A model of crack based on dislocations in smectic A liquid crystals. School of Physics, Beijing Institute of Technology

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1 A model of rak based on disloations in smeti A liquid rystals Tian You Fan, Zhi Yi Tang Shool of Physis, Beijing Institute of Tehnology Beijing , China Abstrat A plasti rak model for smeti A liquid rystals under longitudinal shear is suggested. The solution of srew disloation in smeti A is the key in whih the orret result is just obtained by overoming a longstanding puzzle [19]. We further use the disloation pile-up priniple and the singular integral equation method, onstrut the solution of the rak in the phase. From the solution we an determine the size of the plasti zone at the rak tip and the rak tip opening (tearing) displaement, whih are parameters being relevant to the loal stability/instability of materials. The results may be useful in developing soft-matter mehanis. Key words: Smetis A, srew disloation, disloation pile-up, plasti rak, loal instability It is well-known that liquid rystals and quasirystals are the important phases in ondensed matter, and fasinating from a fundamental point of view. Liquid rystals are also signifiant on aount of a number of urrent and potential appliations. They belong,marosopially, to an intermediate phase between isotropi liquid and solid. The mehanial 1

2 properties of the phase are very interesting and have been extensively studied, see de Gennes and Prost [1], Kleman and Oswald [], Oswald and Pieranski [3] for example. The need of basi researh and engineering appliations requires promoting the study further, espeially of three-dimensional elastiity, plastiity, defets and dynamis. Among defets, disloations and foal onis have been investigated, see for example the above referenes [1-3], Landau and Lishitz [4], Fujii et al [5]. In addition, Brostow et al [6] have arried out numerial simulation on rak formation and propagation in polymer liquid rystals. Beause liquid rystals inluding smeti ones an be lassified as monomer liquid rystals (MLCs) irrespetively of the fat whether they an or annot polymerize and polymer liquid rystals (PLCs). That lassifiation is due to Samuski [7] and has been used by a number of authors [8, 9]. The present model applies to MLCs, smeti A phase in partiular. To develop the pioneering work of Brostow et al [6], Fan [10] studied the novel defet in liquid rystal very reently. In the respet of dynamis, the hydrodynamis, elastodynamis and wave propagation were disussed in the papers and books of de Gennes and Prost [1], de Gennes and Kleman [11], Oswald and Pieranski [3] and Landau and Lishitz [4]. In addition, on the plastiity of liquid rystals, de Gennes and Prost [1], Oswald and Pieranski [3] have done the pioneering work. However the plasti theory of liquid rystals has not been developed so far, either mirosopially or

3 marosopially. This leads to some diffiulties on the study of plastiity of liquid rystals. On the three-dimensional elastiity for liquid rystals, there is a preliminary work, see for example Fan [1], but more diffiult topis have not yet been engaged. Owing to the diffiulty mentioned above, we suggest a phenomenologial model to ondut a plastiity problem for smeti liquid rystals. An analyti solution of plastiity-rak interation are presented, whih may provide a basis for fundamental researh in soft matter physis and hemistry. It is well-known that there are inherent onnetions between disloations and raks, for example, piling-up of disloations an result in a rak, whih may strongly influene the mehanial behaviour of liquid rystals. Craks affet the plastiity of liquid rystals, but the mehanism is not so lear. A oupling between frature and plastiity makes the problem more ompliated. Fan [10] disussed a plasti rak in a smeti liquid based on the disloation pile-up model. This model has been used for rystals by Bilby et at [1, 13], and for quasirystals by Fan et al [14, 15]. But the work arried out in Ref [10] ontains some approximations, in whih the movement of disloations do not stritly follow the Peah-Koehler fore rule, the reason for this lies in the limitation of methodology adopted. In Ref [10] work onduted is edge disloation pile-up model. The present work takes srew disloation pile-up model in whih the movement of the disloations an stritly obey the Peah-Koehler fore rule. 3

4 Consider a smetis A liquid rystal whose layers are in the xy plane, and a rak disloation group with length l along x axis ( ) subjeted to a uniform shear stress σ = τ yz shown in Fig.1, in whih the single srew disloation has Burgers vetor b = (0,0, b). Beause the deformation is assumed to be independent from variable y, the figure depits any transverse setion of the body. At the rak tip there is an srew disloation pile-up with length d, whose value is unknown and to be determined. We all the pile-up as sliding disloation group. Within the zone defined by y = 0, l x l+ d, a ounter diretion shear stress τ is applied, the value of whih represents the yield strength of the materials marosopially. In other words the disloation pile-up zone is the plasti zone. The physial meaning of τ an be referred to the monograph [1] (p.499). 4

5 Fig.1 A plasti rak in a smetis A under longitudinal shear The model an be formulated by the following (equivalent) boundary onditions: x + y σ = 1/ ( ) : ij 0 ( ) = 0, < : σ yz = τ ( ) 0, : σ yz τ τ y x l y = l < x < l+ d = + (1) A quantitative desription of the problem needs to introdue relevant basi governing equations. The mathematial model (1) requires the non-linear (plasti) problem to be linearized although it ontains a unknown onstant d. This greatly simplifies the mathematial solution, and allows us to hoose linear governing equations. Aording to de Gennes and Prost [1] or Landau and Lifshitz [4] the stress tensor for smeti liquid rystals (exatly speaking for smetis A) 5

6 is u σ xx = σ yy = K1 z ' u σ ρ zz = 0B z u σzx = σ xz = K1 x u σzy = σ yz = K1 y σ xy = σ yx = 0 () in whih ' u u is the displaement of z omponent, B = B C / A, z ρ 0 and K 1 are materials onstants, see [4]. Physially the onstant ' B desribes material modulus relating to deformation arising from displaements, while K 1 desribes material modulus relating to deformation arising from urvature. It is evident that to the stress tensor () only the symmetry part, aording to the Landau s terminology, is onsidered, the visous part is omitted here by following the methodology adopted in the liquid rystal study ommunity. Substituting equations () into the equilibrium equations σ ij x j = 0 (3) yields the final governing equation u =0 (4) where x y = +. Equation (4), whih an also be derived from the energy minimization, is idential to those given by de Gennes and Prost [1], Oswald and Pieranski [3] and Landau and Lishitz [4]. 6

7 The solution of the plasti rak is redued to solve equation (4) under boundary onditions (1), whih an be alled the boundary value problem (4),(1). To solve the boundary value problem (4),(1) we an refer to the experienes in rystals[13,14], quasirystals [15,16], we an also refer to the experiene in liquid rystals for edge disloation more reently, i.e., to transform the proble into the following singular integral equations L f( ξ ) d ξ ( x) = τ ξ x A (5) in whih f ( ξ ) is a disloation density funtion, ξ the disloation soure point oordinate, and x the field point oordinate on the real axis, L represents interval ( ( l+ d), l+ d), and τ ( x) the shear stress distribution at the region z = 0, x l+ d, i.e., τ ( x) ( ) τ, x < l = ( ) τ + τ, < < + l x l d (6) and the key is onstant A how an be hosen? This depends upon the solution of srew disloation of smeti A liquid rystals. Kleman [17] and Pershan [18] have derived the solution for a long time, but it is wrong, this results in a longstanding puzzle in study, whih an named de-gennes-kleman-pershan paradox. Fan and Li [19] overome the puzzle, and gave a orret solution. 7

8 b A= KD, D = π πα 3 π bk R π αβ ( R + r ) + ln + αγ ( R r ) π r0 (7) in whih α 4 b ' = ρ0b π 3π β = γ = π + 56π (8) (Note that the onstant A here should not be onfused with the material onstant A in equation ()). In terms of the singular integral equation theory of Muskhelishvili [18] (p.51), the integral equations (5) under ondition (6) has the solution 1 x+ ( l+ d) ξ ( l+ d) dξ f( x) = τξ ( ) π A x ( l+ d) L ξ + ( L+ d) ξ x 1 x+ ( l+ d) 1 l ( ) = i os τ τ π π A x ( l+ d) l d + τ 1 ( l + d) lx 1 ( l + d) + lx + osh osh π A ( l+ d)( l x) ( l+ d)( l+ x) (9) (the details of the mathematial alulation are quite lengthy and are omitted here), in whih i = 1, and A is defined by equation (7). Beause the disloation density f( x ) should be a real funtion, the fator multiplying the imaginary number i in the first term of right-hand side of formula (8) must be zero, this leads to τ l l d τ π = + 1 ( ) os 0 8

9 i.e., ( ) πτ d = l se 1 τ (10) This determines the plasti zone size. From solution (9) we evaluate amount of disloations N( x ) suh as N( x) f( ξ) dξ 0 x = (11) Substituting (9) (oupled with (10))into (11) we an get values of Nl ( + d) and () Nl, so the amount of disloation motion is ( ) blτ l+ d τ l πτ δ = bnl [ ( + d) Nl ( )] = (ln ) = ln se( ) (1) π πρ τ ' A l 0B This is the rak tip opening (tearing) displaement, whih is an important parameter. We suggest the following frature riterion δ = δ (13) whih an be used for determining the thermodynami stability/instability of the material, δ is the ritial value of the rak tip opening displaement, whih an be measured by experiments, and is a material onstant of the liquid rystals. The equation (13) desribes a ritial state of equilibrium of the plasti rak. When δ < δ, the rak does not propagate, but whenδ > δ, the rak will propagate. By using this ( ) riterion, the limiting value of the applied stress τ or the limit value of the rak size l an be determined. The above treatment is a maro-desription(or the ontinuum model),but 9

10 a miro-desription (or a miro-mehanism) an be given as follows. By introduing the de Gennes theory (refer to [1]), the yield stress is τ πγ ~ aktln( v / v) 0 B 0 1 (14) where 1 K γ K Ba / η, η ~, v = 10 s m, v = 1s m B (15) and a0 represents the thikness of the layer of the smeti, whose value is almost equivalent to the magnitude of a Burgers vetor (10 nm), k B is the Boltzman onstant, T the absolute temperature, v 0 and v 1 the flutuation frequenies, respetively. Substituting expression (14) into equations (10)and (1)respetively one reveals the physial sense of the plasti zone size (or disloation sliding width) and rak tip opening displaement (or amount of disloation motion) in-depth. Owing to the limitation of spae a detailed disussion is not given here. Crak and plastiity are diffiult topis in liquid rystals. One of reasons for this lies in there being lak no theory of plastiity, at least, there is an absene of marosopi plasti onstitutive equation so far. Here we have adopted a phenomenologial model to disuss the problem. In this way we obtain some physial quantities for desribing the oupling between frature and plastiity. The methodology developed here is generally effetive for other problems in smetis and other lasses of liquid 10

11 rystals. Results obtained by the present author and o-workers will be reported in other papers, e.g. [19]. The model suggested in Ref [10] has its limitation itself, for example, the Peah-Koehler fore for the onfiguration is not stritly along the x-diretion, only has a omponent along the diretion. This means the movement of disloations is not stritly along the x-diretion, so the disloation pile-up. So that the assumption of the movement of disloations along the x-diretion is only approximate, the assumption of the disloation pile-up along the diretion is also approximate. For the srew disloation under the ation of longitudinal shear, the movement and pile-up are stritly along the x-diretion, and the solution is exat. Referenes [1] P. G. de Gennes and J. Prost, The Physis of Liquid Crystals, Clarendon, London, 1993, p. 490,493, 499. [] M. Kleman and P. Oswald, J. Physique, 43 (198), p.655. [3] P. Oswald and P. Pieranski, Smeti and Columnar Liquid Crystals, Taylor & Franis, London, 006. p.13. [4] L. D. Landau and E. M. Lifshitz, Theory of Elastiity, 3d Edition, Pergamon, Oxford, 1986, p.177,181. [5] S. Fujii, S. Komura, Y. Ishii and C-Y. D. Lu, J. Phys.: Condens Matter, 3 (011), p [6] W. Brostow, A. M. Cunha, J. Quintanila and R. Simoes, Maromol. 11

12 Theory Simul., 11 (00), p.308. [7] E. T. Samulski, Faraday Disuss. Chem. So., 79 (1985), p.7-0. [8] W. Brostow, Properties of polymer liquid rystals: hoosing moleular struture blending, Polymer, 31(1990), p.979. [9] M. Hess, High Performane Polymers, in Performane of Plastis, Brostow W, ed, Hanser, Munih-Cininnati, 000, Ch. 1,p.519. [10] T. Y. Fan, Plasti rak in a smeti A liquid rystals a possible model, Phil Mag Lett, 9(01), p.153. [11] P. G. de Gennes and M. Kleman, Liquid Crystals and Plasti Crystals, 1 (1974), p.9. [1] T. Y. Fan, Displaement potentials and ellipti disk-shaped rak in three-dimensional smeti B liquid rystals, unpublished work (011). [13]B. A. Bilby, A. H. Cottrell and K. H. Swinden, Pro. R. So. A, 7(1963), p.304. [14] B. A. Bilby, A. H. Cottrell, E. Smith and K. H. Swinden, Pro. R. So. A, 79 (1964), p.1. [15] T. Y. Fan, H-R. Trebin, U. Messershmidt and Y. W. Mai, J. Phys.: Condens. Matter, 16 (004), p.59. [16] T. Y. Fan, Mathematial Theory of Elastiity of Quasirystals and Its Appliations, Springer-Verlag, Heidelberg, 010, p.307. [17] M. Kleman, J. Physique, 35(1974), p.595. [18] Pershan P S, J Appl Phys, 45(1974), p

13 [19] T.Y. Fan and X. F. Li, Corret solution of srew disloation in smeti A liquid rystals and mistakes of the lassial solution, Chin Phys B, 01, in press. [0]. N.I. Muskhelishvili, Singular Integral Equations, Groningen, Noordhoff, 1956, p