Evaporation behavior of a thinning liquid film in a spin coating setup: Comparison between calculation and experiment

We present and analyze comprehensive measurements of the evaporation behavior, E, of a thinning liquid film during a hydrodynamic‐evaporative spin coating experiment. E, ω (the rotation speed), and ν (the liquid viscosity) are the main control parameters of the process. The entire film thinning process can be described theoretically quite well if these parameters are known. Values of ν are easily accessible in advance (calculations, literature values, measurements). Values for E can essentially not be found in the literature. They are hard to measure and specific for the experimental conditions. There is also no generally accepted strategy to calculate E. Our experimental results are compared with a theoretical prediction for E based on ideas by Bornside, Macosco, and Scriven, which were presented long ago. Their approach was never tested experimentally. Theory and experiment agree well for many solvents and different ω . This approach permits in advance the quantitative calculation of the evolution of the entire hydrodynamic‐evaporative film thinning process. We also derive a general formula to predict ab initio, with literature data only, the amount of final deposit (film thickness) of solute in the case of spin coating mixtures of volatile solvents and nonvolatile solutes.

The thinning of the liquid film and the spatio-temporal evolution of the solute/solvent composition during hydrodynamic-evaporative spin coating have been analyzed in some detail. [10][11][12][13][14] Meanwhile the process is understood quite well [15][16][17] and if the physico-chemical process parameters are known, in many cases, the final solute coverage can be predicted. In fact, aside from the evaporation rate, all the other relevant spin coating parameters can be obtained rather easily, for example, from literature or from independent measurements. For instance, the density and viscosity of the solvent/solute mixture can be measured or obtained (calculated) from literature data. The rotation speed can be adjusted.
Alas, reliable values for evaporation rates under spin coating conditions cannot be found in the literature. They are considered to be rather specific for the experimental conditions (geometry of the spin coating setup, rotation speed, etc.). Therefore, it has been suggested to measure the evaporation rate of a specific solvent for a specific spin coating configuration. 18,19 This evaporation rate can then be used to calculate the outcome of spin coating processes with the same setup and solvent, but otherwise different process parameters. The relevant evaporation rate can be derived for instance, from the final solute coverage resulting from a "calibration" spin coating experiment. 2,16 This approach is useful. Nevertheless, it is still desirable to know/calculate the evaporation rate in advance without the necessity of a "calibration" experiment. Thus, one would gain for instance substantial flexibility to pre-select suitable solvents to achieve a desired spin coating result.
Already some time ago, Bornside, Macosco, and Scriven (abbreviated in the following as "BMS") proposed how to calculate the evaporation behavior of volatile liquids in a spin coating configuration. 7,20,21 For the calculation, only readily accessible literature and process parameter data are necessary. Up to now, this theoretical approach has never been tested/confirmed thoroughly by experiment. Presumably this is the case, because (1) reliable experimental evaporation data were not available and (2) a concise theoretical description of the spin coating process was not existing until recently. Indeed, evaporative film thinning in spin coating configurations has been measured. 14,[22][23][24][25] However, until recently, it was not clear how meaningful these data were. For instance, it has been discussed whether there exists something like a "simple" evaporation rate. 26,27 It was unclear how much the increase of the solute concentration during evaporative film thinning might affect the evaporation rate itself (e.g., via a "skin formation" 28 ) and thus modify the film thinning process and in particular the resulting final solute deposition. 29 Also, the impact of film dewetting (hole formation) during film thinning (in particular in the late stages of film drying) was not known. 30 With on-line imaging of the film thinning during the spin coating process, it is possible to investigate the film thinning behavior with high precision. 17,30 We applied this experimental approach and in the following we will present precise experimental data on the evaporation behavior derived from the film thinning. The results will be analyzed in view of the predictions of the BMS approach. We will address in particular whether the BMS theory can be used to quantitatively describe the evaporative contribution to the liquid film thinning in a spin coating process. In addition we will discuss in which cases it is possible to apply this approach for solvent/solute mixtures. Last not least we will show how the new insights can be used to predict the final solute coverage, that is, the main purpose of most spin coating processes.

Substrates
As substrates served silicon wafer pieces of ≈ 2 cm × 2 cm with natural oxide surfaces (oxide layer thickness ≈ 2 nm) or with artificial oxide layers of (50 ± 1) nm thickness. In both cases the surface roughness was ≈0.5 nm.

Substrate surface preparation
The substrates were first cleaned in an ultrasonic bath by a sequence of immersions (for 10 minutes each) in: (1) de-ionized water, (2) ethanol, (3) acetone, (4) ethanol, and (5) de-ionized water. In a second cleaning step, they were immersed for 30 minutes in piranha solution (H 2 O 2 (35%)/H 2 O(65%) and H 2 SO 4 (96%), 1:3 volume ratio). Finally they were again immersed and sonicated for 10 minutes in de-ionized H 2 O and stored therein. Just before use they were dried by blowing with dry N 2 (purity: 5.0).

Optical imaging
For the in-situ observation of the spin coating process, a modified optical microscope (Axio Scope A1 from Zeiss) was used. The light source was a blue diode laser (6 W, 445 nm, LDM-445-6000, LASERTACK, de-speckled by a combination of liquid light guide and a rotational diffusor). Microscopy was performed from the top in interference enhanced reflection mode. 31 A high speed monochromatic camera (1000 fps) and suitable image triggering and processing provided single frames during the film thinning. 32 Interferometric data (brightness variations during film thinning) revealed the film thinning behavior 33 (for more details, see also Supplement).

Spin coating
The home-made spin coater allowed a precise adjustment of the rotation speed. The substrate holder was a planar, round Teflon disk with a diameter of 3 cm. It had a small hole in the center, where a negative pressure could be applied to fix the substrates. The combination of spin coater and microscope was not encased in a closed box. Instead, the rotating sample surface was exposed to the normal laboratory air convection environment. There was no artificial, extra air flow directed towards the substrate surface. There was also no strong environmental air flow, for example, due to air conditioning or a chemical hood. The microscope and the spin coater were arranged as sketched in Figure 1.
In the experiment, a drop (0.2 mL) of the liquid is placed on the solid, planar substrate, which is already rotating at a constant speed, . After its deposition, the liquid drop rapidly forms a planar film due to the combination of centripetal and viscous forces. 3 Due to outward liquid flow and/or spin off at the substrate perimeter, the height, h, of this liquid film is continuously decreasing (h = h(t)). With volatile liquids, evaporation also contributes to film thinning. Because viscous shear forces increase rapidly if the film gets thinner evaporative liquid losses increasingly dominate film thinning. For sufficiently thin films, the outward liquid flow practically ceases and film thinning only occurs due to evaporation. This range of essentially purely evaporative film thinning reveals the evaporation rate, E, [14][15][16]18 with a linear decrease of the film thickness: configuration. The sample surface is imaged via reflection microscopy, that is, with monochrome illumination from the top (through the lens). The interference of the light reflected from the surface of the continuously thinning liquid film and from the interface between the film and the substrate surface modulates the intensity of the reflected light as indicated. The time evolution of the film thickness can be derived from these intensity modulations with great precision. More technical details on the experimental setup (synchronization of the frame rate with the sample rotation, background subtraction, etc.) can be found in Reference 30 and in the Supplement E can be determined with great precision with the setup depicted in Figure 1 in combination with Equation (1). In the observed film thinning behavior, the range of purely evaporative thinning can be identified easily, because the transition between the (nonlinear) thinning driven by lateral liquid flow and the linear thinning dominated by evaporation is rather well defined and can be measured experimentally. The transition occurs at (and defines) the so-called transition film height, h tr . The theoretical analysis of hydrodynamic-evaporative film thinning 15,34 shows that h tr is related to E, (viscosity), and (rotational velocity) according to: The transition height, h tr , is the key process parameter, because together with the solute concentration it can be used to calculate the final deposit of the solute (see the Discussion section).
As mentioned above, the evaporation rate, E, can be determined directly from the linear section of the thinning curve as described above by Equation (1). For this publication, we derived E by fitting the entire thinning curve with the theoretical model presented by Karpitschka et al. 15 This zero-order approach assumes that E and remain constant during the whole process. Detailed theoretical and experimental investigations show that this approach describes the thinning curve with excellent precision for the experimental conditions applied here (pure solvents and mixtures with low solute concentrations). In particular, it effortless yields precise values for E. In any case, fitting the complete curve results within the measurement errors in the same values for E as those obtained directly from the slope of the thinning curve in the linear range.

THEORETICAL CALCULATION OF E
Some time ago, Bornside, Macosko, and Scriven presented a theoretical description of the steady state evaporation behavior of thin volatile films in a spin coating configuration. 7,20,21,35 They predict an evaporation rate, E, with * : Here, solv is the solvent density in its liquid phase and vap is its density in the vapor phase. x solv and x solv, ∞ are the solvent mass fractions in the liquid phase and faraway from the liquid in the vapor phase, respectively. k is the mass transfer coefficient.
Assuming Raoult's law to describe the vapor-liquid equilibrium of the solvent and the ideal gas law to estimate its density, 7,20 Equation (3) can be written as: P * solv is the vapor pressure of the pure solvent and M solv is its molecular weight. R g is the ideal gas constant and T is the temperature.
According to Bornside, Macosco, and Scriven for a planar plate rotating at speed , the mass transfer coefficient k can be replaced leading to: The constant c is a function of the Schmidt number Sc with c = 0.386 ⋅ Sc 0.462 and Sc = air ∕D solv,air . 6 D solv, air is the diffusion coefficient of the solvent (solv) in air (air) and air is the kinematic viscosity of air. * In contrast to the theoretical approach taken in this report, which starts with pure solvents and modifies the findings to solutions, BMS focused in their theoretical description right away on the evaporation rates of polymer solutions. They assumed, for instance, nonlinear changes of the viscosity and of the partial pressure of the solvent during the process. Their approach is complicated and not easily comprehensible. This limits its value, in particular for users, who want to apply spin coating without caring about its physico-chemical background. In hindsight, the method applied by BMS is absolutely apprehensible. Detailed experimental data about the film thinning were barely available at their time, not to mention a consistent theory on spin coating, which does exist meanwhile. 15 With the approximation c = 0.386 ⋅ Sc 0.462 ≈ 0.4 ⋅ ( air ∕D solv,air ) 1∕2 , Equation (5) can be simplified to: For pure solvents x solv = 1 and x solv, ∞ = 0, because the solvent concentration in the air far away from the evaporating surface can be neglected. Therefore (x solv − x solv, ∞ ) = 1. Accordingly, Equation (6) can be simplified further: This is an important result, because all the data necessary to calculate E via Equation (7) are either available from literature or given by the experimental conditions.  F I G U R E 3 Experimentally measured evaporation rates, E, as function of the square root of the sample rotation, 1∕2 . Data are shown for pure n-nonane (NON), pure toluene (TOL), and for various solutions of PS-b-PMMA (MW:55k-b-22k) in toluene (the polymer concentrations are given in mass percentage). The dashed lines show linear fits with slopes indicating the speed-independent evaporation rates e TOL as defined by Equation (8) The region of film thinning dominated by evaporation can easily be identified in all cases. It is the range where the film height decreases linearly, ending with the bare substrate (h=0) for the pure solvents. In the case with the solution containing a nonvolatile component, the decrease ends with a final deposit of film thickness, h(t → ∞) = h f . The solid lines show the slope of the purely evaporative region with dh/dt = −E according to Equation (1). The interpretation of the data with respect to the evaporation behavior is corroborated by the analysis of the entire thinning curve. The dashed lines are theoretical fits to the experimental data according to the scenario of hydrodynamic-evaporative film thinning as described in the supplement and in the literature. 15 In short, this scenario takes into account both, hydrodynamic and evaporative thinning. It assumes ("zero-order theory") only a volatile liquid and neglects the impact of the solute concentration (and changes thereof during the film thinning) on the evaporation behavior. Despite this supposedly rather crude assumption, the agreement between experiment and theory remarkably is good (this is the case even for higher order corrections see References 16 and 17).

Evaporation rates for different rotation speeds
For a better comparison of the data at various rotation speeds, it is convenient to re-scale the evaporation rates and to define a speed-independent evaporation rate, e, by rewriting Equation (7) accordingly: Figure 4 presents evaporation data for many different pure solvents. The data are plotted according to Equation (9) with the physico-chemical data for the different systems obtained from literature (see the Appendix). Motivated by Equation (9), the dashed straight line in Figure 4 has a slope of 0.4/(R g T) with T = 26 • C, the experimentally applied temperature. † For simplicity it is assumed for the fits that E( = 0) = 0. The experimental data are not accurate enough to conclude/predict with sufficient significance by extrapolation the (small) evaporation rates for E( → 0).

F I G U R E 4 Experimentally measured speed-independent evaporation
rates, e, for various solvents as function of the pure solvent properties D ab P * M a ∕ L (see Appendix, Table A1). The dashed line shows a linear fit (Pearson coefficient = 0.996) through the origin. Its slope is 1.6 × 10 −4 mol/J

General remarks
Evaporation rates in a spin coating configuration, E( ), were measured for different rotation speeds, different solvents, and various solvent/solute mixtures by direct observation of the film thinning. The measurements via on-line imaging show that the evaporating liquids form closed and planar films throughout the entire film thinning process. Thus it is assured that the evaporative film thinning agrees with the scenario assumed for a typical spin coating process as it is described by the corresponding theory. 15 It is observed that the evaporation rates are proportional to 1∕2 . This behavior and the absolute numbers measured for E are in good quantitative agreement with a theoretical prediction published some time ago by Bornside et al. 7,20 This means that E can be calculated directly from literature data of the solvent properties and from readily available process parameters. More so, because all the other process parameters are also accessible in advance, it means that it is possible to calculate ab initio the entire film thinning in the course of the spin coating process including its result, the deposition of the solute.

Validity and limitations of the approach
Quite remarkable, the technical specifications of the spin coating setup, such as for instance the size/diameter of the substrate, are not taken into account by the BMS approach. It is rather unlikely that the technical conditions of our setup just by chance lead to the observed quantitative agreement between theory and experiment. Our conditions (see Materials and Methods section) were: (1) sample exposed to a normal laboratory air environment without artificial air flow; (2) substrate dimensions of a few cm in diameter; and (3) deposition of an excess volume of the liquid on the sample already rotating at a fixed speed. It can be assumed, that under similar spin coating conditions the evaporation rates can also be calculated quite accurately via the BMS approach. Which aspects of different technical implementations of spin coating setups may or may not have an influence on the validity of the presented calculation of E can be guessed from the line of argumentation following Equations (3) through (7). According to Equation (3), the evaporation rate depends on the difference of the vapour pressures, x solv − x solv, ∞ , and on the transfer coefficient, k. For pure solvents, x solv = 1. This assumption still holds with good approximation (i.e., error ≤10%) also for solutions, as long as the mass concentration of the solute is ≤10%. 16,17 The assumption of x solv, ∞ = 0 is also reasonable if the environmental vapor pressure of the solvent is low. The content of the first round bracket of Equation (5) means that k depends on the air flow conditions close to the substrate surface created by the process itself (sample rotation). With rotational speeds typically applied in spin coating (e.g., from 500 to 3000 rpm), this means induced air flow speeds near the sample surface in the range of 1 m/s and more (= perimeter rotational speeds for sample sizes of a few centimeters at 10 rps). The size of the sample does not appear in Equations (3) through (7) because its impact on k is generally considered rather weak. [36][37][38] In agreement with our results, this means that the presented calculation of E is reasonable, if the spin coating setup is not placed within a closed compartment of small (vapour) volume (i.e., the environmental solvent vapor pressure, x solv, ∞ remains small). The calculation is also applicable, if there is no external air flow exceeding speeds of 0.1-0.2 m/s 39 and aiming directly at the substrate surface. This means flow from air conditioning or a chemical hood may or may not cause a problem. It depends on where exactly the spin coater is positioned (for instance, placed within the hood closer to the rear corner is a suitable solution, whereas a location in front next to a nearly closed door may cause a problem). Sample sizes between 1 and 10 cm in diameter should be appropriate.
With liquid mixtures of two or more different volatile components, the evaporation behavior and thus the film thinning behavior may not be described correctly by assuming a (constant) evaporation rate calculated with (the approach leading to) Equation (7). Even if the calculation of E with an appropriately modified (for mixtures) Equation (7) yields a correct evaporation rate for thick planar films, Marangoni effects may influence the thinning behavior of thin films. They may cause surface undulations in the late stages of film thinning and thus, for instance, film rupture. Last not least, miscibility gaps may affect E and .
Another aspect has to be considered for solutions of volatile solvents and nonvolatile solutes. Even with small solute concentrations, the viscosity of the solution can be very different to the viscosity of the pure solvent. In this case, the film thinning process can be described quantitatively quite well by taking into account the viscosity of the solution rather than the solvent. 16,17 The result will be correct if the true initial viscosity (of the solution), , and the applied , is taken to calculate/predict the film thinning behavior. For the evaporation rate, it is sufficient to take an E, which is calculated with Equation (7) by assuming the pure solvent. In this case E of the pure solvent is acceptable, because the evaporation rates of pure solvent and solution are quite similar as long as the solute concentration does not exceed 10-20% due to evaporative enrichment. Of course, in the very late stages of film thinning the deviation between experiment and calculation becomes significant, because solute deposition is not explicitly taken into account in modeling the thinning of the film in the zero-order approach. Nevertheless, the amount of solute deposition can be calculated, because it is determined by the conditions at the early stages of the hydrodynamic-evaporative film thinning. The evaporation rate can still be calculated with reasonable precision assuming the pure solvent properties. If the (measurable) true initial solute viscosity is taken into account, it is possible to quite accurately predict the amount of final solute as will be shown in the following.

Estimation of the solute deposition
The final solute film thickness can be calculated with the spin coating parameters via the transition height, h tr (see Equation (2)) as follows: [15][16][17] This equation assumes identical densities of the solvent and the solute and x 0 denotes the volume ratio between solute and solvent in the initial (weighing in) solution.
With different densities of solvent, solv , and solute, solute , Equation (10) has to be modified: If the solute concentration is given as c 0 , that is, as an entity per volume (e.g., moles or particles per volume), deposition leads to a final coverage of the solute in entity per area, Γ 0 : It should be noted that solute denotes the average density of the solute. This is important, for instance, in the case of metal nanoparticles coated with a shell of organic molecules (see example in the Supplement). In this case solute results from the weight of the particle core plus its shell divided by the volume of the entire particle (core and shell).

5.4
The final "master" equation The combination of Equation (11) or Equation (12) with Equation (7) yields a general "master" formula to predict ab initio the final film thickness: ( 13) or the final coverage: The assumptions leading to Equations (11) and (12) indicate an accuracy of about ±10%. Assuming similar margins for Equation (7) based on the experimental results presented in Figure 4, suggests an accuracy of typically ±(20)% or better for the final master Equations (13) and (14). This is consistent with the experimental data of Figures 3 and 4.
In the Supplement, we present two examples, which confirm our assumptions regarding the accuracy of the ab initio predictions made by Equations (13) and (14). We present predictions based on cases of experiments performed by other groups concerning: (1) nanoparticle monolayer deposition 40 and (2) deposition of a do you have any idea precursor for in situ polymerized films. 41 In the examples, we take the relevant spin coating data from the information given in the publications. ‡ Together with these data, literature data, and Equations (13) (or 14), we can reproduce the described experimental findings quite well.

SUMMARY AND CONCLUSION
Hydrodynamic-evaporative spin coating is the process of the evolution of a planar, continuously thinning film of a (partially) volatile liquid on a rotating substrate. Recently this process has successfully been analyzed quantitatively and described theoretically. The film thinning behavior is essentially determined by three control parameters: (1) the evaporation rate, E, (2) the liquid viscosity, , and (3) the rotation speed, . If the liquid is a solution of a volatile solvent and a nonvolatile solute, a final deposit of the solute remains after evaporation of the solvent. The amount of final deposit can be calculated from a fourth parameter, x 0 , the solute concentration of the solution at the beginning of the process. Three of these parameters, , , and x 0 , are readily available prior to the spin coating process. They can either be adjusted, easily measured, or obtained (calculated) from literature data. Values for E, on the other hand, are not readily available for several reasons. It is not easy to measure E under different spin coating conditions. Prior to the recent progress in understanding the process theoretically in depth, the complicated film thinning process with its time-dependent combination of evaporative and hydrodynamic contributions suggested that E changes during the process. E was also considered to be rather specific for the experimental setup. Not the least because of these reasons, there was no generally accepted approach on how to calculate E.
Yet, already some time ago Bornside, Macosco, and Scriven suggested a way to calculate E. Up to now, their proposal has never been adapted appropriately and tested by experiment. We measured evaporation rates, E, of many different volatile solvents and mixtures of volatile solvents and non-volatile solutes under different spin coating conditions and analyze them in view of the proposal by Bornside, Macosco, and Scriven. We find that their theoretical calculation of E agrees remarkably well with our measured E for many different solvents and spin coating conditions. We combined their approach with the recently available theoretical description of spin coating. Thus the entire spin coating process can be calculated/modeled quantitatively in advance. This is of great practical importance, because in the case of spin coating solutions containing nonvolatile solutes, this solute is deposited in the course of the process. In most cases, this deposition of the solute is the main purpose of the spin coating process. Not the least, this is because through the variation of the ‡ Many spin coating experiments are described in the literature, mostly to prepare films of specific thicknesses, composition, and so on, to be used for certain applications. However, it is very hard to find publications, which contain all the relevant parameters necessary to unambiguously reproduce the experiments (meaning: which completely specify the process as we define and investigated it in previous publications). process parameters the amount of solute deposition can be controlled and adjusted within wide margins. We present a "master" formula to calculate in advance the final solute deposit. This ab initio calculation is solely based on data readily available prior to the process. Thus the desired solute coverage can be optimized/adjusted in advance.

APPENDIX A. PHYSICO-CHEMICAL PROPERTIES OF THE SOLVENTS
See Table A1.

APPENDIX B. SPEED-INDEPENDENT EVAPORATION RATES OF THE SOLVENTS
See Table B1. Note: The measured evaporation rates, e, are also presented in Figure 4 of the main text. The theoretical evaporation rates, e th , are calculated using Equation (8) (see below and main text) with the physico-chemical properties given by Table A1 and