How patchy can one get and still condense? The role of dissimilar patches in the interactions of colloidal particles

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1 Page 1 of 100 How patchy can one get and still condense? The role of dissimilar patches in the interactions of colloidal particles Paulo Teixeira Instituto Superior de Engenharia de Lisboa and Centro de Física Teórica e Computacional Lisbon, Portugal Work done in collaboration with: José Maria Tavares and Margarida Telo da Gama

2 1. Criticality of strongly dipolar fluids is still unsolved problem (image by J.-J. Weis). A related, more general issue is interplay between condensation and association. It is now possible to custom-fabricate matter that exhibits both selfassembly and the usual phase transitions (condensation, freezing, etc): patchy colloids (images by Y. S. Cho et al.). Page 2 of 100 We want to study a model that retains the essential symmetry of dipolar forces leading to association, but leaves out apparently inessential features (long range and complex angular dependence). As a first step, we generalise work by Sciortino and co-workers by applying Wertheim s theory, as formulated by Jackson et al., to patchy particles decorated with different interaction sites.

3 2. Page 3 of 100 Hard spheres of diameter σ and volume v s, each decorated with three sticky spots: two A s and one B. AA, BB or AB bonds may form. Bonding free energy from Wertheim s theory: βf b βf b N = 2 ln X A + ln X B X A X B , X i is the probability of having a sticky spot of type i not bonded. Law of mass action yields: X A + 2η AA X 2 A + η AB X A X B = 1, X B + η BB X 2 B + 2η AB X A X B = 1, where η (N/V )v s is the packing fraction, and (v ij = v b, g ref (r) = 1) ij = 1 g ref (r) [exp(βɛ ij ) 1] dr v b [exp(βɛ ij ) 1]. v s v ij v s Free energy per particle is: which is a function of (η, T ) only. βf = βf HS + βf b,

4 Lowest-energy structures (without loops): Linear chains Dimers Page 4 of 100 Hyperbranched polymers

5 Different types of junctions X-junction is always attractive: ɛ j = ɛ BB < 0 Page 5 of 100 Y-junction is attractive only if ɛ j = ɛ AB + ɛ AA /2 < 0 ɛ AB /ɛ AA > 1/2

6 Phase diagrams for ɛ BB = 1 and variable ɛ AB Coexistence densities vs temperature Page 6 of 100 Liquid phase has low density. Two-phase coexistence for ɛ AB 0: X-junction condensation. At large ɛ AB, T c saturates, ρ c 0.

7 Degrees of non-association of A sites at coexistence Page 7 of 100 Liquid more strongly associated than vapour. For ɛ AB 0, fewer AB bonds form in the liquid phase and more in the vapour phase two-phase region shrinks. At large ɛ AB, more AB bonds and Y-junctions: less compact aggregates.

8 Degrees of non-association of B sites at coexistence Page 8 of 100 Liquid more strongly associated than vapour. For ɛ AB 0, more BB bonds form: X-junctions. At large ɛ AB, B sites are fully associated into AB bonds.

9 3.2. Phase diagrams for ɛ AB = 0.75 and variable ɛ BB Coexistence densities vs temperature Page 9 of 100 Liquid phase has low density. Increasing ɛ BB broadens region of two-phase coexistence. T c and ρ c saturate at both large and small ɛ BB, but ρ c remains finite.

10 Degrees of non-association of A sites at coexistence Page 10 of 100 Liquid more strongly associated than vapour. At large ɛ BB, A sites are fairly strongly associated in the liquid phase, more weakly so in the vapour phase. For ɛ BB 0, A-site association is almost complete in the liquid phase, and fairly strong in the vapour phase.

11 Degrees of non-association of B sites at coexistence Page 11 of 100 Liquid more strongly associated than vapour. At large ɛ BB there is strong association of B sites in either phase. For ɛ BB 0, there are few bonded B sites in the vapour phase, and not so many in the vapour phase.

12 3.3. Critical points: ɛ BB 0, different ɛ AB Critical density Page 12 of 100 Complex non-monotonic dependence on ɛ AB : for ɛ AB large ɛ AB hyperbranched clusters. 0 chains, at

13 Critical temperature Page 13 of 100 T c decreases monotonically with ɛ AB. A critical point exists for all ɛ BB.

14 Degree of non-association of A sites Page 14 of 100 If ɛ AB 1, most bonds are AB with a few AAs: we are in the hyperbranched cluster limit and the critical density is low. if ɛ AB 1 we have mostly AAs, hence chains, connected by a few AB bonds (Y-junctions): the number of BB bonds drops as ɛ BB 0, the critical clusters are expected to be large and the critical density low.

15 Degree of non-association of B sites Page 15 of 100 If ɛ AB 1, most bonds are AB with a few AAs: we are in the hyperbranched cluster limit and the critical density is low. if ɛ AB 1 we have mostly AAs, hence chains, connected by a few AB bonds (Y-junctions): the number of BB bonds drops as ɛ BB 0, the critical clusters are expected to be large and the critical density low.

16 3.4. Critical points: ɛ AB 0, different ɛ BB Critical density Page 16 of 100 For ɛ BB 0.75, ρ c stays close to its maximum, so the critical clusters are probably relatively small. However for ɛ BB < 0.5, ρ c drops to very small values.

17 Critical temperature Page 17 of 100 T c decreases as ɛ BB decreases. For ɛ BB = 0 there is no critical point if ɛ AB < 1 3.

18 Degree of non-association of A sites Page 18 of 100 For ɛ BB < 0.5 most A sites are bonded. At large ɛ AB exactly half of all A sites are bonded.

19 Degree of non-association of B sites Page 19 of 100 For ɛ BB < 0.5 hardly any B sites are bonded. At large ɛ BB there is full association of B sites.

20 X-junction driven criticality: ɛ AB = 0, ɛ BB 0 Page 20 of 100 Tc = ɛ BB ln b, [ η c = 9v b 8v s (B 3 + 6B 2 2) 2 ] 1 5 exp ( ln b 5ɛ BB A critical point is always present, with lower and lower critical density and temperature Y-junction driven criticality: ɛ BB = 0, ɛ AB 0 Tc = ɛ AB 1 3, b η c = v ( ) b exp bɛ AB v s 3ɛ AB 1. On decreasing ɛ AB a critical point of vanishingly small density and temperature is obtained only up to ɛ AB = 1. 3 ).

21 5. Page 21 of 100 We have applied Wertheim s theory to patchy colloids with three sites: two A s and one B. Their interaction strengths are ɛ AA, ɛ BB and ɛ AB. We fixed ɛ AA and found first-order condensation with well-defined limits as ɛ BB or ɛ AB are varied. For ɛ AB /ɛ AA 1 there is complete association of the Bs at the critical point. This is consistent with the formation of highly ramified clusters akin to hyperbranched polymers, which grow as the bond gets stronger. For ɛ BB /ɛ AA 1 there is a high degree of association of Bs in either phase. The vapour phase consists mostly of BB dimers, and the liquid phase of BB dimers connected by AA bonds. For ɛ BB 0 or ɛ AB 0, long AA chains form with either AB or BB branches. These are the relevant limits to strong-dipolar-fluid criticality. When ɛ BB = ɛ AB = 0 we recover the limit of two identical As. This limit is non-trivial: If ɛ AB = 0, the critical point exists all the way to ɛ BB /ɛ AA = 0. This corresponds to X-junction condensation. If ɛ BB = 0, there is no critical point for ɛ AB /ɛ AA < 1/3. This corresponds to Y-junction condensation.

22 Acknowledgement Financial support from the Foundation of the University of Lisbon and the Portuguese Foundation for Science and Technology (FCT) under Contracts nos. POCI/FIS/55592/2004 and POCTI/ISFL/2/618, is gratefully acknowledged. Page 22 of 100