Critical behaviour of the Ising ferromagnet confined in quasi-cylindrical pores: a Monte Carlo study

The critical behaviour of the Ising ferromagnet confined in pores of radius R and length L is studied by means of Monte Carlo computer simulations. Quasi-cylindrical pores are obtained by replicating n-times a triangular lattice disc of radius R, where L = na and a is the spacing between consecutive replications. So, spins placed at the surface of the pores have less nearest-neighbours (NN) as compared to 8 NN for spins in the bulk. These "missing neighbour" effects undergone by surface spins cause a strong suppression of surface ordering, leading to an ordinary surface transition. Also, the effect propagates into the bulk for small tubes (R ≤ 12) and the effective critical temperature of the pores is shifted towards lower values than in the bulk case. By applying the standard finite-size scaling theory, subsequently supported by numerical data, we concluded that data collapse of relevant observables, e.g., magnetization (m), susceptibility, specific heat, etc., can only be observed by comparing simulation results obtained by keeping the aspect ratio C ≡ R∕L constant. Also, by extrapolating "effective" R-dependent critical temperatures to the thermodynamic limit (R → ∞, C fixed), we obtained T(C)(∞) = 6.208(4). As suggested by finite-size scaling arguments, the magnetization is measured at the critical point scales according to [|m|]Tc R(β/ν) is proportional to [R/L](1/2), where β and ν are the standard exponents for the order parameter and the correlation length, respectively. Furthermore, it is shown that close to criticality the axial correlation length decreases exponentially with the distance. That result is the signature of the formation of (randomly distributed) alternating domains of different magnetization, which can be directly observed by means of snapshot configurations, whose typical length (ξ) is given by the characteristic length of the exponential decay of correlations. Moreover, we show that at criticality ξ = 0.43(2)R.