Shear viscosity and the nucleation of antikaon condensed matter in protoneutron stars
aa r X i v : . [ a s t r o - ph . H E ] D ec Shear viscosity and the nucleation of antikaon condensed matterin protoneutron stars
Sarmistha Banik, Rana Nandi and Debades Bandyopadhyay
Astroparticle Physics and Cosmology Division,Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700064, India
Abstract
We study shear viscosities of different species in hot and neutrino-trapped dense matter relevantto protoneutron stars. It is found that the shear viscosities of neutrons, protons and electrons inneutrino-trapped matter are of the same orders of magnitude as the corresponding shear viscositiesin neutrino-free matter. Above all, the shear viscosity due to neutrinos is higher by several ordersof magnitude than that of other species in neutrino-trapped matter.Next we investigate the effect of shear viscosity in particular, neutrino shear viscosity on thethermal nucleation rate of droplets of antikaon condensed matter in protoneutron stars. Thefirst-order phase transition from hadronic matter to antikaon condensed matter is driven by thethermal nucleation process. We compute the equation of state used for the calculation of shearviscosity and thermal nucleation time within the relativistic mean field model. Neutrino shearviscosity enhances the prefactor in the nucleation rate by several orders of magnitude comparedwith the T approximation of earlier calculations. Consequently the thermal nucleation time in the T approximation overestimates our result. Furthermore, the thermal nucleation of an antikaondroplet might be possible in neutrino-trapped matter before neutrino diffusion takes place. PACS numbers: 97.60.Jd, 26.60.-c,52.25.Fi,64.60.Q- . INTRODUCTION A first order phase transition from nuclear matter to some exotic form of matter might bepossible in (proto)neutron stars. It could be either a nuclear to quark matter transition ora first order pion/kaon condensation. Consequently, it might have tremendous implicationsfor compact stars [1] and supernova explosions [2]. Here the focus is the first order phasetransition proceeding through the thermal nucleation of a new phase in particular, antikaoncondensed phase in hot and neutrino-trapped matter. After the pioneering work by Kaplanand Nelson on antikaon ( K − meson) condensation in dense baryonic matter formed in heavyion collisions as well as in neutron stars [3], several groups pursued the problem of antikaoncondensation in (proto)neutron stars [4–18]. In most cases, the phase transition was studiedusing either Maxwell construction or Gibbs’ rules for phase equilibrium coupled with globalbaryon number and charge conservation [19]. The first order phase transition driven bynucleation of antikaon condensed phase was considered in a few cases [20, 21]. In particular,the calculation of Ref.[21] dealt with the role of shear viscosity on the the thermal nucleationof antikaon condensed phase in hot and neutrino-free compact stars [21]. It is to be notedhere that the first order phase transition through the thermal nucleation of quark matterdroplets was also investigated in (proto)neutron stars [20, 22–27] using the homogeneousnucleation theory of Langer [20, 22, 28]. The thermal nucleation is an efficient process thanthe quantum nucleation at high temperatures [25, 27].We adopt the homogeneous nucleation theory of Langer [28, 29] for the thermal nucle-ation of antikaon condensed phase. Nuclear matter would be metastable near the phasetransition point due to sudden change in state variables. In this case thermal and quantumfluctuations are important. Droplets of antikaon condensed matter are formed because ofthermal fluctuations in the metastable nuclear matter. Droplets of the new and stable phasewhich are bigger than a critical radius, will survive and grow. The transportation of latentheat from the surface of the droplet into the metastable phase favours a critical size dropletto grow further. This heat transportation could be achieved through the thermal dissipationand viscous damping [29–31].A parametrised form of the shear viscosity was used in earlier calculations of the nu-cleation of quark matter [25]. Recently, the influence of thermal conductivity and shearviscosity on the thermal nucleation time was studied in a first-order phase transition from2uclear to antikaon condensed matter in hot neutron stars [21]. The shear viscosity dueto neutrinos was not considered in that calculation. It would be worth studying the effectof shear viscosity on the thermal nucleation rate of droplets of antikaon condensed matterin neutrino-trapped matter relevant for protoneutron stars. Besides shear viscosities dueto neutrons, protons and electrons, this involves the contribution of neutrinos to the totalshear viscosity. Shear viscosities of pure neutron and neutron star matter were calculatedby several groups [32–37]. We also performed the calculation of shear viscosity in neutronstar matter using the equation of state (EoS) derived from relativistic field theoretical mod-els [38]. Transport properties of degenerate neutrinos in dense matter were estimated byGoodwin and Pethick [39].We organise the paper in the following way. We describe models for EoS, shear viscositiesof different species including neutrinos and the calculation of thermal nucleation rate in Sec.II. Results of this calculation are discussed in Sec. III. Sec. IV gives the summary andconclusions. II. FORMALISM
The knowledge of the EoS for nuclear as well as antikaon condensed phases is essentialfor the computation of shear viscosity and thermal nucleation rate. We consider a first-order phase transition from the charge neutral and β -equilibrated nuclear matter to K − condensed matter in a protoneutron star. Those two phases are composed of neutrons,protons, electrons, electron type neutrinos and of K − mesons only in the antikaon condensedphase. Both phases are governed by baryon number conservation and charge neutralityconditions [19]. Relativistic field theoretical models are used to describe the EoS in nuclearand antikaon condensed phases. The baryon-baryon interaction is mediated by the exchangeof σ , ω and ρ and given by the Lagrangian density [11, 13, 40–42] L N = X B = n,p ¯ ψ B ( iγ µ ∂ µ − m B + g σB σ − g ωB γ µ ω µ − g ρB γ µ t B · ρ µ ) ψ B + 12 (cid:0) ∂ µ σ∂ µ σ − m σ σ (cid:1) − U ( σ ) − ω µν ω µν + 12 m ω ω µ ω µ − ρ µν · ρ µν + 12 m ρ ρ µ · ρ µ . (1)3he scalar self-interaction term [11, 42, 43] is U ( σ ) = 13 g m N ( g σN σ ) + 14 g ( g σN σ ) , (2)The effective nucleon mass is given by m ∗ B = m B − g σB σ , where m B is the vacuum baryonmass. The Lagrangian density for (anti)kaons in the minimal coupling scheme is given by[9, 12–14] L K = D ∗ µ ¯ KD µ K − m ∗ K ¯ KK , (3)where the covariant derivative is D µ = ∂ µ + ig ωK ω µ + ig ρK t K · ρ µ and the effective mass of(anti)kaons is m ∗ K = m K − g σK σ . For s-wave ( p = 0) condensation, the in-medium energyof K − mesons is given by ω K − = m ∗ K − (cid:18) g ωK ω + 12 g ρK ρ (cid:19) . (4)The condensation in neutrino-trapped matter sets in when the chemical potential of K − mesons ( µ K − = ω K − ) is equal to the difference between the electron chemical potential ( µ e )and neutrino chemical potential ( µ ν e ) i.e. µ K − = µ e − µ ν e . The critical droplet of antikaoncondensed matter is in total phase equilibrium with the metastable nuclear matter. Themixed phase is governed by Gibbs’ phase rules along with global baryon number conservationand charge neutrality [19].We adopt the bulk approximation [22] which does not consider the variation of the mesonfields with position inside the droplet. We solve equations of motion self-consistently in themean field approximation [40] and find effective masses and Fermi momenta of baryons.Pressures in neutrino-trapped nuclear matter ( P N ) and antikaon condensed matter ( P K )are given by Ref.[13]. Here we calculate zero temperature equations of state because it wasnoted earlier that the temperature of a few tens of MeV did not modify the EoS considerably[21].Next we discuss the calculation of shear viscosity in protoneutron star matter. It wasnoted in earlier calculations that the main contributions to the total shear viscosity inneutron star matter came from electrons, the lightest charged particles, and neutrons, themost abundant particles [32–34, 36, 37]. Neutrinos are trapped in protoneutron stars andtheir contribution might be significant in transport coefficients such as shear viscosity. Inprinciple, we may calculate shear viscosities for different particle species ( n , p , e and ν e )in neutrino-trapped matter using coupled Boltzmann transport equations [33, 36, 38]. We4an immediately write a set of relations between effective relaxation times ( τ ) and collisionfrequencies ( ν ( ′ ) ij ) following the Ref.[38] X i,j = n,p,e,ν e ( ν ij τ i + ν ′ ij τ j ) = 1 , (5)which can be cast into a 4 × η ν = 15 n ν p ν cτ ν " π
12 + λ η X k = odd k + 1) k ( k + 1) [ k ( k + 1) − λ η ] . (6)For neutrino shear viscosity, we only consider scattering processes involving neutrinos andother species. Various quantities in Eq.(6) are explained in Ref.[39] and given below. Theneutrino relaxation time ( τ ν ) is, τ − ν = X i = n,p,e τ − νi , , (7) τ − νi = E F i ( k B T ) π < I i > , (8)and λ η is defined as λ η = τ ν X i λ iη τ − νi (9) λ iη = R d Ω d Ω d Ω [3( ˆ p · ˆ p ) − h| M | i i δ ( p + p − p − p ) R d Ω d Ω d Ω h| M | i i δ ( p + p − p − p ) (10)= 1 − p ν I i < I i > + 38 p ν I i < I i > , (11)5 I i > = 8 G F πp ν " C V i C A i (cid:18) p ν E i (cid:19) + ( C V i + C A i ) ( (cid:18) p i E i (cid:19) + 25 (cid:18) p ν E i (cid:19) ) − (cid:18) m i E i (cid:19) ( C V i − C A i ) (12) I i = 32 G F πp ν " C V i C A i (cid:18) p ν E i (cid:19) + ( C V i + C A i ) ( (cid:18) p i E i (cid:19) + 127 (cid:18) p ν E i (cid:19) ) − (cid:18) m i E i (cid:19) ( C V i − C A i ) (13) I i = 128 G F πp ν " C V i C A i (cid:18) p ν E i (cid:19) + ( C V i + C A i ) ( (cid:18) p i E i (cid:19) + 103 (cid:18) p ν E i (cid:19) ) − (cid:18) m i E i (cid:19) ( C V i − C A i ) . (14) Here < | M | > is the squared matrix element summed over final spins and averaged overinitial spins for a scattering process and C V and C A are vector and axial vector couplingconstants.For non-relativistic nucleons ( m i /E i ) ≃
1, ( p i /E i ) ≪ p ν /E i ) ≪ λ iη reduces to [39] λ iη = C V i + C A i C V i + 2 C A i (15)However, we do not assume non-relativistic approximation in this calculation. The totalshear viscosity is given by η total = η n + η p + η e + η ν , (16)where η i (= n,p,e ) = n i p F i τ i m ∗ i . (17)The relaxation time of i-th species ( τ i ) is calculated using Ref. [36, 38]. Effective massand Fermi momentum of i-th particle species are denoted by m ∗ i and p F i , respectively. Theelectron effective mass is taken as its chemical potentials due to relativistic effects.We are interested in a first order phase transition driven by the nucleation of dropletsof antikaon condensed phase in the neutrino-trapped nuclear matter. Droplets of antikaoncondensed phase are born in the metastable nuclear matter due to thermal fluctuations.Droplets of antikaon condensed matter above a critical size ( R c ) will grow and drive thephase transition. According to the homogeneous nucleation formalism of Langer and others,6he thermal nucleation per unit time per unit volume is given by [28, 29]Γ = Γ exp (cid:18) − △ F ( R c ) T (cid:19) , (18)where △ F is the free energy cost to produce a droplet with a critical size in the metastablenuclear matter. The free energy shift of the system as a result of the formation of a dropletis given by [24, 26] △ F ( R ) = − π P K − P N ) R + 4 πσR , (19)where R is the radius of the droplet, σ is surface tension of the interface separating twophases and P N and P K are the pressure in neutrino-trapped nuclear and antikaon condensedphases, respectively as discussed above. We obtain the critical radius of the droplet fromthe maximum of △ F ( R ) i.e. δ R △ F = 0, R C = 2 σ ( P K − P N ) . (20)This relation also demonstrates the mechanical equilibrium between two phases.We write the prefactor in Eq. (18) as the product of two parts - statistical and dynamicalprefactors [29–31] Γ = κ π Ω . (21)The available phase space around the saddle point at R C during the passage of the dropletthrough it is given by the statistical prefactor (Ω ),Ω = 23 √ (cid:16) σT (cid:17) / (cid:18) R C ξ (cid:19) , (22)Here ξ is the kaon correlation length which is considered to be the width of the interfacebetween nuclear and antikaon condensed matter. The dynamical prefactor κ is responsiblefor the initial exponential growth rate of a critical droplet and given by [30, 31] κ = 2 σR C ( △ w ) (cid:20) λT + 2( 43 η + ζ ) (cid:21) . (23)Here △ w = w K − w N is the enthalpy difference between two phases, λ is the thermalconductivity and η and ζ are the shear and bulk viscosities of neutrino-trapped nuclearmatter. We neglect the contribution of thermal conductivity because it is smaller comparedwith that of shear viscosity [21]. We also do not consider the contribution of bulk viscosityin the prefactor in this calculation. 7e can now calculate the thermal nucleation time ( τ nuc ) in the interior of neutron starsas τ nuc = ( V Γ) − , (24)where the volume V = 4 π/ R nuc . We assume that pressure and temperature are constantwithin this volume in the core. III. RESULTS AND DISCUSSION
Nucleon-meson coupling constants in this calculation are taken from Glendenning andMoszkowski parameter set known as GM1 [48] and those are obtained by reproducing thesaturation properties of nuclear matter such as binding energy
E/B = − . n = 0 .
153 fm − , asymmetry energy coefficient a asy = 32 . K = 300 MeV and effective nucleon mass m ∗ N /m N = 0 . K − optical potential depth at normal nuclear matter density. The strengthof antikaon optical potential depth is obtained from heavy ion collision experiments and K − atomic data [49–54]. It has a wide range of values. On the one hand, the analysisof K − atomic data predicted the real part of the antikaon optical potential to be as largeas U ¯ K ( n ) = − ±
20 MeV at normal nuclear matter density [49, 50]. On the otherhand, theoretical models including chirally motivated coupled channel models as well as adouble pole structure of Λ(1405) yielded a less attractive antikaon optical potential depth[55–58]. Here we consider an antikaon optical potential depth of U ¯ K ( n ) = −
120 MeV atnormal nuclear matter density and the corresponding kaon-scalar meson coupling constantis g σK = 1 . M ⊙ in earlier calculations using the Maxwellconstruction [12]. This is consistent with the recently observed 2 M ⊙ neutron star [59].Using the above mentioned nucleon-meson and kaon-meson coupling constants, we calcu-late the EoS of neutrino-trapped nuclear and antikaon condensed phases at zero temperaturein a self-consistent manner. A temperature of a few tens of MeV might modify the EoS atvery high densities in (proto)neutron stars compared with the zero temperature EoS as itwas noted in Ref.[17]. However, the effect of finite temperature on the threshold of an-8ikaon condensation is negligible [17]. In this calculation, the EoS enters in Eq. (19) asthe difference between pressures in two phases and in Eq. (23) as the enthalpy differencebetween two phases. Here we exploit the zero temperature EoS for the the calculation ofshear viscosity and thermal nucleation time. The thermal nucleation of exotic phases wasearlier investigated using zero temperature EoS in Ref.[21, 26].First we calculate shear viscosities of neutrons, protons and electrons in neutrino-trappednuclear matter using Eq. (5) in the same fashion as it was done in Ref.[21]. We take leptonfraction Y L = 0 . T = 1, 10, 30 and 100MeV in Fig. 2. The shear viscosity is found to increase with baryon density. Furthermore,the shear viscosity decreases with increasing temperature. The temperature dependence ofshear viscosities is a complex one and quite different from the characteristic 1 /T behaviourof a Fermi liquid [36]. It is observed that the shear viscosities of neutrons, protons andelectrons in neutrino-trapped nuclear matter are of the same orders of magnitude as thoseof the neutrino-free case [38].Next we calculate the shear viscosity due to neutrinos. As a prelude to it, we compare ef-fective relaxation times corresponding to different species in neutrino-trapped nuclear matterin Fig. 3. Relaxation time is plotted with normalised baryon density at a temperature T=10MeV in Fig. 3. Relaxation times of different particle species due to scattering under strongand electromagnetic interactions are much much smaller than that of neutrinos undergoingscattering with other species through weak interactions. Consequently, particles excludingneutrinos come into thermal equilibrium quickly on the time scale of weak interactions. Wecalculate the shear viscosity due to neutrinos only treating others as background and it isshown as a function of normalised baryon density for temperatures T = 1, 10, 30 and 100in Fig. 4. Like Fig. 2, the neutrino shear viscosity decreases with increasing temperature.However, the neutrino shear viscosity is several orders of magnitude larger than shear vis-cosities of neutrons, protons and electrons shown in Fig. 2. It is the neutrino shear viscositywhich dominates the total viscosity of Eq. (16) in neutrino-trapped matter. We performthe rest of our calculation using the neutrino viscosity in the following paragraphs.9e calculate the prefactor (Γ ) according to Eqs. (21)-(23). The dynamical prefactor notonly depends on the shear viscosity but also on the thermal conductivity and bulk viscosity.However, it was already noted that the thermal conductivity and bulk viscosity in neutrino-trapped nuclear matter were negligible compared with the shear viscosity [39]. We onlyconsider the effect of shear viscosity on the prefactor. Besides transport coefficients, theprefactor in particular, the statistical prefactor is sensitive to the correlation length of kaonsand surface tension. The correlation length is the thickness of the interface between nuclearand kaon phases [25, 30] having a value ∼ ξ ) for kaons [22, 30]. We perform our calculationwith antikaon droplets with radii greater than 5 fm. The other important parameter inthe prefactor is the surface tension. The surface tension between nuclear and kaon phaseswas already estimated by Christiansen and collaborators [60] and found to be sensitive tothe EoS. We perform this calculation for a set of values of surface tension σ = 15, 20, 25and 30 MeV fm − . The prefactor (Γ ) is shown as a function of temperature in Fig. 5. Itis shown for a baryon density n b = 4 . n which is just above the critical density 3.9 n for antikaon condensation at zero temperature [13], and surface tension σ = 15 MeV fm − .The prefactor was also approximated by T according to the dimensional analysis in manycalculations [26, 30]. The upper curve in Fig. 5 shows the prefactor of Eq. (21) includingonly the contribution of neutrino shear viscosity whereas the prefactor approximated by T corresponds to the lower curve. It is evident from Figure 5 that the approximated prefactoris very small compared with our result.Now we discuss the nucleation time of a critical droplet of antikaon condensed phase inneutrino-trapped nuclear matter and the effect of neutrino shear viscosity on it. The thermalnucleation rate of the critical droplet is calculated within a volume with R nuc = 100 m inthe core of a neutron star where the density, pressure and temperature are constant. Thethermal nucleation time is plotted with temperature for a baryon density n b = 4 . n inFig. 6. Furthermore, this calculation is done with the kaon correlation length ξ = 5 fm andsurface tension σ = 15, 20, 25 and 30 MeV fm − . The size of the critical droplet increaseswith increasing surface tension. Radii of the critical droplets are 7.1, 9.4, 11.7 and 14.1fm corresponding to σ = 15, 20, 25 and 30 MeV fm − , respectively, at a baryon density4 . n . The nucleation time of the critical droplet diminishes as temperature increases forall cases studied here. However, the temperature corresponding to a particular nucleation10ime for example 10 − s, increases as the surface tension increases. There is a possibility thatthe condensate might melt down if the temperature is higher than the critical temperature.So far, there is no calculation of critical temperature of antikaon condensation in neutrino-trapped matter. However, the critical temperature of antikaon condensation was investigatedin neutrino-free matter in Ref.[18]. We compare thermal nucleation times corresponding todifferent values of the surface tension with the early post bounce time scale t d ∼
100 ms inthe core collapse supernova [2] when the central density might reach the threshold densityof antikaon condensation. The time scale t d is much less than the neutrino diffusion time ∼ t d . For σ = 15 MeV fm − , the thermalnucleation time of 10 − s occurs at a temperature 16 MeV. It is evident from Fig. 6 thatthe thermal nucleation time is strongly dependent on the surface tension. Further thermalnucleation of an antikaon droplet is possible so long as the condensate might survive the meltdown at high temperatures[18]. Our results of thermal nucleation time are compared withthe calculation taking into account the prefactor approximated by T in Fig. 7 for surfacetension σ = 15 MeV fm − and at a density n b = 4 . n . The upper curve denotes thecalculation with T approximation whereas the lower curve corresponds to the influence ofneutrino shear viscosity on the thermal nucleation time. The results of the T approximationoverestimate our results hugely. For a nucleation time of 10 − s at a temperature T=16 Mev,the corresponding time in the T approximation is larger by several orders of magnitude. IV. SUMMARY AND CONCLUSIONS
We have studied shear viscosities of different particle species in neutrino-trapped β -equilibrated and charge neutral nuclear matter. We have derived equations of state ofnuclear and antikaon condensed phases in the relativistic mean field model for the calcula-tion of shear viscosity. It is noted that neutrons, protons and electrons come into thermalequilibrium in the weak interaction time scale. The shear viscosity due to neutrinos iscalculated treating other particles as background and found to dominate the total shearviscosity.Next we have investigated the first-order phase transition from neutrino-trapped nuclearmatter to antikaon condensed matter through the thermal nucleation of a critical droplet of11ntikaon condensed matter using the same relativistic EoS as discussed above. Our emphasisin this calculation is the role of the shear viscosity due to neutrinos in the prefactor and itsconsequences on the thermal nucleation rate. We have observed that the thermal nucleationof a critical antikaon droplet might be possible well before the neutrino diffusion takes place.Furthermore, a comparison of our results with that of the calculation of thermal nucleationtime in the T approximation shows that the latter overestimates our results of thermalnucleation time computed with the prefactor including the neutrino shear viscosity. Thoughwe have performed this calculation with antikaon optical potential depth of U ¯ K ( n ) = − [1] N. K. Glendenning, Compact Stars, (Springer-Verlag, New York, 1997).[2] I. Sagert et al., Phys. Rev. Lett. 102 (2009) 081101.[3] D.B. Kaplan and A.E. Nelson, Phys. Lett. B , 57 (1986);A.E. Nelson and D.B. Kaplan, Phys. Lett. B , 193 (1987).[4] G.E. Brown, K. Kubodera, M. Rho and V. Thorsson, Phys. Lett. B , 355 (1992).[5] V. Thorsson, M. Prakash and J.M. Lattimer, Nucl. Phys. A572 , 693 (1994).[6] P.J. Ellis, R. Knorren and M. Prakash, Phys. Lett. B , 11 (1995).[7] C.-H. Lee, G.E. Brown, D.-P. Min and M. Rho, Nucl. Phys. A585 (1995) 401.[8] M. Prakash, I. Bombaci, M. Prakash, P.J. Ellis, J.M. Lattimer and R. Knorren, Phys. Rep. , 1 (1997).[9] N.K. Glendenning and J. Schaffner-Bielich, Phys. Rev. C , 025803 (1999).[10] R. Knorren, M. Prakash and P.J. Ellis, Phys. Rev. C , 3470 (1995).[11] J. Schaffner and I.N. Mishustin, Phys. Rev. C , 1416 (1996).[12] S. Pal, D. Bandyopadhyay and W. Greiner, Nucl. Phys. A674 , 553 (2000).[13] S. Banik and D. Bandyopadhyay, Phys. Rev. C , 035802 (2001).[14] S. Banik and D. Bandyopadhyay, Phys. Rev. C , 055805 (2001).[15] S. Banik and D. Bandyopadhyay, Phys. Rev. C , 065801 (2002).[16] S. Banik and D. Bandyopadhyay, Phys. Rev. D , 123003 (2003).[17] J.A. Pons, S. Reddy, P.J. Ellis, M.Prakash, and J.M. Lattimer, Phys. Rev. C
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10 12 14 16 180.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 l og η ( g c m - s - ) n b /n Y L =0.4 T=1 MeVT=10 MeVT=30 MeVT=100 MeV
FIG. 2. Total shear viscosity in neutrino-trapped nuclear matter except the contribution ofneutrinos is plotted with normalised baryon density for different temperatures. l og τ ( s ) n b /n Y L =0.4T=10 MeV τ p τ n τ e τ ν Fig. 3. Relaxation times corresponding to different species in neutrino-trapped nuclear matter areshown as a function of normalised baryon density at a temperature T = 10 MeV and Y L = 0 .
20 22 24 26 28 300.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 l og η ν ( g c m - s - ) n b /n Y L =0.4 T=1 MeVT=10 MeVT=30 MeVT=100 MeV
Fig. 4. Neutrino shear viscosity is shown as a function of normalised baryon density at differenttemperatures.
10 15 20Temperature (MeV)10 -9 -6 -3 Γ ( c f m -4 ) n b =4.235n T Y L =0.4 σ=15 MeV/fm FIG. 5. Prefactor including the contribution of shear viscosity is plotted as a function of temper-ature at a fixed baryon density and surface tension and compared with that of T approximation. -3 -2 -1 l og τ nu c ( s ) σ=15 MeV/fm σ=20 MeV/fm σ=25 MeV/fm σ=30 MeV/fm U K =-120 MeVn b =4.235 n Y L =0.4 FIG. 6. Thermal nucleation time is displayed with temperature for different values of surfacetension. -3 -2 -1 l og τ nu c ( s ) T σ=15 MeV/fm U K =-120 MeVn b =4.235 n Y L =0.4 FIG. 7. Same as Fig.6 but our results for a fixed surface tension are compared with the calculationof T approximation.approximation.