systems.

References [ 1 ] B.B. Mandelbrot, Les Objets Fractals: Forme. Hasard et D~mension ( Flammarion, Pans, 1975 ). [2] Y. Gefen. Y. Meir, B.B. Mandelbro! and A. Aharony, Phys. Rev Lett. 50 ( 1983)145. [3] Y. Gefen. ~.. Aharony, Y. Shapir and ll.B. MandelbroL J. Phys. A 17(1984) 435. [4] Y. Gefen, A. Aharony and B.B. MandelbroL J. Phys. A 17 (1984) 1277. [ 5 ] G. Bhanot, H. Neuberger and J A. Shapiro, Phys. Rev. Left. 53 (1984) 2277. [ 6 ] P.Y. Lm and Y.Y. Goldschmtdl. J. Phys. A 20 ( 1987 ) 2159. [7] B. Bonnier. Y. Lerc)¢r and C. Me,~.ers. Phys. Rex. B 37 (1988) 3205 ioi D. Dhar, J.Malh. Ph.~s. 19 (Iq78) 5. [9] 8. Pammal. G. Toulouse and J. Vann,menus. I. Ph.ss (Paris) 45 ( 19841 389.

355

[ I'3] R. Rammal and G. Toulouse. J. Ph)s. (Pans) 44 (iq$3) LI3 [ I I I K. Kremcr, Z. Phys. B 45 ! 1951 ) 148 [ 12 i B. Den'ida, J. Phys. A 15 ( 1982 ) L! 19. 113 ] A.B. Harris, Z. Phys. B 49 (1983) 347. [ 14] A.K. Roy, B.K. Chakrabanl, J. Phys. A 20 ( 198,'7) 215. [15]J.-P. Bouehaud and A. Georges, Phys. Rev. B 39 (19891 2846. [ 16] A. Ahamny and A. Brooks Hams. J. Slat. Ph)~. 54 (198q) 1091. [ 171 J- Vannimcnus, J.-P. Nadal and H. Manta. ,L Ph.vs. A 17 (1984) L351. [ 18] B. Hu, Phys. Rev. B ,~3 (1986) 6503. [ 19] D.J. Klein and W.A. Seitz. J. Phys. Lett. 45 (1984) L241. [ 20 ! D. Dhar and J. Vanmmenus, J. Phys. A 20 ( 1987 ) 199. 121 ] E. Eisennegler, K. ICremer and K. Binder, J. Chem. Phys. 77 (1982) 6296 1221 E. Bouchaud and J. Vannimenus, Ecole Normale Supe.P.eure preprin! (1989). [ 23 ] M. Knezevic and J. Vannimenus. Phys. Roy. Letl.. 56 ( i 986 ) 1591: Ph)~. Rev. B .35 ( 1987)4988. [ 24 ] ].A. Given and B.B. Mandelbrot, J. Phys. ,~ 16 ( 1983 ) L565 [25].1. Vannimenus and M. Knezev,c, Europhys Lett. 3 ! 1957) 21. [26]S. Elezovic, M.Knezetlc and S. Mflosevtc. J Phys A 20 (1987) 1215. [27] D. Ohar. J. Ph')s A49 (1988) 397. [28] M. Knezextcand J Vannimenus, J Ph.~s 4, 20( 1987 ~ L~ .e [29]l.S. ChangandY. Shap,r.J Ph.~s.~,3l (198S~L903. [30] A Maman. LIm~ers:h o1" Padoxa prcpnnt I I'~$q~