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Casimir and Critical Casimir effects An overview Sergio Ciliberto Laboratoire de Physique, ENS de Lyon, UMR5672 CNRS, Lyon, France lundi 2 novembre 2015 Casimir and Critical Casimir effects An overview Outline (1) Introduction to Casimir effect (2) Experiments on electromagnetic Casimir effect (3) Introduction to critical Casimir effect (4) Experiments in wetting films and binary mixtures (5) Casimir effects and colloids (7) Conclusons and perspectives lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force In 1930 London showed how the leading behavior of the Van de Waals force between neutral molecules could be understood from quantum mechanics, using an effective interaction potential between the dipoles M2 M1 VLondon hν α1 α2 ∝ R6 R where ν the frequency, α the electronic polarisability and R the distance bewteen de molecules In 1948 Casimir and Polder introduced retardation, resulting in a potential which for large distances falls off faster, VCasimir−P older hc α1 α2 ∝ R R6 M2 M1 R tdelay = R/C Both results can be understood by dimensional considerations. In 1948 Casimir showed that zero-point energy was responsible for the intermolecular force. He used this idea to compute the force between two parallel perfectly conducting plates lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 Two parallel, perfectly conducting and uncharged metallic plates in vacuum attract each other due to the quantum fluctuations of the electromagnetic fields, S >> L lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 Two parallel, perfectly conducting and uncharged metallic plates in vacuum attract each other due to the quantum fluctuations of the electromagnetic fields, S >> L lundi 2 novembre 2015 Boundary conditions on the components of electromagnetic fields EII , B⊥ = 0 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 Two parallel, perfectly conducting and uncharged metallic plates in vacuum attract each other due to the quantum fluctuations of the electromagnetic fields, S >> L Boundary conditions on the components of electromagnetic fields EII , B⊥ = 0 The fluctuation modes of the fields in the space within the two plates can only have a specific set of L-dependent allowed wave-vectors: k⊥ = π n/L with n = 1, 2, . . .. lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 Two parallel, perfectly conducting and uncharged metallic plates in vacuum attract each other due to the quantum fluctuations of the electromagnetic fields, S >> L Boundary conditions on the components of electromagnetic fields EII , B⊥ = 0 The fluctuation modes of the fields in the space within the two plates can only have a specific set of L-dependent allowed wave-vectors: k⊥ = π n/L with n = 1, 2, . . .. The “unbalance” between the pressure exerted by the allowed modes within the plates and the one exerted by the modes outside them is at the origin of the Casimir effect lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 Two parallel, perfectly conducting and uncharged metallic plates in vacuum attract each other due to the quantum fluctuations of the electromagnetic fields, S >> L Boundary conditions on the components of electromagnetic fields EII , B⊥ = 0 The fluctuation modes of the fields in the space within the two plates can only have a specific set of L-dependent allowed wave-vectors: k⊥ = π n/L with n = 1, 2, . . .. The “unbalance” between the pressure exerted by the allowed modes within the plates and the one exerted by the modes outside them is at the origin of the Casimir effect This statement can be made quantitative by calculating the size-dependent energy E(L) of the electromagnetic fields in the vacuum within the plates lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 The size-dependent energy S >> L lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 The size-dependent energy S >> L lundi 2 novembre 2015 In the vacuum this quantity is divergent and meaningless. Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 The size-dependent energy S >> L In the vacuum this quantity is divergent and meaningless. What is observable is the change in the zero-point energy when matter is introduced In this way we can calculate the Casimir forces. lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 The size-dependent energy S >> The expansion in decreasing powers of L takes the form : L lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 The size-dependent energy S >> The expansion in decreasing powers of L takes the form : L is the energy associated to the electromagnetic field of the vacuum in the absence of the plates within the volume of space V = SL lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 The size-dependent energy S >> The expansion in decreasing powers of L takes the form : L is the energy associated to the electromagnetic field of the vacuum in the absence of the plates within the volume of space V = SL is the sum of the energies associated to the introduction of each single plate in the vacuum lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 The size-dependent energy S >> The expansion in decreasing powers of L takes the form : L is the energy associated to the electromagnetic field of the vacuum in the absence of the plates within the volume of space V = SL This is the Casimir term which represents the interaction energy between the two plates, due to their simultaneous presence in space. is the sum of the energies associated to the introduction of each single plate in the vacuum lundi 2 novembre 2015 Electromagnetic Casimir and van der Waals force Casimir H B 1948 Proc. K. Ned. Akad. Wet. 51 793 S >> The expansion in decreasing powers of L takes the form : This is the Casimir term which represents the interaction energy between the two plates, due to their simultaneous presence in space. L is universal. and does not depend on the specific material of the plates, but only on geometrical properties (S and L) and on fundamental constants The higher order term in the expansion: is a material-dependent parameter which describes the deviations of the plates from the perfectly conducting behaviour Finite temperature corrections can be usually neglected lundi 2 novembre 2015 First experimental verification of the Casimir force The amplitude of the force : S=100X100 μm2 S �c −11 F � � 6 10 N 4 L and L= 1 μm It can be measured using : a torsion pendulum : S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997). an AFM lundi 2 novembre 2015 : U. Mohideen and Anushree Roy Phys. Rev. Lett. 81, 4549 (1998) ; Phys. Rev. D 60 ,111101 (1999) First experimental verification of the Casimir force Corrections for real measurements • Sphere-plane instead of plane-plane interactions 3 π R�c o Fc (d) = − 360 d3 • Finite conductivity of the plates p Fc (d) = o Fc (d) � 1+f For metals Plasma Frequency : ωp � 1016 rad/s and λp = 2πc/ωp � 100 nm lundi 2 novembre 2015 � λp d2π � �� � λp −1 f � 10 at d = 10 nm 2π d � � λp f � 0.7 at d = 100 nm 2π d First experimental verification of the Casimir force Corrections for real measurements • Sphere-plane instead of plane-plane interactions 3 π R�c o Fc (d) = − 360 d3 plates • Finite conductivity of the � � Fcp (d) • = Fco (d) 1+f � λp � 10−1 at d = 10 nm 2π d � � λp f � 0.7 at d = 100 nm 2π d �� f 2 The roughness of the surfaces with variance σr Fcr (d) = Fcp (d) • λp d2π � � 1+O � σr2 d2 �� Finite temperature corrections are negligible lundi 2 novembre 2015 � 3� 2 kB T d O g where g(T, d) � � 2 ∗ 10−4 nm−1 d �c First experimental verification of the Casimir force After that all these corrections are taken into account Phys. Rev. D 60 ,111101 (1999) lundi 2 novembre 2015 Other experiments on electromagnetic Casimir Bressi et al. PRL 88, 041804-1(2002) R. Decca et al. Phys. Rev. D, 75 :077101,2007. S. de Manet al., PRL 103, 040402 (2009) J. N. Munday et al, Nature 457, 170 (2009) attractive repulsive Rico F. Tabor et al., PRL 106, 064501 (2011) lundi 2 novembre 2015 Correction of finite conductivity and temperature J. Laurent et al. Phys. Rev. B 85, 035426 (2012) lundi 2 novembre 2015 Experimental verifications of the Casimir force lundi 2 novembre 2015 The critical Casimir effect Thirty years after the seminal paper by Casimir, M. E. Fisher and P. -G. de Gennes published a note : ”On the phenomena at the walls in a critical binary mixture” R. Acad. Sci. Paris,Ser. B 287, 20 (1978) in which it was shown that Casimir-like effects (i.e., fluctuation-induced forces) arise also in statistical physics when a medium in which fluctuations of a certain nature take place is spatially confined Let us consider a mixture of two fluids A and B of concentration CA and CB lundi 2 novembre 2015 The critical Casimir effect Phase Diagram The binary mixture of two fluids Homogenous phase at low temperature Increasing the temperature the liquid demixes into an A- and a B-rich solution and becomes inhomogeneous in the test tube where the two solutions are typically separated by an interface. The transition between the mixed and the demixed phases occurs at the solid first-order transition line. The lower point is the critical point and the transiion becomes second order lundi 2 novembre 2015 The critical Casimir effect Phase Diagram The relevant observable is the local concentration mean concentraction < cA (x) >= cA The order parameter of the transition can be identified with The correlation defines the spatial correlation length The correlation length is fixed by the fluctuations of the order parameter lundi 2 novembre 2015 The critical Casimir effect Phase Diagram local concentration mean < cA (x) >= cA order parameter The spatial correlation length what are the consequences of confinement ? • We intoduce a plate into de mixture • The plate shows preferential adsorption for one of the two components, say, A. • The presence of the plate in the mixture induces a local increase of < δcA (x) > The plate imposes boundary conditions on the order parameter lundi 2 novembre 2015 The critical Casimir effect Phase Diagram glass plate L<ξ what are the consequences of confinement ? • We intoduce a plate into de mixture • The plate shows preferential adsorption for one of the two components, say, A. • The presence of the plate in the mixture induces a local increase of < δcA (x) > The plate imposes boundary conditions on the order parameter lundi 2 novembre 2015 Analogies and differences between electromagnetic and critical Casimir effects. lundi 2 novembre 2015 Analogies and differences between electromagnetic and critical Casimir effects. glass plate bulk energy L lundi 2 novembre 2015 surface energy The wall interact only if L < ξ Casimir energy Analogies and differences between electromagnetic and critical Casimir effects. glass plate bulk energy surface energy Casimir energy The wall interact only if L < ξ L The scaling function ΘII depends only on some gross features of: (a) the bulk, e.g. symmetries of the interaction, kind of order parameter (b) the surfaces of the wall, e.g. the symmetries of the bulk which they break (c) The shape of the walls (a) and (b) define the bulk and surface universality classes of the confined system lundi 2 novembre 2015 Analogies and differences between electromagnetic and critical Casimir effects. glass plate bulk energy L surface energy Casimir energy The wall interact only if L < ξ The scaling function ΘII depends only on the features of the universality classes of the confined system and on the confining geometry The scaling function can be computed by means of suitable representative models belonging to the same bulk and surface universality classes of the system lundi 2 novembre 2015 Analogies and differences between electromagnetic and critical Casimir effects. glass plate L The critical Casimir Force The amplitude of the force : S=100X100 μm2 , T=300K and Critical Electromagnetic lundi 2 novembre 2015 L= 1 μm F � S kB T /L3 � 4 10−11 N F � S � c/L4 � 6 10−11N Experiments on critical Casimir The first experimental tests have been performed around 2000 (1) Thin wetting films close to critical point in binary mixtures and in 4He (2) lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture Experiments on critical Casimir The first experimental tests have been performed around 2000 (1) Thin wetting films close to critical point in binary mixtures and in 4He liquid-gas phase transition vapour condensation liquid film formation on a substrate L diverges at Po The liquid is confined between the solid substrate and the liquid-vapour interface L is determined by the van der Waals interactions Close to the end point the Casimir force adds up to the van-der-Waals forces in determining the equilibrium distance L lundi 2 novembre 2015 Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) Two different critical binary liquid mixtures have been used methanol + hexane (MH) and 2-methoxy-ethanol + methylcyclohexane (MM), onto the Si wafer lundi 2 novembre 2015 Experimental details Liquid mixtures properties methanol + hexane (MH) and 2-methoxy-ethanol + methylcyclohexane (MM), Upper critical point (MH) correlation length versus temperature ν � 0.6 which is the correct 3D Ising model exponent lundi 2 novembre 2015 Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) Two different critical binary liquid mixtures have been used methanol + hexane (MH) and 2-methoxy-ethanol + methylcyclohexane (MM), onto the Si wafer lundi 2 novembre 2015 Experimental details Measure of the film thickness Phase-modulated ellipsometer o Incident angle near the Brewster angle � 76 Ellipticity ρ = [Im(rp /rs )]θB where rp and rs are the complex reflection amplitudes for p (in plane) and s (out of plane) polarizations rp � 0 at θB extremely good film thickness resolution of 0.02nm averaged over the focused laser beam diameter of 0.25mm lundi 2 novembre 2015 Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) Two different critical binary liquid mixtures have been used methanol + hexane (MH) and 2-methoxy-ethanol + methylcyclohexane (MM), onto the Si wafer lundi 2 novembre 2015 Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) L* lundi 2 novembre 2015 Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) Free Energy per unit area substrate liquid lundi 2 novembre 2015 liquid vapor gravity non critical critical Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) Free Energy per unit area substrate liquid liquid vapor gravity Van der Waals interaction between the substrate and the vapor phase for sufficiently thin films (L < 10–20 nm) in the nonretarded regime W is the Hamaker constant lundi 2 novembre 2015 non critical critical Repulsive structural interaction due to the presence of the solid wall δ is of the order of the molecular diameter Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) Free Energy per unit area substrate liquid liquid vapor Critical ωc lundi 2 novembre 2015 � L ξ± gravity � = kB T θ non critical +,− L2 � critical L ξ± � Wetting film in binary mixtures substrate liquid liquid vapor Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) non critical gravity ωc The minimum of F gives: � L ξ± � critical = kB T θ+,− L2 � L ξ± � At T far from Tc θ+,− (L/ξ) � 0 this equation can be used for calibration For small H , L is large : lundi 2 novembre 2015 At large H , L ≈ L* gravity is negligible the only free parameter is now δ which is determined by the crossover Wetting film in binary mixtures lundi 2 novembre 2015 Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) Wetting film in binary mixtures Ashis Mukhopadhyay and Bruce M. Law PRL 82, 772 (1999) and PRE 62, 5201 (2000) y = L/ξ ν � 0.6 which is the correct 3D Ising model exponent lundi 2 novembre 2015 Wetting film in binary mixtures lundi 2 novembre 2015 Experiments in He PRL 83, 1187 (1999) lundi 2 novembre 2015 Experiments in He h CF liquid films G CV vapor d liquid film 1 1 1 = + C CV CF lundi 2 novembre 2015 CF Experiments in He Effective dielectric constant for capacitor 1 as a function of temperature. Inset (a) shows the film thickness calculated using Eq. (3), where a dip is found 2.6 mK below Tλ lundi 2 novembre 2015 Experiments in He t = (T − Tλ )/Tλ ν = 0.66 is the exponent of the XY model for 4 He film on Cu, γ0 � 2.6 K nm3 and d1/2 � 20nm lundi 2 novembre 2015 Experiments in He comparison with theory The data points are obtained by Monte Carlo simulation of the XY model on the lattice (Vasilyev O, Gambassi A, Maciolek A, and Dietrich S 2007 EPL 80 60009) The solid red line represents the experimental results (Garcia R and Chan M H W 1999 Phys. Rev. Lett. 83, 1187) lundi 2 novembre 2015 Experiments on critical Casimir The first experimental tests have been performed around 2000 (1) Thin wetting films close to critical point in binary mixtures and in 4He (2) lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture Direct measure on a collidal bead inside a binary mixture Hertlein C, Helden L, Gambassi A, Dietrich S and Bechinger C 2008 Nature 451 172 Bulk phase diagram of the binary liquid mixture of water and 2,6-lutidine (dimethylpyridine C7H9N) lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture Hertlein C, Helden L, Gambassi A, Dietrich S and Bechinger C 2008 Nature 451 172 R � 1µm lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture � � Φ(z) P (z) = Po exp − kB T Φ(z) = −kB T log(P (Z)/Po ) lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture Hertlein C, Helden L, Gambassi A, Dietrich S and Bechinger C 2008 Nature 451 172 lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture Hertlein C, Helden L, Gambassi A, Dietrich S and Bechinger C 2008 Nature 451 172 electrostatic interaction lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture Hertlein C, Helden L, Gambassi A, Dietrich S and Bechinger C 2008 Nature 451 172 gravity electrostatic interaction lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture Surface treatement of the cell: 1) rinsed by NaOH : hydrophilic (-) 2) Hexamethyldisilizane vapors : hyfrophobic (+) Two kind of polystyrene beads : hydrophilic (-) and hyfrophobic (+) (-)(-) L kB T Φ(z) = A exp(−kz) + S 2 ϑ|| ( ) L ξ lundi 2 novembre 2015 (-)(+) Direct measure on a collidal bead inside a binary mixture Surface treatement of the cell: 1) rinsed by NaOH : hydrophilic (-) 2) Hexamethyldisilizane vapors : hyfrophobic (+) Two kind of polystyrene beads : hydrophilic (-) and hyfrophobic (+) (-)(-) (-)(+) Theoretical curves are obtained by Monte Carlo simulation of 3D Ising model which belongs to the same universality class of binary mixtures. O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, Phys.Rev. E 79, 041142 (2009) Phys. Rev. E 80, 039902 (2009). lundi 2 novembre 2015 Direct measure on a collidal bead inside a binary mixture (-)(-) (+)(+) lundi 2 novembre 2015 (-)(+) Direct measure on a collidal bead inside a binary mixture (+)(+) Off-critical composition. Lutidine mass fraction 0.2 Temperatures close to the demixing line. lundi 2 novembre 2015 Casimir Force in action Temperature tuned force Patterned surfaces lundi 2 novembre 2015 Torque on ellipsoidal bead Horizontal force on a bead Casimir force in action Patterned surfaces F. Soyka et al., PRL 101, 208301 (2008) lundi 2 novembre 2015 Conclusions • We have discussed several experimental problems related to the measure of the electromagnetic and critical Casimir • Concerning the critical Casimir. We have shown the importance that it has in films and colloids close to the critical points. • The direct measure has been performed in a few experiment and always mediated by the mesure of the potential. • Casimir forces is a temperature driven force. It can be useful to tune MEMS, to tune interactions between colloids. Future developments • • • • What are the fluctuations of the Casimir force ? What about the transient Casimir effect ? The role of viscosity The energetics lundi 2 novembre 2015 Out of Equilibrium fluctuations in confined phase transitions Experiments: Bellon Ludovic, Crauste Caroline, Anne Le Cunuder, Ignacio Martinez Devailly Clémence, Laurent Justine, Petrosyan Artyom, Theory: Holdsworth Peter, Francesco Puosi David Lopez lundi 2 novembre 2015 Experimental set-up Optical trap - Camera rapide AOD 75 MHz lundi 2 novembre 2015 Experimental set-up Optical trap silica bead laser beam - Camera rapide AOD 75 MHz lundi 2 novembre 2015 k 2 U (x) = x 2 Examples of traps lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 ξo � 3nm Distance between particles ∆x Probability density function of the particle distance 60 lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 Fit of the PDF with the Casimir Potential ∆X = 2R + ∆Xs and ∆Xo = 2R + ∆Xs Fitting parameters R=0.8mm 61 lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 ξo � 3nm Casimir induced synchronisation Non synchronised T<Tc 62 lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 ξo � 3nm Temperature dependent synchronisation Non synchronised T<Tc Synchronised T≅Tc 63 lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 ξo � 3nm T<Tc Temperature dependent synchronisation Non synchronised T<Tc Synchronised T≅Tc T≅Tc 64 lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 ξo � 3nm T<Tc Temperature dependent synchronisation Non synchronised T<Tc Probability of not being synchronized Synchronised T≅Tc T≅Tc Probability of being synchronisized 65 lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 Measure of Casimir forces and potentials ξo � 3nm Temperature dependent synchronisation Non synchronised T<Tc Synchronised T≅Tc 66 lundi 2 novembre 2015 Two silica beads trapped by two laser beams in a critical mixture of water/C12E5 The work performed by the trap �τ T Wα = − 0 k(α − x2 (t)) ◦ dt where α stands for either ϕ or r and τ is the ramp duration < Wϕ > < Wr > 67 lundi 2 novembre 2015 Conclusions • We have discussed several experimental problems related to the measure of the electromagnetic and critical Casimir • Concerning the critical Casimir. We have shown the importance that it has in films and colloids close to the critical points. • The direct measure has been performed in a few experiment and always mediated by the mesure of the potential. • Casimir forces is a temperature driven force. It can be useful to tune MEMS, to tune interactions between colloids. Future developments • • • What are the fluctuations of the Casimir force ? What about the transient Casimir effect ? The energetics lundi 2 novembre 2015 69 lundi 2 novembre 2015