Laser-Induced Liquid-Vapor Phase Transitions in Thin Films at Solid Surfaces : Nanomechanical Investigations on the Sub-Nanosecond Scale

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Laser-Induced Liquid-Vapor Phase Transitions in Thin Films

at Solid Surfaces

Nanomechanical Investigations on the Sub-Nanosecond Scale

Dissertation

zur Erlangung des akademischen Grades

des Doktors der Naturwissenschaften (Dr. rer. nat.) an der Universit¨at Konstanz,

Mathematisch-Naturwissenschaftliche Sektion, Fachbereich Physik

vorgelegt von Florian Lang

Tag der m¨ undlichen Pr¨ ufung: 19.10.2007

Referent: Prof. Dr. Paul Leiderer

Referent: Prof. Dr. Thomas Dekorsy

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1 Introduction

Liquid-vapor phase transitions at rapidly heated solid surfaces are of fundamental importance for a wide range of applications. In microfluidic systems liquid flow is frequently controlled by electrical heaters which induce bubble generation in microchannels. The most established implementation in this respect are thermal ink jet printers [1]. Applications based on laser heating of the solid surface cover optical switches, laser surgery, laser desorption spectroscopy, and Steam Laser Cleaning.

Nonlinear optical switches consisting of absorbing micro- or nanoparticles in a suspension rely on bubble formation around the particles, which results in a change of the index of refraction and total reflection of the beam [2]. In laser surgery wavelengths which are directly absorbed in water in order to induce an explosive boiling process are widely used [3]. However, treatments like the selective retina therapy are based on bubble formation at laser heated pigments [4, 5, 6, 7].

Furthermore, gold nanoparticles can be coated with antibodies so that they attach to specific receptors and serve as selective absorbers for laser treatment [8]. Various laser desorption spectroscopy techniques, which have evolved as a powerful tool for the analysis of large bio molecules, also rely essentially on material ejection induced by explosive boiling [9]. A comprehensive up-to-date review of the subject can be found in reference [10]. Besides, the phase transition can be employed for effective removal of submicron particles from sensitive surfaces by Steam Laser Cleaning [11, 12, 13]. In the first step of this process, a thin liquid layer is deposited on the contaminated substrate. Subsequently, the substrate is heated by a laser pulse, which induces the phase transition at the surface as a result of heat conduction into the liquid. During the explosive boiling process, intense acceleration forces are exerted onto the contaminants, which eventually cause the particles to detach from the substrate.

The widespread applications motivated numerous studies addressing various aspects of the phase transition, such as the threshold laser fluence necessary for bubble detection in bulk liquid, growth velocities of bubbles, and the generated

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pressure waves (Chapter 3). In spite of the considerable efforts, the early stages of the phase transition remain poorly understood. Indeed, theoretical models in this regard exist, but they are mostly based on thermodynamic equilibrium concepts making their validity questionable. Therefore, experimental investigations on the topic continue, even though they are ambitious since a spatial sensitivity on the nanometer scale has to be combined with a temporal resolution below one nanosecond. Lately, molecular dynamics simulations provide additional insights, but studies which allow a direct comparison to experiments are still lacking. Yet, such a complementary approach could eventually result in a detailed picture of the initial mechanisms. In this context, open questions relate to the attainable superheating in the liquid, the generated pressures, the intrinsic timescales of the phase transition, and the underlying formation mechanism of the vapor phase. Further- more, the influence of the heating rate, the surface structure, and the thickness of the liquid layer on a substrate are of particular interest.

In the experiments presented in the following, the liquid-vapor transition in thin isopropanol films on commercial silicon wafers is studied. From the application-oriented point of view, silicon is one of the technologically most relevant materials for Steam Laser Cleaning. A more detailed understanding of the mechanisms for particle removal allows further optimization of the technique.

Regarding the fundamental processes involved in the phase transition, the high surface flatness of the silicon wafers suppresses inhomogeneous nucleation at irregularities allowing measurements at a nearly ideal interface. Since the isopropanol is condensed onto the substrate from a flow of supersaturated gas, a high purity of the liquid is ensured. The thin isopropanol layers, with thicknesses on the order of 100 nm, respond very sensitively to the pressure generated during the evaporation process. Therefore, the liquid film can be utilized as a highly resolving nanomechanical sensor, which is the foundation of this thesis.

The contents are structured as follows: Subsequent to the introduction, the theoretical concepts for the presented experiments are discussed in Chapter 2.

Chapter 3 puts the studies in context to previous experimental and theoretical investigations. The Chapters 4 and 5 constitute the main part and delineate the performed experiments. First, the setup and the experimental implementation are described. Afterwards, the results are discussed starting with a rather general description of the observed behavior followed by a more profound analysis, which partially relies on numerical calculations. In Chapter 6, the investigations are summarized and assessed from a more general point of view. Finally, approaches for future experiments are suggested in Chapter 7.

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Parts of this thesis have already been published:

F. Lang, P. Leiderer, and S. Georgiou. Phase Transition dynamics measurements in superheated liquids by monitoring the ejection of nanometer-thick films. Appl.

Phys. Lett., 85 (14):2759-2761, 2004

F. Lang and P. Leiderer. Liquid-vapor phase transitions at interfaces: sub- nanosecond investigations by monitoring the ejection of thin liquid films. New J.

Phys., 8 (14), 2006

A fellowship (HPMT-GH-00-00177-13) by the Marie Curie training site op- erating at IESL-FORTH and financial support by the Deutsche Forschungsge- meinschaft (project number: LE 315/22-1) and the Optik Zentrum Konstanz are gratefully acknowledged.

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2.1 Interfacial Tension and Contact Angles

The formation of an interface between different substances requires a certain amount of energy. This effect influences various aspects of the conducted experiments, such as the wetting behavior of the deposited liquid on the substrate and the vapor formation in the superheated liquid. The underlying physical rea- son for the occurrence of this interfacial energy are the cohesion forces between the different molecules. In bulk the interaction forces in each direction are identical. As a consequence, the net force on each molecule is zero. In contrast, at an interface, the symmetry around the molecules is broken, and a resulting force occurs. The arising effect between two substances i and j is described by the interfacial tension σij, which can be interpreted either as a force per unit length acting perpendicular to the contact line or alternatively as an energy density per surface area. In the special case of an interface between a substance and vacuum, σ is called surface tension.

Figure 2.1: Contact angle of a liquid droplet on a substrate.

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2.2 Liquid-Vapor Phase Transitions

For a solid surface in contact with a liquid and a gas, the interfacial tensions are manifested by the contact angle. Fig. 2.1 displays the situation for a liquid droplet on a substrate. The treatment of a gas bubble in a liquid is analogous.

The total interfacial energy of the system is given by

Wtot =σ12A12+σ23A23−σ13A12. (2.1) If external forces like gravitation are negligible, the droplet shape is a spherical cap in order to minimize the area of the liquid-gas interface. The contact angle Θ is calculated by minimizing Wtot, which results in the Young equation

cos Θ = σ13−σ12

σ23

. (2.2)

An alternative derivation can be accomplished from the force balance at the three phase contact line. Surfaces for which the contact angle of a liquid is below 90^◦ are called lyophilic, whereas they are called lyophobic if the contact angle is larger.

The case of Θ = 0^◦ is called complete wetting.

The curvature of the interface is associated with a pressure difference, which can be calculated using the Laplace equation

∆P =σ23· 1

r1

+ 1 r2

, (2.3)

with r1 and r2 denoting two radii of curvature measured in two planes perpendicular to each other. Since the surface tension tends to pull the interface flat, the pressure under a concave surface is reduced, whereas it is increased under a convex surface like in a bubble.

2.2 Liquid-Vapor Phase Transitions

Phase transitions are characterized by an abrupt change in a materials physical properties caused by a small variation of the external conditions. They are ac- companied by a change in the order of the system. According to Ehrenfest [14], phase transitions can be classified based on the lowest derivative of the system’s Gibbs free energy G which exhibits a discontinuity. First order phase transitions are characterized by a jump in the entropy S = −^∂G∂T and, consequently, the occurrence of latent heat at the transition point. In contrast, no heat is released or absorbed during phase transitions of second or higher order, which are, therefore, called continuous. The liquid-vapor phase transition investigated in this thesis is a first order transition except if it occurs at the critical point, where it is continuous.

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2.2.1 Statical Thermodynamic Description

2.2.1.1 Landau Theory

The Landau (L.D. Landau 1937) theory [15] is an empirical phenomenological approach originally designed to describe second order phase transitions. With a small extension, it can also be used to model first order transitions. The starting point of the Landau theory is the definition of an order parameter Φ, which is zero for temperatures above the transition temperature Ttrans and unequal zero for temperatures below. Since fluctuations of the order parameter are neglected, the applicability very close to the transition point is limited. A quantitative estimation of the validity range is given by the Ginzburg criterion, which compares the magnitude of the thermal fluctuations to the mean value of Φ. Although Landau’s model provides quantitatively wrong results in some aspects, it offers a good qualitative description. Furthermore, it avoids the difficulties of calculating the Gibb’s free energy from statistical mechanics, which so far has only be done exactly for a few special systems (e.g. for the Ising model by Onsager 1944).

Figure 2.2: Gibb’s free energy as a function of the order parameter Φ for a second order phase transition. 1. T < Ttrans, 2. T =Ttrans, 3. T > Ttrans.

It is assumed that nearby the transition temperature the order parameter is small. As a consequence, Landau suggested to expand the Gibbs free energy for second order transitions by powers of Φ in the vicinity of the transition point as

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follows:

G= Z

dV g ; g =a·(T −Ttrans) Φ²+uΦ⁴ +A(∇Φ)²−hΦ , (2.4) with a, u, and A denoting positive constants and h being an external field. Odd powers of Φ do not occur since the Gibb’s free energy has to be symmetric in Φ if no external field is present. Fig. 2.2 illustrates g(Φ) for different temperatures in the case that no external field is applied and that Φ is locally constant. In equilibrium, the system is in the state for whichG(Φ) has a minimum. If the temperature is increased towards the transition temperature Ttrans, the order parameter Φ approaches zero continuously according to the power law Φ ∝ |T−Ttrans|^β with the critical exponent β = 1/2. Moreover, it can be demonstrated that the suscepti- bility χ = ^∂Φ_∂h

h=0 diverges following the function χ ∝ |T −Ttrans|⁻^γ with the critical exponent γ = 1. Other variables describing the system like the specific heat capacity show a similar behavior. However, a detailed treatment of critical exponents and scale laws relating the different exponents to each other is beyond the scope of this thesis.

In order to model first order phase transitions, which are in the focus of the following investigations, with the Landau theory, the equation for the Gibb’s free energy 2.4 has to be modified. An easy expansion to describe a first order transition is

g =a·(T −T0)²Φ²+u(T) Φ⁴+cΦ⁶+A(∇Φ)²−hΦ , (2.5) with A, a, and c being positive constants and u < 0 in contrast to equation 2.4.

Other parametrizations for g(Φ) are possible but provide similar results.

Fig. 2.3 shows g(Φ) from equation 2.5 under the same assumptions that have already been used in Fig. 2.2 (h = 0, Φ=const.). The phases with Φ = 0 and Φ6= 0 are in equilibrium at the transition temperature Ttrans =T0+u·4ac^u² . The main characteristic for a first order phase transition is the existence of a local energy minimum in a certain temperature range, which is separated from the absolute minimum by an energy barrier. Therefore, the order parameter changes discontinuously during a first order transition.

Furthermore, superheating or supercooling into metastable states may occur due to the energy barrier. If the system is in the ordered phase initially with Φ6= 0, and the temperature is increased, it can be superheated above Ttransuntil curve 4 is reached. The temperature corresponding to curve 4 defines the thermodynamic limit of superheating - the spinodal. If the system is cooled down from the high temperature phase with Φ = 0, it can be supercooled until curve 2 is reached.

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Figure 2.3: Gibb’s free energy as a function of the order parameter Φ for a first order phase transition. 1. T < Tsupercool, 2. T ≈Tsupercool, 3. T ≈Ttrans, 4. T ≈Tsuperheat, 5. T > T_superheat.

So far, only the case ofh= 0 has been discussed. If an external field is applied, the two phases with positive and negative order parameter Φ are not energetically equivalent any more. Furthermore, the energy barrier stabilizing the metastable state at a given temperature betweenT2 andT4 disappears if the magnitude of the external field exceeds a certain limit. This limit defines the spinodal curve in the h-T-diagram. Considering liquid-vapor phase transitions in particular, the order parameter is identified with the density relative to the density at the transition point Φ = n−ntrans. This quantity is directly related to the volume of the system commonly used in representations of phase diagrams. In addition, the pressure of the system can be directly related to the external field h. A further discussion of the spinodal curve based on thermodynamic equations of state and phase diagrams is given in the next section.

2.2.1.2 The Spinodal Limit of Metastability

In order to illustrate the spinodal metastability limit, Fig. 2.4 depicts the P-v phase diagram for a pure substance in the area of the liquid-vapor transition. The figure shows an isotherm below the critical temperature. Using Maxwell’s construction, the horizontal line ACE is placed in such a way that the two areas it encloses with the isotherm are of equal size. The saturation curve is found by constructing the

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Figure 2.4: P-v phase diagram for the liquid-vapor transition. The solid line represents an isotherm for T < Tc. The horizontal line is placed according to the Maxwell construction so that it confines areas of equal size with the isothermal line.

outer intersection points A and E for different isotherms. It is also known as the binodal curve and defines where the phase transition occurs in thermodynamical equilibrium. Nevertheless, starting from the liquid phase the isotherm can be followed beyond the point A after which the liquid becomes metastable. At the point B the mechanical stability criterion − ^∂P∂V

T,N > 0 fails, and the limit of metastability is reached. It should be noted that five more stability criteria exist

∂T

∂S

P,N >0, ^∂T_∂S

µ,V >0, − ^∂P∂V

µ,S >0, _∂N^∂µ

P,S >0, _∂N^∂µ

T,V >0 and that all of them fail simultaneously [16]. Similarly, the binodal can be passed at the point E starting from the vapor phase until the point D is reached. The entire spinodal curve is found by determining the points B and D for different temperatures. In the following discussion, spinodal refers to the part of the curve left of the critical point, which is denoted as the liquid spinodal in Fig. 2.4 since this is the relevant part for the liquid-vapor phase transition.

Fig. 2.5 shows the saturation curve and the spinodal for the liquid-vapor transition in a schematic P-T-phase diagram. The kinetic limit of superheating indicated in the diagram will be discussed in Section 2.2.2.2. A liquid is superheated if its temperature is higher than the saturation temperature for the prevalent pressure.

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This can be achieved by different paths in the phase diagram. Two main pos- sibilities are distinguished. The first one is to lower the pressure by isothermal expansion. If a phase transition is induced along this way, it is called a cavitation process. Experimentally determined cavitation pressures for alcohols at room temperature are on the order of 10 MPa, which is significantly below theoretically predicted values though [17]. The second possibility is isobaric heating as indicated by the horizontal path in Fig. 2.5. A phase transition along this path is called a boiling process. The experiments described in this thesis exclusively in- vestigate phase transitions by isobaric heating. Though, strictly speaking a minor pressure increase occurs during the heating process due to the thermal expansion of the substrate and the liquid. A possible influence of this effect on the phase transition is estimated in Section 2.6, which treats thermoelastic processes.

Figure 2.5: Schematic P-T phase diagram for the liquid-vapor transition. From the starting point A the two paths corresponding to cavitation and boiling are indicated by arrows.

The pressure dependence of the spinodal temperature can be determined quantitatively by applying the condition for the limit of metastability ^∂P_∂V

T,N = 0 to an equation of state for the system. However, the approach is problematic since the validity of the thermodynamic equations of state in the metastable region, which is experimentally hardly accessible, remains questionable. On the other

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hand, different equations of state provide similar results for the spinodal curve.

Figure 2.6: Pressure dependence of the spinodal temperature calculated from the Berth- elot and the van der Waals equation of state. Pressure and temperature are plotted in reduced units relative to their critical values. The experimental values for methanol and ethanol are taken from [18].

Fig. 2.6 pictures the pressure dependence of the spinodal temperature calculated based on the van der Waals

Pr+ 3

v_r² vr− 1 3

= 8

3 ·Tr (2.6)

and the Berthelot equation of state

Pr+ 3

Tr·v_r² vr− 1 3

= 8

3 ·Tr, (2.7)

wherePr= _P^P

c denotes the reduced pressure,Tr = _T^T

c the reduced temperature, and vr = _v^v

c the reduced specific volume. The experimentally determined superheating limits plotted in the diagram imply that the prediction from the van der Waals equation underestimates the attainable superheating, whereas the curve calculated from the Berthelot equation represents an unexceedable upper limit, which is consistent with the conception of a spinodal. Recapitulating the results in Fig. 2.6,

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the spinodal temperature is estimated to be 0.9Tc in a broad pressure range from 10⁻³Pc to 0.1Pc, which is consistent with an estimation by Blander and Katz at normal pressure [19]. For isopropanol, this implies a spinodal temperature of 457 K in the pressure range from 0.05 atm to 5 atm, which is 102 K above the boiling temperature at normal pressure.

2.2.2 Kinetics

The transition from metastable states or instable states at the spinodal to equilibrium requires the formation of a new phase. In the following, the mechanisms and the formation rate during the initial stage of this process are discussed. If the system is quenched towards the thermodynamical stability limit fast, in particular near the critical point, the spinodal can be reached, and relaxation occurs via spinodal decomposition. Otherwise, the relaxation process starts in the metastable area of the phase diagram by nucleation due to thermal fluctuations.

2.2.2.1 Spinodal Decomposition

At the spinodal, the compressibility of the system χ = −V¹

∂P

∂V

T,N becomes negative (2.2.1.2), and arbitrarily small density fluctuations grow. A thorough theoretical treatment can be accomplished based on the density functional theory, which provides a generic description of density variations in the system. A detailed presentation is given by Onuki [20]. The most characteristic feature is the existence of one preferred Fourier component km for the density fluctuation during the initial stage of the process. However, km decreases for longer timescales due to nonlinearities. With respect to the relevant timescales, the disintegration into two phases sets in instantaneously after the spinodal curve is reached. The duration of the decomposition process is determined by the diffusivity D and the correlation length ξ of the system: t∝ ^ξD². Molecular dynamics simulations indicate that the entire process lasts typically 100 ps [16, 21, 22].

2.2.2.2 Nucleation Theory

The classical nucleation theory (CNT) described in this section originates from the work of Volmer and Weber (1926) and was initially developed to describe droplet condensation from supercooled vapors. Nevertheless, it can likewise be used to model the vapor formation in superheated liquids. The CNT restricts density fluctuations to be spherical vapor bubbles, which causes a loss in gen- erality but has the advantage of providing analytical equations for the relevant

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quantities. Moreover, the results of the CNT depend solely on macroscopic material properties, which are easily accessible experimentally. Generally, three basic nucleation processes are distinguished. Homogeneous nucleation refers to the formation of vapor bubbles in bulk liquid exclusively due to thermal fluctuations. In contrast, bubble formation at an idealized solid-liquid interface is called heterogeneous nucleation. Finally, inhomogeneous nucleation arises if vapor inclusions preexist either dissolved in the liquid or trapped in roughnesses at an interface.

Homogeneous Nucleation Homogeneous nucleation takes place if the energy barrier for the formation of stable vapor bubbles in bulk liquid can be overcome by thermal fluctuations. According to Skripov [17], the Gibbs free energy ∆G required for the formation of a spherical bubble with radius r, is given by:

∆G= 4πr²σ− 4

3πr³(Pv−Pl) + (µv−µl)mv, (2.8) where σ denotes the surface tension and mv the mass of the vapor in the bubble.

Pv refers to the pressure in the vapor,Pl to the one in the liquid, andµv and µl

to the corresponding chemical potentials. Equation 2.8 features a surface contribution proportional to r² which counteracts the bubble formation and a volume contribution proportional to r³ which favors the nucleation process. A schematic representation of ∆Gas a function of the bubble radius is given in Fig. 2.7.

Figure 2.7: Gibbs free energy of a vapor bubble. Bubbles smaller than re collapse, whereas larger bubbles grow spontaneously.

Bubbles smaller than the equilibrium radiusrecollapse, whereas bubbles larger than re grow spontaneously and initiate a macroscopic phase transition. The equilibrium radius as well as the height of the energy barrier ∆Ge decrease with

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increasing temperature. As shown by Carey [18], re is given by the following expression:

re(Tl, Pl) = 2σ(Tl, Pl) Psat(Tl)·η(Tl)−Pl

with η=exp

Pl−Psat(Tl) ρl(Tl)RsTl

. (2.9)

Tl represents the temperature in the liquid, Psat(Tl) the saturation pressure, Rs

the specific gas constant, and ρl the density of the liquid. Fig. 2.8 visualizes the dependence of re on the temperature for bubble formation in isopropanol as calculated from equation 2.9. The required material data of isopropanol is given in the Appendix (Chapter 10). Remarkably, the equilibrium radius is larger than zero at the spinodal temperature of isopropanol Tsp = 457 K, and the energy barrier ∆Ge persists as well. The phase transition process continues to be initiated by thermal fluctuations indicating that the CNT does not include spinodal decomposition. This shortcoming is eliminated in more sophisticated theoretical approaches based on the density functional theory.

Figure 2.8: Unstable equilibrium radius re for homogeneous nucleation in isopropanol calculated from the CNT. At a particular liquid temperature vapor bubbles larger than r_e grow spontaneously, whereas smaller ones collapse.

The nucleation rate and the kinetic limit of superheating are obtained from the number of vapor seeds generated by thermal fluctuations which reach the size of the critical radius re. Considering an idealized system in thermal equilibrium,

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the number Nn of vapor bubbles per unit volume containingn molecules is given by a Boltzmann distribution [18]

Nn =Nl·exp

−∆G(r) kBTl

, (2.10)

with Nl denoting the total number of liquid molecules per unit volume. It should be noted that the development of such a distribution in the system is expected to be associated with a characteristic relaxation time. Thus an incubation period should become observable before the phase transition sets in after very fast heating processes. Such an incubation time is predicted by molecular dynamics simulations for homogeneous nucleation in a superheated liquid [22] and has already been observed experimentally for nucleation processes in other systems [23, 24].

From equation 2.10 the volumetric homogeneous nucleation rate is calculated to be

Jhom(Tl, Pl) =Nl

3σ(Tl, Pl) πm

¹₂

·exp

"

−16πσ(Tl, Pl)³ 3kBTl(ηPsat(Tl)−Pl)²)

#

, (2.11) with m denoting the mass of a liquid molecule [18]. The predicted nucleation rate features a steep increase over numerous orders of magnitude in a narrow temperature range of just a few Kelvin as displayed in Fig. 2.9.

In order to determine the kinetic limit of superheating, which has already been schematically indicated in Fig. 2.5, the probability to generate a spontaneously growing nucleus must be calculated. For this purpose, the nucleation rate from equation 2.11 has to be integrated over the superheated volume and over the ob- servation time of the experiment taking into account the space and time dependent temperature distribution. An expectation value of ¹₂ defines the kinetic nucleation limit of superheating. This limit does not constitute a rigorous threshold though.

Nucleation occurs over a certain temperature range, which depends to some de- gree on the heated volume and the duration of the heating process. However, the dependence is minor because of the steep increase in the nucleation rate as a function of temperature. As a consequence, the kinetic limit is frequently determined by simply imposing a nucleation rate Jhom on the order of about 10¹²m⁻³s⁻¹ as the limit of superheating [19]. In the case of isopropanol, this appraisal results in a kinetic threshold temperature at normal pressure of approximately 463 K, which is slightly above the estimation for the spinodal temperature though (2.2.1.2). This inconsistency may be attributed to the limitations of the CNT but can, as well, be caused by the extrapolation of the material data into the metastable region.

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Figure 2.9: Homogeneous nucleation rate in isopropanol atP_l= 101.3 kPa as calculated from the CNT.

Heterogeneous Nucleation Heterogeneous nucleation refers to the generation of vapor bubbles by thermal fluctuations at a perfectly smooth solid-liquid interface. This scenario is technically more important since heat transfer is typically accomplished through a solid wall. It is also expected to be a relevant mechanism in the experimental investigations of isopropanol at a laser heated silicon substratein this thesis.

If heat is supplied to the liquid through a solid substrate, a temperature gradient develops, and the phase transition is initiated directly at the solid-liquid interface. However, for the sake of simplicity, most theoretical descriptions of the heterogeneous nucleation process consider a liquid of homogeneous temperature and pressure. Under this assumption, the model is analogous to the classical theory for homogeneous nucleation. The extension with respect to the surface energies at the solid surface is mathematically included by the contact angle Θ between the solid and the vapor bubble, which is idealized as a spherical cap. Note that as in Fig. 2.1 the angle Θ is measured towards the liquid covered part of the solid, which is on the outside of the cap though. The heterogeneous volumetric

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nucleation rate under these conditions is calculated to be Jhet =N

2 3

l

1 + cos (Θ) 2F(Θ)

3σF(Θ) πm

¹₂

·exp

−16πσ³F(Θ) 3kBTl(ηPsat(Tl)−Pl)²)

(2.12) with F (Θ) = ¹₄(1 +cosΘ)²(2−cosΘ) [18]. An increase in the contact angle causes the nucleation to occur at lower temperatures. For a precise calculation of the heterogeneous nucleation threshold, the temperature dependence of the contact angle has to be taken into account. Nevertheless, an estimation can be obtained by assuming a constant value for Θ. In this case, the heterogeneous nucleation rate for a specific angle features qualitatively the same behavior as the homogeneous nucleation rate.

Fig. 2.10 indicates the heterogeneous nucleation limit of isopropanol as a function of the contact angle. For angles up to 80^◦, the threshold temperature remains nearly constant, but decreases noticeably for higher values. Since isopropanol wets the silicon wafer almost perfectly, the contact angle of a vapor bubble is close to zero. Therefore, the heterogeneous and the homogeneous nucleation limits are approximately identical.

Figure 2.10: Heterogeneous nucleation limit of isopropanol at Pl= 101.3 kPa as a function of the bubble contact angle.

Inhomogeneous Nucleation Inhomogeneous nucleation occurs if gas inclusions are already present in the liquid before the nucleation process sets in. The nomen- clature in the literature is not distinct. Sometimes this conception is included in

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the term ”heterogeneous nucleation”. Preexisting gaseous nucleation seeds can cause phase transitions to occur at just a few Kelvins of superheating as typically observed during boiling in a pot. Furthermore, the highest experimentally observed cavitation pressure for water at room temperature is 270 bar [25], which is in clear contrast to theoretical predictions ranging from 500 bar [26] to values that are even significantly higher [27, 28]. This deviation is also attributed to entrapped nucleation seeds.

Vapor seeds for nucleation can either preexist in bulk liquid or trapped in pits or irregularities at solid-liquid interfaces. Based on thermodynamic consid- erations, gaseous nucleation seeds in bulk should collapse or grow and raise to the surface depending on their initial size as discussed in the paragraph about homogeneous nucleation. Several models have been suggested to account for the existence of stable vapor inclusions. The explanations comprehend stabilization by ions or organic substances [29] as well as continuous generation by cosmic ra- diation [30]. At a solid-liquid interface vapor can be inclosed in surface grooves if the advancing contact angle Θ is larger than the groove angle 2γ[18]. In this case, the vaporization process may be initiated at the liquid-gas interfaces of the vapor filled cavities. For a simplified theoretical description, the cavities are assumed to be conical with an opening angle of 2γ and a mouth radius R. In addition, the liquid-vapor interface is assumed to have a spherical surface, which can be characterized by a single radius of curvature. Under these conditions, a nucleation site in contact with a liquid of homogeneous temperature Tl becomes active if the following condition is satisfied [18]:

Tl−Tsat(Pl)> 2σTsat(Pl) (ρl−ρv)hlvrmin

. (2.13)

hlv denotes the mass specific latent heat, ρl and ρv the density of the liquid and the vapor, and rmin the minimum curvature radius of the vapor-liquid interface during the growth process of the vapor inclusion out of the cavity. If the contact angle is large 2γ < Θ ≤ 90^◦, the radius of curvature rmin can be approximately set equal to the mouth radius R. Otherwise, a model by Lorenz et al. [31]

can be used to estimate rmin. In case that heating is accomplished across the interface, the temperature in the liquid is not homogeneous but a gradient evolves.

In this regard, the model of Hsu [32] shows that not just a lower, but also an upper boundary for the mouth radius R of activated nucleation sites exists. The upper limit occurs as for large values of R the bubble becomes sufficiently large to protrude beyond the superheated layer at the interface.

Concluding the discussion of the nucleation theory, it should be added that

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inhomogeneous nucleation occurs in most experimental superheating studies to some extent. Whether it becomes the predominant mechanism of the phase transition depends on several factors, such as the density of the preexisting nucleation seeds and the rate at which the liquid is heated. In the experimental investigations of this thesis, inhomogeneous nucleation sites at the solid-liquid interface are kept to minimum by employing silicon wafer substrates of high surface flatness.

2.2.3 Bubble Growth in Superheated Liquids

After a vapor bubble larger than the equilibrium radius re has been generated by a nucleation process, the growth process in the surrounding superheated liquid starts. Since the focus of the studies in this thesis is on the nucleation processes and not on the long-term growth, the presentation of the growth dynamics is restricted to a basic theoretical description and the estimation of characteristic growth velocities. However, experiments which exclusively probe nucleation dynamics without any influence of the growth process are hardly realizable.

The first description of bubble dynamics in a liquid was given by Rayleigh [33]. In his approach, he examined a spherical bubble in a liquid of uniform pressure and temperature. The surface tension and the viscosity of the liquid were neglected. Furthermore it was assumed that the liquid is incompressible.

The influence of the surface tension and the viscosity can be easily included in the theory though, which leads to the Rayleigh-Plesset equation [34, 35] for the dynamics of the bubble growth

rd² dt²r+ 3

2 d

dtr 2

= 1 ρl

Pv−Pl− 2σ r − 4µ

r d dtr

. (2.14)

r refers to the radius of the bubble and µ to the viscosity of the liquid. Nev- ertheless, the Rayleigh-Plesset model obviously does still not constitute an all- embracing theory of bubble growth. Several aspects are neglected, for instance by assuming that the pressure in the bubble is homogeneous and that the liquid is incompressible. In addition, the phase transition at the interface of the vapor bubble and the associated heat flow are not explicitly included. Miscellaneous extensions have been proposed to correct for the drawbacks of the model [36, 37].

Nevertheless, the Rayleigh-Plesset theory provides considerable insights into the bubble growth dynamics. Since a general analysis is rather complex, two limiting cases are examined more detailed in the following.

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Figure 2.11: Asymptotic bubble growth velocity in the inertia controlled regime.

2.2.3.1 Inertia Controlled Growth

During the first stages of bubble growth in a highly superheated liquid, the process is inertia controlled. The heat transfer to the bubble interface does not constitute a limiting factor, and the growth is governed by the momentum transfer to the surrounding liquid. The temperature of the vapor in the bubble is approximately equal to the one in the liquid Tv ≈ Tl, and the pressure in the bubble is close to the maximum value possible, which is given by the saturation pressure Pv ≈ Psat(Tl). The constant pressure inside the bubble during the inertia controlled regime implies that the pressure drop caused by the expansion is compensated by vapor formation at the interface. Under these assumptions, the contribution of the surface tension in equation 2.14 is negligible. If, in addition, the viscosity is set to zero for simplicity, the process can be modeled using the equation originally derived by Rayleigh. The solution of this differential equation shows that the bubble growth velocity approaches an asymptotic limit [35] given by

d

dtrasym = s2

3

Pv−Pl

ρl

. (2.15)

Accordingly, the bubble radius increases linearly in time after the initial stage of the growth process. Fig. 2.11 illustrates the asymptotic bubble expansion velocity

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in isopropanol as a function of the liquid temperature. For superheatings close to the spinodal temperature, which is approximately 0.9Tc = 457 K (2.2.1.2), a maximum growth velocity of 49 m/s is predicted.

2.2.3.2 Heat Transfer Controlled Growth

In the final stages of the growth process, the expansion velocity is limited by the heat transfer towards the bubble interface. The growth velocity is significantly smaller than during the inertia controlled regime. The pressure in the bubble has dropped to a value close to the pressure of the surrounding liquid Pv ≈Pl, and the vapor temperature is approximately equal to the boiling temperature of the liquid Tv ≈ Tsat(Tl). Under these conditions, the growth velocity can be determined from the energy balance between the heat flow to the bubble and the latent heat of vaporization [35]

d dtr=

r3 π

kl

hlvρv(Tsat)

Tl−Tsat

√Dt , (2.16)

where Ddenotes the thermal diffusivity andkl the thermal conductivity. Integra- tion of this equation shows that the bubble radius increases proportional to the square root of time r ∝√

t for heat transfer controlled growth.

The transition from the inertia controlled to the heat transfer controlled regime takes place smoothly. At intermediate times, the process is influenced by an interplay of both effects. A simple interpolation has been proposed by Mikic et al. [38]. For high superheatings, this approach becomes inaccurate though, and improved theories have, therefore, been developed [39].

2.2.3.3 Bubble Growth at a Heated Solid Surface

In inhomogeneously heated liquids, the modeling of the bubble growth process is significantly complicated by the nonuniform temperature field and the lack of the spherical symmetry. Depending on the local temperature, evaporation and condensation may occur at different parts of the bubble boundary, influencing the vapor pressure in the bubble. Moreover, a theoretical approach has to incorporate the local and temporal variation of temperature due to the heat transfer in the vicinity of the bubble. For that purpose, not only heat conduction has to be considered but also convection, liquid motion caused by the bubble dynamics, and latent heat flow generated by the phase transition.

In view of these difficulties, the following description of bubble dynamics near a heated surface is restricted to a qualitative picture of the process. As in the case of

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an extensive uniformly heated liquid, the two regimes of inertia controlled and heat transfer controlled bubble growth can be distinguished. However, the occurrence of either one is influenced by additional factors, such as the temperature of the solid surface, the heat flux across the interface, the contact angle, and the roughness of the surface [18]. In the inertia controlled case, the bubble expands rapidly, and its shape is nearly hemispherical. Contrariwise, for heat transfer controlled growth, the pressure in the bubble and the inertia forces are less dominant. Therefore, the surface tension causes the bubble to be nearly spherical. During the growth process, the bubble is attached to the interface due to the interfacial tension along the contact line to the solid. As the bubble becomes larger, this force can be overcome by buoyancy and inertia forces from the surrounding liquid, and the bubble may detach from the interface [18]. An analytical model for bubble growth near a heated surface including both regimes was proposed by van Stralen et al.

[40].

2.3 Liquid Condensation from a Supersaturated Gas Flow

The liquid condensation from a supersaturated gas is analogous to the processes described in the section on liquid-vapor phase transitions (2.2). In thermodynamic equilibrium at a flat interface, the pressure in the gas phase is equivalent to the saturation vapor pressure. At a curved interface though, the influence of the Laplace pressure (Eq. 2.3) has to be considered. Therefore, the vapor pressure P depends on the radius of curvature, which is described by the Kelvin equation

P(r1, r2) =P0·exp σVm

RT · 1

r1

+ 1 r2

. (2.17)

P0 denotes the vapor pressure at a flat surface,Vm the molar volume of the liquid, and r1andr2 two radii of curvature in perpendicular planes. For a convex surface, the vapor pressure is increased, whereas it is reduced for a concave interface.

In the experiments of this thesis, the deposition of thin liquid films is achieved by conducting a flow of heated gas saturated with isopropanol vapor over the substrate. At the substrate surface, the gas cools down and becomes supersaturated.

Subsequently, a heterogeneous nucleation process sets in. Shortly afterwards, the individual nucleation centers coalesce, and a closed liquid film is formed on the surface.

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2.4 Laser Materials Interaction

Even though the liquid which is used to saturate the heated air flow contains at least 99.9 % isopropanol, the deposited film may contain a water content on the order of a few percent due to the humidity of the carrier gas. Since the inflowing air at T = 20^◦C is conducted through a silicate filter before it is heated to 45^◦C and saturated with isopropanol vapor, its initial relative humidity is well below 50 %.

The ratio of isopropanol to water in the vapor that is conducted to the sample can be calculated from the saturation vapor pressures at T = 20^◦C and T = 45^◦C (Appendix 10.1). For high isopropanol concentrations, the condensating liquid is expected to have an even lower water content than the vapor [41], which gives an estimation for the mass specific isopropanol content in the deposited film of at least 98 %.

2.4 Laser Materials Interaction

2.4.1 Optical Material Properties

The optical material properties in the visible, the near IR and the near UV are governed by the interaction of the electromagnetic wave with the valence and free electrons. Since the wavelength is significantly larger than the atomic distances, the material can be characterized by macroscopic quantities. For this purpose, the wavelength dependent complex index of refraction

m(λ) =n(λ) +ik(λ) , (2.18)

or the complex dielectric constantǫ=ǫR−iǫI =m² can be used. A dependence of the optical properties on the polarization shall not be discussed here since effects like optical activity and birefringence are of no importance for the understanding of the following. The light absorption in the material is determined by the extinction coefficient k. In an absorbing material, the light intensity I decreases according to the Lambert-Beer law

I(z) =I0·exp (−αz) =I0·exp

− z dopt

, α= 4πk

λ . (2.19)

The variable αis referred to as the absorption coefficient, and the optical penetration depth dopt = 1/α indicates the length scale on which the energy of the electromagnetic wave is deposited. The optical properties of silicon and isopropanol at room temperature for the wavelengths used in the experiments of this thesis are given in Table 2.1. It is noted that for high intensities the optical properties may

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depend on the local amplitude of the electromagnetic wave. For the experiments of this thesis, these nonlinear effects are nonsignificant though.

wavelength silicon isopropanol

λ n k dopt n k dopt

248 nm 1.47 3.57 6 nm 1.42 ≈0 ≈ ∞

488 nm 4.37 7.9·10⁻² 490 nm 1.38 0 ∞

532 nm 4.15 4.4·10⁻² 960 nm 1.38 0 ∞

635 nm 3.88 1.9·10⁻² 2.7µm 1.38 0 ∞

Table 2.1: Optical properties of silicon and isopropanol at room temperature for the wavelengths used in the experiments [42, 43] (partly extrapolated).

2.4.2 Dynamics of Optical Absorption

The investigation of laser induced thermal effects on the sub-nanosecond timescale raises questions regarding the thermalization times of the substrate. Initially, the energy of the absorbed light is transferred to the electrons of the system.

In the next step, the electrons thermalize within approximately 100 fs [44] after which they have adopted a Fermi-Dirac distribution. At this point, a distinct temperature can be attributed to the electron system. The temperature of the lattice is still significantly lower though. Thermal equilibrium of the material is reached after the electron-phonon relaxation and the thermalization of the phonon system are completed, which typically takes a few picoseconds [45]. However, the generated electron-hole pairs still exist. Their recombinations cause so called delayed Auger heating, which for the most part takes place within 100 ps but may continue for a few hundred picoseconds at a continuously decreasing rate [46].

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2.4 Laser Materials Interaction

Compared to the timescales of the experiments in this thesis in which nanosecond pulses are used, and a time resolution of 200 ps is achieved, the relevant thermal relaxation processes occur significantly faster. Therefore, the absorbing material remains in quasi stationary thermal equilibrium at all times, and the laser pulse can be considered as a fast heating pulse.

2.4.3 Optical Properties of Layered Systems

The light transition from one medium into another is generally associated with a splitting of the beam in a reflected and a transmitted part. The reflectivity R and the transmissivity T for s-polarized (linear polarization withE perpendicular to the plane of incidence) and p-polarized (E k to the plane of incidence) light can be calculated with a generalized form of the Fresnel equations comprehending absorbing materials

Rs=

√ǫ1cosθ1−√ǫ2cosθ2

√ǫ1cosθ1+√ǫ2cosθ2 2

(2.20) Rp =

√ǫ2cosθ1 +√ǫ1cosθ2 2

(2.21)

Ts= 1−

√ǫ1cosθ1+√ǫ2cosθ2 2

(2.22)

Tp = 1−

√ǫ2cosθ1 +√ǫ1cosθ2 2

. (2.23)

ǫ1 andǫ2 denote the complex dielectric constants of the two media andθ1 and θ2 the corresponding propagation angles, which are connected by the generalized law of refraction

cosθ2 = r

1−ǫ1

ǫ2

sin²θ1. (2.24)

In this extended description, complex angles may arise. In the common case of perpendicular incidence from vacuum or air into another medium, the Fresnel equations for the two directions of polarization are identical, and for the reflectivity they reduce to

R^⊥ = (n−1)²+k²

(n+ 1)²+k² . (2.25)

Finally, it is pointed out that for nonabsorbing materials, equation (2.24) yields the well known Snell’s law, and the Fresnel equations can be simplified to the form usually given in optics textbooks.

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With regard to systems of several planar layers, a general model is complicated. In particular, serious difficulties arise if the propagation direction of an electromagnetic wave is nonperpendicular to the interface of an absorbing layer.

The planes of constant amplitude are parallel to the interface, while the planes of constant phase are tilted, resulting in an inhomogeneous wave. Furthermore, for originally s-polarized light, the polarization of the electrical field becomes el- liptical, and, for originally p-polarized light, this effect occurs for the magnetic field. The easiest description of such a planar layered system is achieved with the transfer matrix method [47, 48], which automatically includes multiple reflections and interference. For this concept, two incident and two outgoing waves are considered at each interface. The parallel and perpendicular components of the wave vector in each layer are calculated from the generalized law of refraction (2.24).

At each interface, the continuity requirements for the electrical field parallel to the interface and for the normal component of the dielectric displacement have to be satisfied. For the transition from layer i to layer i+ 1, these are written in form of a multiplication with the transfer matrix T_iⁱ⁺¹:

E_i+1⁺ E_i+1⁻

=T_iⁱ⁺¹ E_i⁺

E_i⁻

, (2.26)

where the indices + and - indicate light propagation in forward or backward direction. The components of the transfer matrix depend on whether the light is s- or p-polarized and can be found in the more extensive presentation of the transfer matrix method in reference [49]. The propagation in the individual layers can also be described by means of a matrix Φi. For the complete system, the overall transfer matrix is determined by multiplying the individual matrices:

T =T0¹·

N

Y

i=1

ΦiT_iⁱ⁺¹. (2.27)

The reflectivity and the transmissivity are calculated from the electrical field in the incident and the last medium. In order to determine the energy absorption in each layer, the Poynting theorem can be used. Since the calculation of the Poynting vector in an absorbing material is problematic, vacuum layers of in- finitesimal thickness can be introduced at the interfaces to avoid these difficulties [50]. However, in the case of perpendicular incidence, the absorbed energy can be determined by multiplying the incident fluence with the transmissivity and using the Lambert-Beer law (Eq. 2.19).

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2.5 Heat Transport

The energy deposited in the absorbing substrate material by a laser pulse is subsequently redistributed by heat conduction. In the case of silicon substrates, the heat transport is initially dominated by the diffusion of the charge carriers [51].

From the charge carrier diffusion coefficient in silicon of Dcc = 18 cm²/s [52] and the electron-phonon thermalization time of 1 ps measured by Goldman and Pry- byla [45], a diffusion length on the order of 10 nm is estimated. Admittedly, even after thermalization, heat conduction in the electron system may contribute by the delayed Auger heating processes (Section 2.4.2). The recombinations of electron- hole pairs, which typically occur on a timescale of 100 ps, are associated with energy transfer to free charge carriers [46]. Thereby, the density of the electrons and holes is reduced, while simultaneously their temperature rises. In addition, an increase of the diffusivity at very high charge carrier densities is expected [51].

However, laser fluences which are at least close to the ablation threshold are necessary to reach this regime [53]. Hence, the heat transport by charge carriers is circumstantial for the experiments of this thesis. The thermal diffusion length in the phonon system is significantly larger on the relevant timescales as shown in the next paragraph, and the illumination spot diameters are bigger than 1 mm.

Additionally, the optical penetration depth for the measurements conducted with a pulsed Nd:YAG laser is about 1µm and exceeds the typical length scale for charge carrier heat transport by far.

Therefore, the dominant mechanism for the described experiments is the heat diffusion in the phonon system. A physically exact model, taking the finite propagation velocity of phonons into account, has to be based on a hyperbolic differential equation [54]. Since the thermal diffusion length is small compared to the propagation distance of sound for the investigated time scales, a good approximation can be achieved based on the heat diffusion equation. For the one-dimensional case, this parabolic partial differential equation takes the following form:

ρ(z, t)CP(z, t)∂T(z, t)

∂t = ∂

∂z

κ(z, t)∂T(z, t)

∂z

+ ∂

∂tq(z, t) , (2.28) with ρ denoting the density, CP the mass specific heat capacity, and κ the heat conductivity. _∂t^∂q(z, t) refers to the external heating rate per unit volume. The thermal diffusion length is defined as follows:

ddif f(t) = 2√

Dt, (2.29)

with D = _ρC^κ

p denoting the diffusion coefficient. It is a measure for the spacial temperature distribution width after the time t which develops from an initially

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δ-shaped peak. The characteristic heat conduction properties of silicon and isopropanol at room temperature are summarized in Table 2.2. For precise simulations though, the temperature dependence of the material data, which is illustrated in the Appendix (Chapter 10), has to be taken into account.

quantity silicon isopropanol density

2330 785

ρ [kg/m³] heat capacity

710 2610

Cp [J/(kg K)]

heat conductivity

148 0.14

κ[W/(mK)]

therm. diffusion coeff.

89000 68

D [nm²/ns]

therm. diffusion length

1890 52

ddiff (t=10 ns) [nm]

Table 2.2: Heat conduction properties of silicon and isopropanol. The values refer to room temperature [55, 56, 57, 58, 59, 60].

At interfaces between different materials and at grain boundaries of the same material, the heat flow is limited. Thus, a discontinuity in the temperature develops. This concept is well known for liquid helium in contact with a solid, where it is termed Kapitza resistance [61], but the effect also occurs at higher temperatures. The heat conduction properties of an interface can be characterized by the heat transfer coefficient (HTC)

ξ= ∂Q/∂t

A∆T , (2.30)

where ^∂Q_∂t is the rate of heat transfer, ∆T the temperature difference across the boundary, and A the area of the interface. An extensive review on the topic of the thermal boundary resistance has been published by Swartz and Pohl [62].

From the theoretical viewpoint, two basic models for the thermal resistance have been suggested. The acoustic mismatch (AM) model is based on the jump in the acoustic impedances Zi =ρici at the interface. This conception is in analogy to light propagation between materials of different refractive indices, whereby the incident electro-magnetic wave is partially reflected. The boundary has no intrinsic properties in the AM model, and no scattering occurs during the transition.

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2.5 Heat Transport

Figure 2.12: Temperature distribution in a silicon substrate heated by a Nd:YAG laser pulse (F=150 mJ/cm², λ = 532 nm, gaussian temporal profile, FWHM=7 ns). t=0 ns corresponds to the pulse maximum.

As a consequence, the AM model predicts no thermal resistance at grain boundaries, which disagrees with experimental observations. In contrast, the diffuse mismatch (DM) model presumes all phonons to be scattered at the interface. The scattering probability to each side is solely determined by the density of phonon states in each material. As in the AM theory, the structure of the interface is not included in the DM model. Recapitulating, both treatments just provide rough in- dications and do not incorporate the complex interaction mechanisms of phonons and real interfaces. A more precise description of heat transfer across interfaces is still topic of ongoing experiments and theoretical work. Furthermore, concep- tional questions arise with regard to temperature gradients on the nanoscale. The smallest dimension for which a local temperature is meaningful ought to be defined by the phonons’ mean free path length. In this context, it is discussed if

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a material interface constitutes a natural boundary separating different regions of temperature. A detailed discussion of these topics can be found in the recent review by Cahill et al. [63].

The temperature calculations in this thesis are conducted by numerically solv- ing the one-dimensional heat diffusion equation (2.28) using a finite elements algorithm. Specifics about the implementation can be found in the PhD theses of V.

Dobler [50] and J. Bischof [64]. The heat per unit volume q supplied by the laser pulse is determined with the methods presented in the discussion of laser material interactions (Section 2.4). The algorithm includes the influence of temperature changes on the material properties. For the density, the heat capacity, and the heat conductivity, the dependencies are illustrated in the Appendix (Chapter 10).

The temperature dependence of the optical constants are incorporated in the simulation by fitting and also partly extrapolating the data in [65, 66, 67, 68, 43, 42].

Furthermore, the numerical calculation considers predefined HTCs between different layers. The simulation has proven to be very reliable in different applications, for instance an excellent agreement of the calculated and the measured laser fluence required to induce melting of a silicon surface is found. The calculated temperature evolution for a laser heated silicon sample is illustrated in Fig. 2.12.

In the case that the silicon surface is in contact with isopropanol, and the same laser energy is absorbed, the peak temperature is reduced by about 13 K due to heat transfer to the liquid. Remarkably, less than 0.6 % of the absorbed pulse energy is conducted into the liquid before the isopropanol at the interface reaches the spinodal temperature for the investgated fluence.

2.6 Thermoelasticity

The heating by the laser pulse causes a rapid thermal expansion of the substrate material. Even though the resulting surface acceleration is not in the focus of this thesis, a potential influence on the investigated phase transition has to be evalu- ated. Therefore, the pressure variation in the adjacent isopropanol layer caused by the thermoelastic acceleration is estimated. In the most simplistic approach, the substrate expansion is calculated based on the linear thermal expansion coefficient α:

d(t) = α ρCp

(1−R)F Z t

−∞

f(t^′) dt^′, (2.31) whereR is the reflectivity of the system,F the incident laser fluence, andf(t) the temporal pulse shape normalized according toR^∞

−∞f(t^′)dt^′ = 1. The temperature

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2.6 Thermoelasticity

dependence of the material parameters is neglected in this picture, which is a reasonable approximation in the case of silicon. For a temporally gaussian laser pulse with FWHM=τ, the following results for the maximum surface displacement dmax and the maximum acceleration amax are calculated:

dmax = 13 pm mJ

cm² ·F , (2.32)

amax = 1.7·10⁷ m s²

ns² mJ/cm² · F

τ² . (2.33)

If, in addition, the lateral expansion of the material is considered, the displacement and also the acceleration increase by a factor of 2 (ν+ 1) [69], with ν denoting the Poisson number, which is about 0.2 in the case of silicon.

Thus, for characteristical laser parameters used in the experiments (τ = 7 ns, F = 150 mJ/cm²), a surface displacement of dmax = 4.7 nm and a maximum acceleration of amax = 1.2·10⁸m/s² are obtained. Experimentally determined expansions and accelerations are significantly smaller [50] so that the theoretical values can be considered as upper limits.

The pressure in the adjacent isopropanol layer which is generated by the thermal expansion has to be estimated by two different approaches, depending on whether the liquid layer is thick or thin compared to the sound propagation length in isopropanol during the laser pulse (c·τ = 8.5µm). For large thicknesses, the expanding substrate can be modeled as a sound emitter in bulk material. In the case of a harmonically oscillating source the pressure P is related to the amplitude A and the angular frequency ω by P = A·Z ·ω, with Z = ρc referring to the acoustic impedance. The substrate displacement can be considered as one half oscillation of such a harmonic source with A = 2.4 nm and ω = ^π_τ, which gives a maximum pressure Pmax ≈ 1 MPa. If the liquid layer thickness is small though, the pressure is generated by the inertia forces of the deposited liquid F =mIP Aa, which leads to Pmax =ρhamax. With regard to the following experiments, this scenario is the relevant one. For a typical isopropanol layer thickness of 100 nm, a maximum pressure of Pmax ≈ 9 kPa is estimated. In view of significantly larger vapor pressure changes within a few Kelvin (Appendix 10.1) and a nearly pressure independent spinodal temperature Fig. 2.6, a substantial influence of the thermoelastically generated pressure in the liquid on the phase transition can be excluded.

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3.1 Superheating Experiments

The first experimental investigations of superheated liquids were reported by Du- four in 1861 [70, 71]. In his experiments he slowly heated water droplets in a pool of oil and observed superheating temperatures up to 448 K. This method of using two immiscible liquids inhibits heterogeneous and inhomogeneous nucleation, which usually occur at solid-liquid interfaces. Later on the principle was implemented in the bubble-column apparatus [72, 73], which was employed in a large number of superheating studies. In this setup a droplet of the test liquid is inserted at the bottom of a vertical column filled with a host liquid. The host liquid is immiscible with the test liquid, has a significantly higher boiling temperature and a higher density. A constant temperature gradient from the bottom to the top of the column is generated for example by increasingly closely spaced heating wires. Therefore, the droplet of the test liquid is quasistatically heated as it slowly ascends due to buoyancy. When the droplet reaches a certain height, it evaporates explosively. From the position in the moment of the phase transition the limit of superheating can be determined with an accuracy of about 1 K. The observed maximum superheating temperatures were in excellent agreement with the predictions of the CNT for homogeneous nucleation. The only exception in this respect was water for which the highest experimentally found temperature was about 20 K below the calculated value [18]. Nevertheless, the results indicate that the CNT provides a convenient description of the phase transition under quasi-equilibrium conditions despite of its simplifications, such as just considering density fluctuations that are spherical bubbles and using macroscopic thermodynamic quantities to describe nanoscopic gas seeds. The maximum superheating temperatures are generally close to the spinodal estimated from the Berthelot equation of state (Fig. 2.6), which holds as an upper limit though.

Another aspect which has been studied with the bubble-column setup is the

Laser-Induced Liquid-Vapor Phase Transitions in Thin Films at Solid Surfaces : Nanomechanical Investigations on the Sub-Nanosecond Scale