REPELLENT SYSTEM S IN LIFSHITZ VAN DER WAALS MODEL By LONG MA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2017
2017 Long Ma
Thanks for my parents and friends who helped me make it through
4 ACKNOWLEDGMENTS I thank my parents for their endless encouragement and positivity, my friends both short and long distance for helping me through. Finally, I thank my advisor Dr. Sigmund, and the Sigmund group at large for their efforts regarding this w ork and for continuing to teach me new things.
5 TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 7 LIST OF FIGURE S ................................ ................................ ................................ ......................... 8 LIST OF ABBREVIATIONS ................................ ................................ ................................ ........ 10 ABSTRACT ................................ ................................ ................................ ................................ ... 11 CHAPTER 1 INTRODUCTORY ................................ ................................ ................................ ................ 12 2 VAN DER WAALS FORCES AND LIFSHITZ MODEL ................................ ................... 16 Background ................................ ................................ ................................ ............................ 16 The Development of the van der Waals Theory ................................ .............................. 16 Catalog of the vdW Forces ................................ ................................ .............................. 18 va n der Waals Forces in Quantum Mechanics ................................ ................................ 19 Modern View of the vdW Interaction ................................ ................................ ............. 20 The Lifshitz Van Der Waals (LW) Model ................................ ................................ ............ 23 Comparison between the Lifshitz van der Waals Forces and the Traditional ................................ ................................ ................................ ..... 23 The Parallel Plate Model of the Lifshitz van der Waals Interaction ............................... 24 Hamaker Constant in Lifshitz Model and Its Physical Perspective ................................ 25 Dielectric Response Function ................................ ................................ ................................ 27 The Ninham Response Function .............. 28 Cauchy Plot and the Relationship between the Dielectric Function and the Refraction index ................................ ................................ ................................ ........... 30 The Attraction Repulsion Transition In Silica Alkanes Cellulose Systems ......................... 32 The Assumptions for the LW Model in Silica Alkane Cellulose System ...................... 34 Optical Data Collection for the Materials in Silica Alkane Cellulose System ............... 35 The Dielectr ic Response Functions in Silica Alkane Cellulose Systems ....................... 35 The Influence of the Separation Distance on Lifshitz van der Waals Int eraction .......... 40 Conclusion ................................ ................................ ................................ ....................... 42 The Design Of Repellent Systems For Martian Dust ................................ ............................ 42 Limits of the Parallel Plate Lifshitz Model in Application ................................ ................... 44 3 DISCUSSION AND SUMMARY ................................ ................................ ......................... 57 APPENDIX A REPULSIVE VAN DER WAALS SYSTEMS IN LIQUID AND GASEOUS PHASES .... 59
6 B MULTIPLE LAYERS IN LIFSHITZ VAN DER WAALS MODEL ................................ ... 63 LIST OF REFERENCES ................................ ................................ ................................ ............... 66 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ......... 70
7 LIST OF TABLES Table P age 2 1 T he Cauchy plot data for nitrogen ga s in 15C ................................ ................................ .. 46 2 2 The NP constants derived from the Cauchy plot ................................ ................................ 47 2 3 The r efra ction index data for silica ................................ ................................ .................... 47 2 4 The NP constants derived from the Cauchy plot ................................ ................................ 48 2 5 T he C UV C IR UV IR 0 data collection for n alkanes ................................ .............. 49 2 6 The average optical constants of cellulose derived from Cauchy plots ............................. 50 2 7 The calculated Hamaker constants and Lifshitz van der Waals forces in silica alkane cellulose particle systems with equal separation distance. ................................ ................ 51 2 8 The NP constants for the material s of silicon, silica and the magnetite. ............................ 54 2 9 The optical constants collection for the materials in Martian dust repelling systems ........ 54 2 10 The calculated Hamaker constants A in alumina silica iron compounds systems ............. 56 A 1 The NP constants for the material of silica, diiodomethane and the magnetite. ................ 61 A 2 The NP constants for the materials of silicon, silica and the air. ................................ ........ 61
8 LIST OF FIGURES Figure page 2 1 The Lifshitz van der Waals model for the interaction between two semi infinite parallel plates across the media c. The separation distance is l. ................................ ........ 46 2 2 The Cauchy plot of the N 2 in 15C. ................................ ................................ .................... 47 2 3 The Cauchy plot of the A silica ................................ ................................ .......................... 48 2 4 The Cauchy plot for n alkanes. The numbers next to the lines represent the number of carbon in the chains. copyright 2017 Elsevier ................................ ............................... 49 2 5 The Cauchy plot of cellulose at various thicknesses. Triangle represents 100 layers. The white circle represents 120 layers an d the solid circle represents 140 layers. copyright 2017 Springer ................................ ................................ ................................ 50 2 6 The dielectric response function on the real frequency for the silica alkane cellulose particle system. The green cross mark represents the res ponse function of silica substrate. The red cube represents the response function of cellulose particle. The dark blue, light blue, red, purple and yellow plus signs represent the decane (C10), dodecane (C12), tetradecane (C14), hexadecane (C16) and the oc tane (C8) intervening coating. ................................ ................................ ................................ ............ 51 2 7 The calculated Hamaker constants in silica alkane cellulose systems. .............................. 52 2 8 The dielectric response function with respect to the real frequency for the silica alkane cellulose particle system with sqrab line involved. The green cross represents the response function of silica substrate. The red square represents the response function of cellulose particle. The dark blue, light blue, red, purple and yellow plus signs represent the decane (C10), dodecane (C12), tetradecane (C14), hexadecane (C16) and the octane (C8) intervening coating. The cyan line represents the sqrab curve (sqr ab) in the system. ................................ ................................ .............................. 52 2 9 The calculated van der Waals forces in silica X cellulose systems with fixing the separation distance as 1nm. The black cube represents the forces in silica alkane cellulose systems. The red sphere represents the maximum repulsion in silica X cell ulose system. ................................ ................................ ................................ ................ 53 2 10 The calculated Lifshitz van der Waals forces in silica alkane cellulose system with varied separation distances. The black cube represents the silica octane cellulose system. The red sphere represen ts the silica decane cellulose system. The blue triangle represents the silica dodecane cellulose system. The green triangle represents the silica tetradecane cellulose system. The purple triangle represents the silica hexadecane cellulose system ................................ ................................ ................... 53
9 2 11 The dielectric response functions of the materials in SSM system with respect to the frequency. The light green circle represents silicon. The green cross represents the silica. The purple diamond represents the magnetite and the cyan triangle represents sqrab curve in the system ................................ ................................ ................................ ... 54 2 12 The dielectric response functions of the materials in silica alumina magnetite system with respect to the frequency. The orange circle represents alumin a. The green cross represents the silica. The purple diamond represents the magnetite and the cyan curve represents sqrab curve in the system ................................ ................................ ........ 55 2 13 The dielectric response functions of the materials in the silica alumi na magnetite system with respect to the frequency. The orange circle represents alumina. The green cross represents the silica. The dark red diamond represents the maghemite and the cyan curve represents sqrab curve in the system ................................ ......................... 55 2 14 The dielectric response functions of the materials in the silica alumina magnetite system with respect to the frequency. The orange circle represents alumina. The green cross represents the silica. The dark diamond represents the hematite and the cyan curve represents sqrab curve in the system ................................ ............................... 56 A 1 The dielectric response functions of the materials in SDM system with respect to the frequency. The green cross represents silica. The yellow star represents the diiodomethane. The purple diamond represents the magnetite and the cyan curve represents sqrab curve in the system ................................ ................................ .................. 61 A 2 The dielectric response functions of the materials in SSA system with respect to the frequency. The light green circle represents silicon. The green cross represents the silica. The red dot represents the air and the cya n curve represents sqrab curve in the system ................................ ................................ ................................ ................................ 62 B 1 The parallel plates model for four layer lifshitz van der Waals interaction. B is the substrate material. B1 is the extrinsic coating on the substrate of B. C is the contacting material. m is the intervening medium, l is the separation distance and d is the thickness of the coating. ................................ ................................ ............................... 65
10 LIST OF ABBREVIATIONS vdW force van der Wa als force LW model Lifshitz van der Waals model NP Ninham and Parsegian SP SAM SA s qrab SDM SSA Sphere plane Self assembled monolayer Self assembled Square root ab Silica diiodomethane magnetite Silicon silica air
11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science REPELLENT SYSTEM S IN LIFSHITZ VAN DER WAALS MODEL By Long Ma May 2017 Chair: Wolfgang Michael Sigmund Major: Material s Science and Engineering D ust repellent surfaces reduce the use of detergents or water and tend to have low adhesion to stains. To date, significant breakt hroughs have been accomplished such as the dust repellent elec trostatic shield and ion beam bombarded quartz or Teflon coating. H o wever, the broad introduction to market has not been achieved due to poor long term performance under environmental conditions and the mystery of the fundamental mechanism Controlled nanoscale variations in dielectric properties are known to change adhesi ve forces. Therefore, Lifshitz theory is used in this work to calculate systems with minimized attractive forces or even repulsion. Such systems may have two advantages. First, they do not require additional electrical energy to provide dust repellency ; an d second, they may be even better performing than current systems. This thesis present s the fundamental calculation of the van der Waals force in the parallel plate model and the simplification of the formula by applying the optical constants of the materi als Additional discussion covers the reason of attraction repulsion transition in the silica alkanes cellulose system on the variation in the separation distance and the adjustment of the dielectric response function Furthermore the derivation of the theoretically maximum repulsion in designed systems is addressed. In the end, the setup s of repulsive sys tems to re pel most common compounds found in Martian dust will be exhibited.
12 CHAPTER 1 INTRODUCTORY Dust is annoying and p eople never stop battling wi th dust since it trigge rs troubles in numerous aspects. D ust dropping on the furniture and clothes may cause allerg ic reaction s once people touch or inhale it. Tenacious dust on the wind shiel d lead s to reduced visibility while driving and in combination wi th intense light scattering from a setting or rising sun or oncoming traffic may trigger accidents In the semiconductor industry, dust is avoided by using clean rooms because dust particles in the process degrade s the performance of the products severely. When manufacturing the exquisite digital camera, the designers always consider how to make the lens to better screen off the dust to enhance the quality of photos Furth ermore, dust is not only annoying us in our dai ly life but also hindering from stepping forward i n some significant projects. To upgrade the countries energy supply from unsustainable source s to a renewable one, the Qatar gov ernment spent a plenty of budget on solar farms to collect solar energy. However, because of the rugged enviro nment, dust storm s occur regularly and each time would bring layers of dust onto the solar board impacting the photovoltaic performance and cut ting down the irradiance.  Likewise, wh en astronauts or machines explore new extraterr estr ial land the y are threatened by dust clouds and dust devils that lift the regolith of the moon and plane ts surfaces [2 ] Regolith is angular and abrasive so that it may tunnel in and clog sensitive devices, damage spacesuit s. It is also possible to affect the accuracy of data collection such as the light absorptions and emittance due to scattering S everal methods have been suggested to mitigate the dust hazard. developed an electrostatic shield that can repel the charged dust by applying the electri c force [ 3 ] However, since the shielding system re lied on the electric field and the building structure in particular accuracy to a great area with uneven
13 asperity. Besides, a nother solution to removing the dust comes to the super hydrophobic surface s On such surfaces, water forms spherical droplet once it was dropped. The mobility of the water droplet can carry the dust particle s off the surfaces through rinsing [4 ] However, sin ce such surfaces usually have hairy structures, which are vulnerable to physical abrasion or contact. The performance severely degrades due to the weak durability. the method of using the super hydrophobic coatings to clean the dust is not the optimal choi ce To compete these suggestions and minimize the dust adhesion system that intrinsically maintains the repulsion so that the dust is easier to remove T he fundamental mechanism behind this repulsion lies in the interaction of the surface forces Since d ust adheres to the surface mainl y through three kinds of surface forces -----the capillary force, the electrostatic force as well as the van der Waals fo rce, unveiling the secret of these influences promotes the development of the application of the dust repellent surfaces. Capillary force is formed between particle and surface bridged through the fluid interface. It often plays an essential role in particle adhesion, antifoaming and the fabrication of the colloid self assembled structures. [ 5 ] However, since the dust of interest is generally the pollen dust or the extraterrestrial dust, there is less probability that capillary affection overshadows the contribution of other forces in that water is scarcely existing in these cases Furthermore, capillary forces have been well described in the literature and therefore will not be further discussed in this thesis.    Electrostatic force or the Coulombic force is acting when charged surfaces and the dust particles approach T he planet dust is expected to be electrostatically charged because of the solar UV radiation and the exposure to the solar wind. Dust shield invented by NASA gave rise to a new path for dust mitigation by using the electrostatic interaction. [ 9 ] However, d espite the
14 significance of the Coulombic force in the dust adhesion area, there is no secret about this force after it was observed. The physical fundament in this force has been fully investigated through generations of scientists.   [12 ] Thus, th is force, despite of its potential function, is not the factor that will be controlled for the construction of the new version of the repellent system. In nature, there is an interesting finding that the gecko can lift its body even only on the adhesion o f a single toe and recently, it has been proved that capillary force. [ 13 ] [ 14 ] [ 15 to hook up the surface, they can manage their bodies on the surfaces in the extremely slippery condition. [ 16 ] [ 17 ] The secret of this strong adhesive forces were the van der Waals force. B number of the aligned spatula. W hen they press their feet onto the smooth surfaces, each end of the tiny setae will closely approach the surface, enabling an intensive van der Waals attractive force. Thus, with accumulating the total force provided by all the spatula, the outcome is dramatically disti nguished. Inspired by that, the production of some specific systems that i ncrease or reduce the adhesion through adjusting van der Waals forces draws great interests. Calum J. Drummond and his group [ 18 ] discovered that the fluorocarbon materials always preserved low surface tension and was possible to create such a non sticking surface induced by Lifshitz van der Waals forces. By calculating the dielectric function and plugging in the number of Hamaker constant in the Lifthitz equation, he revealed that the force of the interaction between the polytetrafluoroethylene and the solid was ideally matching the AFM data of the retarded vdW force. It demonstrated that for a surface which has a very low surface energy, such as PTFE, the force kept repulsive until increasing the separation distance by 20nm. In Bohling, Chris tian
15 report [ 19 ], the T eflon AF solvent alumina adhesion was measured using the AFM contact mode. He concluded that in a solvent of 2 propan ol, 1 propanol, 1 butanol and the 2 butanol, the forces also presented repulsive at the separation distance of 10 nm. This outcome extended The objective of this chapter is first to construc t the proper Lifshitz van der Waals model that can be used for addressing the attraction repulsion transition in silica alkane s cellulose systems. The reason why the repulsion shows up will be discussed. Then, by applying this model, the derivation will gi ve rise to several possible repulsive systems which can be designed for the Martian dust mitigation
16 CHAPTER 2 VAN DER WAALS FORCES AND LIFSHITZ MODEL Background The Development of the van der Waals T heory Dating back to the 19 th century, one of the greatest contributions to the modern physics is the establishment of the Maxwell equations. The Maxwell equations were not just the aggregations of some significant formulations which successfully solved the empirical facts in the electromagnetic physics, it gave us a deep insight into the world of electromagnetic field and let us know where the electromagnetic field comes, what is the relationship between the charges and the field and how the electromagnetic wave propagates, interacts and dissipate in the field. The concept of the field, upon its present, broke through a new way to understand the interaction between the charges. With the evolution of the modern physics, especially after the development of the atom model and quantum physics, people noticed t hat even for a continuous uncharged matter, it can be regarded as a body containing thousands of fluctuating positive and negative particles. Macroscopically, the whole body is neutral and all of the polarization and the charges are cancelled inside the in tegral. However, microscopically, almost all of the charges never stop their dancing. They move wildly and for each microscopic particle, the position and momentum are tric current and transient electromagnetic field. Therefore, when these instant field interactions are collected and react to other groups of charges, the effects are accumulated over time and an average number can evaluate the total impact, at this time, a charged fluctuation force was created, which was denoted as van der Waals force.
17 The origin of the van der Waals was dating back to the moment after James. C. Maxwell distilled the Maxwell equations. In 1894, P. N. Lebedev [ 20 ] first announced his specu lation in his PhD thesis. He thought of the molecules and atoms as microscopic electronics that can send and receive the electromagnetic messages by themselves. At that time, he estimated that it was those electromagnetic signals that could give rise to so me macroscopic interactions but these interactions were not just the consequence of a single molecule or atom, it was definitely including plenty of molecules and atoms behaving the same over the time period. His words are cited here tween two radiating molecules, just as between two vibrators in which electromagnetic oscillations are excited, there exist ponderomotive forces: they are due to Af ter that, in 1937, H. C. Hamaker [ 21 ] [ 22 ] published the famous paper indicating the vdW interaction between macroscopic bodies. Compared with previous works, h e distinctly applied the method in which the force was integrated pairwise. The results of his work presented the interaction energy was inversely proportional to the r 6 where r is the interaction distance. The influence from the material and the intervening media were all merging into the constant term, which is now called the Hamaker constant. Bas mathematically realized in the macroscopic substances. Besides, he provided a novel summation solution to address the vdW forces in varied shapes. In 1940s and 1950s, another evolution of vdW interaction occurr ed as investigating the range vdW forces that acted as the governing power for the stability, especially when the balanced electrostatic interaction was sheltered by the ions solution. Th erefore, Derj aguin combined with other scientists reported
18 the famous DLVO theory and that became the fundamental principle in colloid science afterwards. [ 23 ] In this theory, only electrostatic force and the vdW force were considered and this approxima tion leaded to the energy maximum and a secondary energy minimum in terms of distance from zero to the infinite. The shape of the energy curve, combined with the applied thermal energy kT can thermodynamically illustrate the probability of the particle agg lomeration at certain condition. It also practically demonstrated the feasibility of the vdW theory for the microscopic particles interaction in the colloid system. Catalog of the vdW F orces As discussed, the vdW forces come from the integral interactions of molecular electromagnetic and these waves are generated by the correlated movement of the positive and negative charges, which is, in another word, called the dipole fluctuation. Therefore, the types of vd W interaction can be divided according to the or igins of these fluctuation and what kinds of dipole involved. Firstly, if the dipole in the molecule is permanent and the dipole in the corresponding molecule is also permanent, the vdW interaction between the permanent dipole molecules are called Keesom i nteraction. In the Keesom interaction, the orientation is always fixed so that it prefers to create net attractive formation. Secondly, If the dipole in one molecule is permanent but there is no permanent dipole in another, the neutral molecule will crea te an induced momentary dipole under the influence of the permanent dipole. At this time, the interaction between the permanent dipole and the induced one is called Debye interaction. The orientation of the induced dipole is always opposite to the permanen t one. At last, if both of the molecules do not have permanent dipoles, because of the thermal
19 induced transient dipole and instantaneously this dipole induce s another transient dipole in another molecule. The induced dipole induced dipole interaction is called the London dispersion vdW interaction, which is named after the German American physicist -----Fritz London. Compared with other vdW forces, the London dispersion force is the weakest. However, since this kind of interaction occurs in almost all condensed matters and presents between all chemical groups, the London dispersion force is most likely to be the representation of the vdW forces in most of the c ases. Later in this chapter, when talking about the van der Waals force without dispersion forces. van der Waals F orces in Q uantum Mechanics In 1930, Fritz London conducted the experiment in terms of the nobl e gas atoms. [ 24 ] [ 25 ] [ 26 ] He observed the particles in the dilute system were sparsely distributed and the interaction energy in the system can be simply derived by summing the gas molecules pairwise. He assumed this phenomenon was caused by the coulombi c interaction between two molecules. Thus, to further prove his assumption, he applied the perturbation theory and calculated the theoretical number of that interaction. By comparing two solutions to the gas model, he demonstrated the attraction between no ble gas atom was attributed to the specific molecular dispersion attractive forces which was now called the London dispersion van der Waals forces. perturbation theory in qu antum mechanics, but what is the perturbation theory? Mathematically, the idea of perturbation can start with a simple system, say, in this case, it was the noble gas molecule or atom where the interaction energy can be addressed with known mathematical fo rmula. In the molecule, the nuclei and the electrons are two opposite poles which constitute the primitive dipole. The mathematic solution to the electric field of the dipole is inverse three power proportional to the distance based on the Maxwell equation s. However, this single dipole
20 is not isolated in the system. It is mutually perturbed by other dipole which has the same electric field function, which, in the perturbation theory, is called the correlation function. The solution to the dipole dipole inte raction based on the mutual perturbation can be derived by multiplying the correlation function with considering the Taylor expansion in the 1/R where R is the distance between two centrals of the dipole. The result showed that vdW interaction energy is in versely proportional to the 6 power of the interaction distance and this was also coordinated to the Modern View of the vdW I nteraction paper, there were still doubts about the precision of the equation. One argument stated that despite the assumption considered the dispersion in this situation, the so lution seemed to be nothing to do with the instantaneous dipole fluctuation. Another disagreement went against the application of the vdW dispersion equation. Since it was derived only from the dilute gas model where the interaction between the molecules w condensed materials in which the molecules are more interacted with each other. Therefore, in modern view of the vdW interaction, the investigated materials were no longer the summation of single dipole m olecules, it was treated as an electromagnetic integration and the interaction occurs when the generated electromagnetic fields are influencing each other. The earliest proposal of the modern view of vdW interaction was provided by Casimir in 1948 [ 27 ], w box, the electromagnetic wave can travel without energy dissipation. Assuming the boundary of the box satisfied the specific condition of polarization, say, it was absolutely conductive. when the electromagnetic wave hit the wall, the bounced electromagnetic wave turned to a standing
21 of the energy conservation, each time the electr omagnetic wave hit the boundary, the black body successfully accounted for the singularity in measuring the heat capacity of the space and open the door for the q uantum mechanism. But for Casimir, this model simplified the electromagnetic conductive wall which radiates the standing waves. Casimir also started up with the black box but alternatively, this box had infinite length and height as well as the finite width, which let it contain two infinite parallel walls facing against each other. By integrating the electromagnetic energies and do the derivative of the energy with respect to the interaction distance, the force between the parallel surfaces can be solved. More importantly, based on the solution, even though the temperature went to the absolute zero, the number of the force is stil l not zero. The remarkable result of Casimir demonstrated the existence of the zero point energy. It indicated even at the thermodynamically lowest temperature, the microscopic particles still do not lose their ability to move and vibrate, which is also c uncertainty principle. Based on that concept, any energy absorption for a certain particle is regarded as an additional disturbance on the primary motion. In another word, the energy applied to the system is not the absolute since it should overcome the original charge dancing pattern. Moreover, with this understanding, the absorption frequency for certain material can also be addressed. The frequency that was acce pted by the substance should compare with the nature frequency of the molecular fluctuation so that the wave at this frequency can intensively calm down the roar of the charge dancing river and lower the free energy as largely as possible.
22 Although the Ca simir effect made great contribution to the modern view of electromagnetic interaction, the model still had restrictions in many areas. On one hand, the Casimir effect was only valid in the system with conductive walls and the vacuum intervening media. It could not be applied to the real materials. Besides, the interaction distance in the system was also limited to a smaller number compared to the wavelength whereas the retarded effect may become significant. With the development of mathematics as well as t he fundamental Thus, the Lifshitz model popped up. Based on the work of Rytov [ 28 ] about the electrical fluctuation and thermal radiation, Lifshitz made thr 1. The materials of the wall were not just the conductive metal. It could be any materials with specific electromagnetic absorption spectra. 2. The substance in the intervening medium was not just the vacuum. It coul d be replaced by any continuous phases with the fluctuating electromagnetic field by itself. 3. The interaction between the two infinite wall s was not restricted to interatomic distance. It can be extended from the number of the wavelength to the number way b eyond the atomic radius. The retarded effect was taken into the consideration. To start with the derivation of the Lifshitz equation between the two parallel plates separated by the distance of L, the Maxwell equation was introduced by plugging in the addi tional terms of correlation function. After solving the electromagnetic wave function in Cartesian coordinate by applying the specific boundary condition, the vdW force can be yielded with integrating the Maxwell tension along with the normal vector to the surface. As a result,
23 model allowed the electromagnetic force to penetrate into the surface of plates and interacted with the molecules not only on the surface but also embedded in the surface. Furthermore, the model also figured out that the fluctuation which most contributed to the force was that dependent on the location of the interfaces rather than those in the intervening space. Therefore, compared with the ot her two models, the Lifshitz model was more advanced and applicable and a plenty of experiments afterwards also demonstrated its validity in the condensed physics. [ 29 ] [ 30 ] [ 31 ] The Lifshitz Van Der Waals (LW) M odel Comparison between the Lifshitz van der Waals Forces E quation Before starting with the LW model, one should know that there was no contradiction the interaction distance. He acknowledged o ther potential factors that can make a difference but instead of listing them in the terms of the equations, he just treated them in a more implicit way, which is the introduction of the Hamaker constant. While for the Lifshitz model, it successfully addre ssed the issue of the variables and he attributed them to the temperature and the fluctuation frequency of individual materials. However, although it was plugged in additional terms in the equations, the LW equation is still distance dependent. Which is to say, to some degree, the LW of the Hamaker constant and the Hamaker constant can be replaced by the term associated with the temperature and dielectric functi on since the interaction is actually caused by the summation of the individual London dispersion van der Waals force. Thus in modern view of the adhesion
24 model lying in the Lifshitz van der Waals forces, I equ ation. The Parallel Plate M odel of the Lifshitz van der Waals I nteraction As the figure 2 1 showed, the original Lifshitz parallel model was developed based on the l establishment were listed below 1. Both of the plates are semi infinite large, which means they both have three infinite dimensions but the extension of one of the dimensions was infinitesimal to the others 2. The surface of the plate was absolutely smooth. Th ere is no roughness or no topological factors that affect the microscopic interaction 3. Compared with the dimension of the plate, the separation distance is supposed to be much smaller in extension. Therefore, for other cases where the curvature of the surfa ce can be neglected, it can also be regarded as this model. 4. Both a and b are continuous phase and electrically neutral. However, since they are both composed of the dancing charges, the interaction can occur between the transient induced positive domain an d the transient induced negative one. The intensity of the interaction is corresponding the intrinsic electromagnetic field of each kind of material. 5. The intervening medium can be any continuous phase including air and specific liquid. The specific absorpt ion spectra of the medium material can identify its electromagnetic property. plate plate geometry, the Lifshitz van der Waals equation can be presented as
25 (2 1) where G is the free energy of the interaction and the A is the Hamaker constant, l is the separation distance. Since the interaction is between the two interfaces which involve the interface between ac and bc another expression which highlight the specific interface goes to (2 2) In the equation, the Hamaker constant is the term that bridges the interaction and the give a deeper i nsight into the correlation and also prove its consistence with the zero point principle as well as the contribution of the fluctuation of the materials. Hamaker C onstant in Lifshitz Model and Its Physical P erspective the same meaning, in order to obtain the expression of the Hamaker constant in the equation 2 2 32 ] ( 2 3) is response function of the systems l is the separation distance and P is the integral number. To simpli fy this complicated equation, I can complete the integration with respect to the P and l. Then reorganize the distance terms into a function, which leads to the neat equation: ( 2 4)
26 ( 2 5) The term A ac/bc is the Hamaker constant. To further figure out the physical meaning of the Hamaker constant, the details of the dielectric response function shoul d be understood. Actually, if I spread these terms out, it presented its relationship with the imaginary dielectric in discrete frequency ( 2 6 ) ( 2 7 ) ( 2 8 ) wh is the quantum number. By considering all of the terms in the Lifshitz equation, the physical signification of the Hamaker constant can be addressed: 1. The Hamaker constant in the LW model is not just a constant. It increased with increasing the temperature since the higher temperature will excite more molecules emanating the electromagnetic field in the interaction. 2. The Hamaker constant is also related to the separation distance independently. This is force investigation. As increasing the separation distance to a certain number, the forc e will drop down exponentially because the electromagnetic response could not react in phase. 3. The Hamaker constant is also presenting its dielectric correlation with respect to the specific fluctuation frequency. Moreover, since the substantial frequency is quantum discrete, the precision of the LW force can be raised up if a huge number of the frequency is taken into account.
27 Dielectric R esponse Function questionable for the derivation of the dielectric function since it was relevant with imaginary complex of the fluctuation frequency. The first ones who gave the remarkable comments on the dielectric response were Hough and White. [ 33 ] They started addressing the formula by introducing the electric displacement vector D, which including the electric field vector as well as the polarization density. With inserting the time dep endent electric wave function and the Fourier transformed polarization function into the displacement vector equation, the dielectric response function, which behaved as the correlation function between the initial electric field and the modified one, was solved as a complex number. The real part of the function and the imagine part of the function satisfied the Kramers Kronig (KK) transition. [ 34 ] This meant if the imagine part of the term was known for all the frequency, the real part of it can also be de rived according to the specific mathematic relation. Despite knowing this may not assist a lot in the dielectric function derivation, the practical significance will allow the measurement only do with the real part of it so as to simplify the process of te sting. So far, the origin of the dielectric response function and its practical value in measurement is clear However, not only the dielectric is no longer constant, the frequency in this respect was also complex, which drew additional confusion. In fact direct significance for the complex frequency. The introduction of the complex terms just mathematically came from the derivative function solution of the electric displacement equation. e frequency represented the transient excursions of the charge motions, which was supplemented to the regular sinusoidal fluctuation. Moreover, both of
28 the real and the imagine can transit to each other easily if they are all continuous on their own axis. respect to the imagine part or the real part of the frequency, especially after Ninham and Parsegian provided another manufactured solution t o the response function. The Ninham Response F unction The aim of the Ninham complicated dielectric response function so that the available data from the experiment can giv e rise to a straightforward expression for the dielectric response between the two materials. [ 3 5 ] In dielectric function with respect to the frequency over the range of ultraviolet. Then, since the imagine part of the dielectric function can yield the real part solution, the whole dielectric function over the ultraviolet range was available to obtain just from one simple electron loss spectroscopy. However, since the precision required the frequency ranging from zero to the infinite, the function only representing in the ultraviolet could not satisfy the practical application. Therefore, the contribution from other ranges of the frequency is deserved to conce rn. As the equation 4 8 showed, the frequency term is discrete with respect to the dissociated principle quantum numbers. At room temperature, the series of the frequency can be calculated as ( 2 9 ) When m =1 the primary frequency is Thus, the corresponding wavelength for the primary frequency is ( 2 10 )
29 This means the primary term directly stepped over the frequency of microwave and got Moreover, if I keep on ca lculating the frequency and plot them in the log 10 plot, the data points in the ultraviolet should be 10 times more than the sample spots in the IR, which indicated the contribution of the ultraviolet frequency was more significant than that of the IR spec tra, not to say other spectra such as the microwave range. As a result, for further simplification of the UV range. In IR spectrum, since the imagine part o f the dielectric function exhibits the electromagnetic wave attenuation, the absorption frequency at IR can be used to average the total relation. The equat ion below shows the imagine part of the dielectric function ( 2 11 ) i is the relaxation frequency in IR range, i is Insert this equation to the KK relation: ( 2 12 ) Ultimately, the derived terms for IR spectrum was shown below: ( 2 1 3 ) If the equation of 2 11 was plugged into the KK relation of then the corresponding equation was yielded ( 2 14)
30 where the is the infrared oscillation constant which is proportional to the oscillation strength, IR one relaxation frequency in the IR range, the expression for the di electric function was the summation of all of the involved terms. be written down as or (2 1 5 ) Where the C UV UV is the relaxation frequency in UV spectrum. In summary, the Ninham complicated dielec tric response function to the expression containing the measurable terms such as the fluctuation frequency and the specific relaxation frequency for individual material. Nevertheless, to construct the dielectric response function, the last puzzle is for th e derivation of the constants in the equation. Cau chy Plot and the Relationship between the Dielectric Function and the R efraction index To acquire the number of the constants, some non trivial conversions for the equation 4 15 should be made. But befo re that, the statements of the approximation had to be announced: 1. The real part of the dielectric function with respect to the real part of the frequency was equal the complex expression of the dielectric function. ( 2 1 6 ) 2. The significance of the UV spectrum made the dielectric function in this range be representing the whole one.
31 3. The dielectric function was equal to the square of the refraction index function with respect to the real frequency, which means ( 2 17 ) Plug these assumption equations into the UV spectru rearrangement, the more applicable Cauchy function can be generated: ( 2 18 ) ct to the real frequency, is the relaxation frequency in UV spectrum, C UV is the corresponding constant. Based on the Cauchy equation, if the refractive of specific material is acquired and the function of with respect to the is plotted the NP constants can be read from the regression line: the slope is equal to the and the intercept is C UV Since both the C UV and the C IR are oscillation strength related, one can be derived with know ing the number of the other according to the equation: ( 2 19 ) One example of the Cauchy plot application is for the nitrogen. The frequency dependent refractive index was colle cted from the literature [36 ]. T he tables and the diagrams are listing in table 2 1 and figure 2 2 The constants that derived from this plot were shown in the table 2 2 equation with respect to the IR and UV range: ( 2 20)
32 Then, the dielectric response function on the entire frequency spectrum can be approximately yielded. The A ttraction Repulsion Transition I n Silica Alkanes Cellulose Systems Base d on the concl usion of the former section, the dielectric response function s of differe nt materials can be achieved by inserting the NP constants to the equation 2 20. In Lifshitz par allel plate model, if the function of each material was obtained, the dielectric respon se functions of the systems can be calculated with the equation 2 6 and 2 7. Thus the Hamaker constants of any specific systems could be derived by using the equation 2 5. [ 30 ] he talked about the trending alternation of the non retarded Hamaker constants in varied systems with the fixed intervening medium such as the air and the water. He concluded all the systems presenting attractive forces was because the dielectric response function of the intermedium lay below or above b oth the substrate and the proposed that the relative location of the dielectric response function line would lead to the minimum or maximum value of the derived Hamaker constant or van der Waals force. To address the minimum or maximum force, the interacting distance should not be ignored. In this section, a specific system comprising of silica, alkane s and cellulose was picked up because on one hand, the dielectric response functions of the alkanes gave rise to gradual tendency of variation which helped to analyze the appearance of the extreme value. On the other hand, the non branched oligomers can be assumed anchored on the substrate while the other parts of the molecules still had random conformations In this case, the dista nce between the substrate and the contact materials can be well controlled and easily defined. There are several ways to anchor the alkane chains onto the substrate of the silica among which using the corresponding alkyl silane is prior. The alkyl silane is a kind of molecule that
33 can be coated on the solid surface in an ordered alignment without putting in additional energy for the arrangement, which is called self assembled (SA) molecule. In the molecule, a t one end, there are active functional group(s) that are used to firmly bonded on the surface. At the other end, there is always a long carbon chain hanging out so that the molecules can pack tightly and regularly once they are anchored. The two dimensional assemblies not only modified the surface prope rties and provided investigation of specific interaction but also inspired the surface designers to graft varied functional chains so as to create elaborated and exquisite surface structures. The first published SAM was prepared by Zisman in 1946. [ 37 ] He made a monolayer of Gottingen [ 38 ]. They coated chlorosilane derivative on the surface of glass and modified the surface from hydrophilicity to hydrophobicity. In 1980 s, Nuzzo and Allara [ 39 ] picked up the alkanethiolate as the precursor and successfully bonded it onto the gold surface with chemically reacting between the disulfide group and the Au atoms. In 1998, Michael and his colleagues [ 40 ] o monodisperse the gold particles and well control the size of gold clusters in the range of nanometer if the alkanethiolate SAM was initially assembled on the cluster surface. T here are several types of molecules that are favorable to form SAM including f atty acids [41 ] organosilicon derivatives [42 ] organosulfur derivatives [43 ] and organophosphate derivatives. [44 ] The alkyl silane self assembled molecule is belong to the organosilicon derivatives. The alkane chains can be attached onto the surface by forming Si O C bonds. The thickness of the self assemble monolayer can be derived with the knowledge of the specific
34 molecular conformations ecules as the intervening media to fix on the surface and have the thickness of the layer calculated. For the contact material, the cellulose was chosen in that it served as the main constitution of the dust in some circumstances. Cellulose insulation mig ht give away hundreds of cellulose coils suspending in the air. The pollens of the plants can be wrapped up in the dust agriculture might be blown away with wind investigate a system with cellulose dust involved. Besides, since there are different kinds of cellulose, the cellulose mentioned in this chapter was referred to the distinguished one produced in Dr. Bergstrom paper. [ 45 ] The optical constants collection will also be listed in this chapter as the data foundations of mathematical calculation. In the parallel plate Lifshitz model, the silica was the substrate material. Alkane was covered on the surface of the subs trate and served as the intervening media. The cellulose particles were the contact material that interacting with silica across the alkane layer. Since the thickness of the alkane is exquisitely controlled by the specific conformations the series of syst ems can show significant variance in Hamaker constants with varied types of alkane coatings of different thickness es The Hypothesis of the project was with fixing the thickness of the SAM by 1 nanometer, the calculated Hamaker constants in the series of t he systems (silica octane cellulose, silica decane cellulose, silica dodecane cellulose, silica tetradecane cellulose and silica hexadecane cellulose) decreased and converted from positive number to the negative one. The Assumptions for the LW Model in Si lica Alkane Cellulose System 1. The roughness of the silica slide was neglected. The surface was flat enough for the LW interaction. The composition of the substrate material was over 99.9% purity and the
35 constants required in NP the measurement of the refraction function by using the Cauchy plot. [ 46 ] 2. The alkane coatings were treated as the conti nuous phase. The slight changes in the molecular conformations or thickness would not affect its dielectric property. Moreover, the alka ne coating is composed of a monolayer of the alkane molecule. The separation distance of the intervening media for specific silica alkane cellulose system reaches the maximum when all alkane molecules have the all trans conformation Otherwise, the minimum thickness can be derived when all alkane molecules are freely jointed and freely rotated (the most loosely packing). In this case, each carbon atom in the chains is entropy dominated and can make random walk statistically. T he data of the alkane chain was 4 7 ] 3. The radius of the cellulose pa rticles is assumed to be infinitely larger than the thickness of the media, which means the curvature of the particle is much larger than the separation distance (the thickness of the alkane the parallel plate setup for Lifshitz van der Waals interaction derivation. The NP constant data of this specific cellulose particle was collected by Bergstrom. [ 48 ] Optical Data Collection for the Materials in Silica Alkane Cellulose System According to the assumptions, the Optical constants of the three materials w ere collected in table 2 3 , 2 4  2 5  and 2 6  The corresponded Cauchy plots were referred to figure 2 3 2 4 and 2 5 The Dielectric Response Functions in Silica Alkane Cellulose Systems In the figure 2 6 the lines are not continuous and consisted of numerous scattered data points in the frequency ranging from 1E+14 rad/s to 1E+17 rad/s The reason to choose this range is because the data is quantum nu mber related. The first data was corresponding to the data
36 when quantum number 1 was plugged in. After 1531 data points, the dielectric response function approach ed Therefore, t he Hamaker constants in each system can be calculated over the whole scatter frequency range and if the thickness of the alkane layer was fixed as 1nm, the Lifshitz van der Waals forces can also be yielded with respect to types of alkane coatings The table 2 7 exhibited the relationship among types of the systems, the derived Hamaker constants and t he related van der Waals forces. The figure 2 7 gives a more straightforward view of the transition o f the calculated Hamaker contants In the table 2 7 the systems involved were well established on the consistent substrate and the contact material while only different in the intervening alkane layer The homolog of the alkanes has non branched carbon chain s and the specific conformation could make the chains in distinguished arrangement, which ensure d the t hickness of the setup preferably reach the same level. In this case, the only factor that contributed to the value of the i nteraction force is the dielect ric property of the materials. To figure out the correlation between the dielectric property and the Hamaker constants additional mathematic derivation was required. Considering the thickness contribution, the application of the derivative formation of equation 2 4 resulted in force distance relation: ( 2 21) in which the force was inversely proportion to the cubic thickness. If only considering the influence of the thickness, s ince the thickness of the alkane layer stay the same, the variance in conformation cannot explain the positive negative trans ition It should be the intrinsic properties of the materials such as the fluctuation response function that accounted for the derivation.
37 Furthermore, according to the equation 2 5, in specific frequency range the Hamaker constants bec ome negative only w hen either ac or bc is negative, which indicated the repulsive van der Waals forces should meet the condition a > c > b or inversely, b > c > a Besides, since the frequency might cover a wide range of the spectrum within which some of the data points fit the premises but others are not, the outcomes usually depended on the quantitative predominance of the data points in the high frequency range in that the density of data spots was higher than the low er one Theref ore, the highlight in figure 2 6 located at the high frequency range and provided information of the force transition. Physically, since the Hamaker constant is the energy term, the higher frequency of the molecular fluctuation leaded to the large r interaction energy so that the polarizability dominance of the high frequency spectrum makes sense. Moreover, if the materials in the system also satisfied the condition a > c > b the polarizability of the intervening material m was larger than material a while smaller than material b In that case, the m was preferable to stay with a rather than b so that it separated the distance between the two interfaces a/m and m/b. In the end, the system tended to present repulsive interaction. However, although th e condition of a > c > b or b > c > a was the sufficient prerequisite to minimum (or maximum ) of the calculated Hamaker constants and the correlated force values. The solution of the question s also came from the equation 2 5, 2 6 and 2 7 whereas for s for the terms in those equations Firstly, t he equation 2 5 converted to be the form below. ( 2 22)
38 w ithout considering the retarded force, the R(l) term is equal to 1. Then, t he equation 2 6 and 2 7 were modified as ( 2 23) ( 2 24) where a, m and b are the simplified term of the response function of material a, c and b. Assuming at each data point, the response function value of material m was in between a and b, which means > > (2 25) If the Hamaker constant reach ed the minimum (or maximum), the term ac bc should go to the minimum (or maximum) correspondingly, which indicated the term of was supposed to be minimum (or maximum). The transition of this term was ( 2 26) Thus, the Hamaker constant arrived at the minimum when ( ) 0.5 Therefore, the curve of ( ) 0.5 was the minimum solution cu rve for specific system. curve the square root ab curve (sqrab). To find out the solution of the boundary condition between positive and negative value, let the l eft hand side of the equation 2 26 be zero. Thus, or (2 27) ( 2 28) which turned out to be consistent with the premise a > c > b.
39 The calculation above showed that if the response curve of the intervening material in the high frequency range was not only in between the curve of a and b, but also get close to the sqrab even though the curve might deviate a lot in the low frequency range This was because the Hamaker constants was the summation of the value over each d ata point. If the value at high frequency spot s show strong negativity, that could balance the positive contribution from the lower frequency range and end up with global negative effect. The reproduced plots with the sqrab line inv olved was displaying in figure 2 8 In figure 2 8 the line of the substrate material and the contact material cross over at the frequency of 5.88E+15 rad/s. Less than this frequency, all the alkane curves were nailed outside the enclosed area surrounded by the silica and the cellulose boundary the system of silica alkane cellulose should present attraction if the curve of intervening materials still stayed outside the area in the high frequency ran ge I n the highlight range from 7 E+16 to 7 .25 E +16, all the curve s of the alkanes kept inside the area walled between silica and cellulose Thus, the summed interaction in this range should have the negative Hamaker constant. The contradiction between the low frequency range and the high one gave the probability that the system could res ult in negative Hamaker constant value s This will occur only when the accumulation over the high frequency range was derived to balance out the positive co ntribution from t he low frequency range. Furthermore, th e probability strongly depended on how close the curve approaching the sqrab lines in the high frequency range. The closer it wa s, the more likely for the system to overwhelm the attraction in the low frequency range Therefore, since the curve of C10, C12, C14 and C16 approached the sqrab line more closely than C8 and within the 5 silica alkane cellulose systems and the curve of C16 g ot closest to the sqrab line, the system of silica alkane
40 c ellulose gave rise to minimum Hamaker constant, which meant the maximum repulsive interaction in the system. G radually, this repulsion turn ed weaker and weaker with increasing the deviation from the sqrab line. It demonstrated that the variance of the dielectric property of the alkanes can lead to the attraction repulsion transition in silica alkanes cellulose system. repulsion in the system silica X cellulose was calculated. Assuming there is a material that has its dielectric response function superpose d on the sqrab line perfectly the maximum repulsion is supposed to occur in the system with this imaginary material as intervening media. Thus, with fixing the s eparation distance as 1 nm, the maximum repulsive van der Waals force in this system can be derive d based on the equation ( 2 29) In th e figure 2 9 only the systems of silica C12 cellulose, silica C14 cellulose and silica C16 cellulose show the repulsion from the output. The deviation percentages from the maximum repulsion in each system are 95.6%, 89.5% and 78.2% respectively. This difference between the repulsion in real systems and the maximum value in theory redu ced gradually as switching the closer the alkane curve approached the sqrab line, the larger repulsion the system might give. In contrast, although this deviation percentage is much larger than most of the Teflon systems [ 50 ], the value is not considerably varie d from other common systems. [ 51 ] T he Influence of the Separation Distance on Lifshitz van der Waals Interaction Now that the dielectric property of the materials intensively influence d the Lifshitz van der Waals interaction, how about the separation distance? To figure out the impact from tha t
41 factor, the system was picked up as fixing each material component while differing the thickness of the alkane layer. T he derivations of the van der Waals force with respect to the distance in silica alkane cellulose systems were listed in the figure 2 10 : In the figure 2 10 the separation distance range was determined by the specific conformation of the molecule chains. The upper limits were reached when the alkane chain showed the all trans co nformation In this case, the thickness can be derived by inserting the fixed angle and the lengths of carbon carbon bond. The lower limits occurred when the chains were entropy dominated, which indicated every segment of the chain can do the random walk a nd the chains tended to coil up to minimize the thickness. In this case, the thickness can be regarded as the double of the radius gyration (R g 2 =nL 2 /6, n is the number of bonds and the L is the length of the carbon carbon bond). Therefore, based on the def ined range, the van der Waals force can be derived by using the equation 2 21. It was exhibited that t he van der Waals force reduced as increasing the thickness of the octane and decane layer while the force increased as increasing the thickness of the do decane, tetradecane and hexane layer. However, although all the systems approached the zero line as increasing the separation distance, they did not break through the limit, which meant only changing the thickness could not achieve the a ttraction repulsion transition. The effect of separation distance extension was to shrink the absolute value of the interaction force. With the knowledge, the maximum repulsive Lifshitz van der Waals force should pop up when the system shows the largest negative Hamaker con stant in absolute value as well as the smallest separation distance. Thus, the maximum Lifshitz van der Waals force in the silica alkane cellulose systems was derived as 0.163pN/nm 2 when the system of silica hexane
42 cellulose has the smallest intervening la yer thickness. Oppositely, the minimum Lifshitz van der Waal s force can also be calculated. Conclusion Overall, the thickness of th e intervening layer can only change the intensity of the interaction force but not make the attraction repulsion transition even though for those chains who have the extreme conformation s. The dielectric property of the materials is the key thing contributing for the transition. The conversion from attraction to repulsion was det ermined by the response function of each material low frequency range and the high frequency range, the value s in the high frequency range usually dominated. B esides, if the dielectric response function curve of the intervening material approach ed the sqrab line in the specific spectrum, the mathematic derivation usually gave rise to the negative Hamaker constants which in turn, could cancel off the summation of the attraction in the contrasting area s and lead to repulsion. However, despite of the attribution of the dielectric property to the attraction repulsion transition, the intensity is low if only considering this factor. Therefore, to figure out the maximum or minimum repuls ion or attraction in silica alkane cellulose systems, both the separation distance and the dielectric property should be taken into account. The Design O f Repellent System s F or Martian Dust The Martian soil is composed of different kinds of the particles including the rock fragments, agglutinate particles as well as the volcanic depositions. Influenced by the bombardment and the meteorite impact, the most regolith on the Mars contains the component of the iron minerals which suspending in the atmosphere a nd make the outlook of the planet red. Thus, to design a system that repels the Martian dust s, the main goal is to eliminate these iron compounds which are significantly abundant
43 dust s is maximally oxidiz ed. [52 ]. In that the systems are supposed to target at the iron minerals such as the magnetite, maghemite and the hematite which at least exhibit +3 chemical valences To design a repellent system for Martian dust removal, the contact materials should be fixed as these iron compounds. Thus, the intrinsic repulsive van der Waals forces in the systems would contribute to the minimization of the particle surface adhesion. Generally, the way to construct the systems was based on the relative location of the d ielectric response functions of the materials involved. Since the contact materials were selected as these iron compounds, the substrate materials should have their dielectric functions separated from the targets so that the functions of the intervening me dia could lie in be tween and meet the condition of repulsion. By calculating the sqrab lines in the systems, the selections of the intervening materials could be further narrowed down to match the sqrab curve. One example of the designed systems for t he Martian dust mitigation was the silicon silica magnetite (SSM) system. In the LW model, the silicon was the substrate material. the silica was the intervening media and the thickness was restricted to several nanometers. The magnetite served the top layer. The NP constants was measured by the ellipsometer by applying the Cauchy plot for K K transition. Therefore, the NP cons tants were listed in the t able 2 8   while the response function of each materials over the frequency range from 1E+14 to 1E+17 was plotted in the figure 2 11 In the figure 2 11 there was also collision between the low frequency range and the high one. In the range from 1E+14 to 6E+15 rad/s, all the data points did not satisfy the premise of repulsion which indicated the integrated Hamaker constant over this spectrum was positive. However, above the 1E+16 rad/s, since the dielectric property of the in tervening material went
44 back in between the substrate and contact materials. Moreover, it even overlappe d with the sqrab curve. T he predominance of the negative effect in the high frequency range overwhelmed the low one. Thus, the system yielded negative summation overall. The derivation of the Hamaker constant in this system was 6.88E 21 J. Besides, other designed repulsive systems consisting of alumina, silica and the iron minerals could also result in repulsion through specific arrangement. In the systems, the alumina was the substrate, the silica was the intervening media and the iron compounds were th e contact the materials. With the collection of the optical constants of these materials in table 2 9  , the dielectric response function can be derived. Thus, the theoretical number of the repulsion can be addressed. In the figure s 2 12 2 13 and 2 14 despite there is the crossover between the substrate material and the contact materials at the frequency between 1 E+15 rad/s and 1 E+16 rad/s the dielectric response function of the intervening media almost fully overlapped with the sqrab lines in the high frequency range, which ended up with the predominance of the repulsion summation over the whole frequency range. In this case, al l the systems gave rise to negative Hamaker constants after calculation, which indicated the repulsive van der Waals force with the proper separation distance assignment By comparing the derivations with the maximum repulsion, the deviation of the values in real systems from the theoretical one can be further addressed. The comparisons of the calculated Hamaker constants between the alumina silica iron compounds systems and the theoretical value of repulsion were listed in the table 2 10 Limit s o f the Parallel Plate Lifshitz Model i n Application The main limits lie in the shape of the materials. This is because the shape severely
45 ene rgy at the unit projection area, the plate plate interaction was much higher than the sphere sphere one.  [55 ] Therefore, the shape of the materials also makes difference. In the assumption, the interaction distance between the substrate and the top layer materials was always set in the nanometer scale. This was because in this case the retardation of the fluctuation response can be neglected. However, most of the setups in real systems do not match that premise. The roughness and irregular arrangement are always broadening the thickness of the intervening media beyond the limitation. Therefore, t he retardation function R(l) is another factor that shoul d be considered in appl ication. Besides, another thing to be aware of is that the repulsion is determined from the dielectric response function of the materials, which is directly related to the optical constants of specific composition instead of the general type. T he complexit y of the concentration of the chemical components in the materials can shift the interaction a lot even though the materials are identical in category. To further demonstrate the prediction of the fundamental theory, additional measurement of the optical p roperty of the target materials should be involved in the preliminary work in experiment.
46 Figure 2 1 The Lifshitz van der Waals model for the interaction between two semi infinite pa rallel plates across the media c The separation distance is l. Table 2 1 T he Cauchy plot data for nitrogen gas in 15C. [ 36 ] Wavelength(nm) Freq(rad/s) 470 1.0003011 4.01 E+15 9.69 E+27 0.000602251 500 1.0003002 3.77 E+15 8.53 E+27 0.00060051 600 1.0002982 3.14 E+15 5.89 E+27 0.000596489 700 1.0002970 2.69 E+15 4.31 E+27 0.000594108 800 1.0002962 2.36 E+15 3.29 E+27 0.000592568 900 1.0002957 2.09 E+15 2.59 E+27 0.000591527 1000 1.0002954 1.88 E+15 2.10 E+27 0.000590787 1200 1.0002949 1.57 E+15 1.46 E+27 0.000589807 1400 1.0002946 1.35 E+15 1.07 E+27 0.000589227 1600 1.0002944 1.18 E+15 8.17 E+26 0.000588847 1800 1.0002943 1.05 E+15 6.45 E+26 0.000588587 2000 1.0002942 9.42 E+14 5.23 E+26 0.000588407
47 Figure 2 2 The Cauchy plot of the N 2 in 15C. Table 2 2 The NP constants derived from the Cauchy plot Items Numbers C UV 5.88E 4 C IR 7.61E 6 UV (rad/s) 2.57E+16 IR (rad/s) 2.14E+14 Table 2 3 The r efraction index data for silica [ 49 ] Wavelength(nm) 294 1.50866 5.25 E+31 1.27605 340 1.49824 3.83 E+31 1.24472 394 1.49040 2.80 E+31 1.22128 446 1.48533 2.15 E+31 1.20620 502 1.48150 1.68 E+31 1.19484 532 1.47992 1.49 E+31 1.19016 618 1.47659 1.10 E+31 1.18032 666 1.47526 9.42 E+30 1.17640
48 Figure 2 3 The Cauchy plot of the A silica Table 2 4 The NP constants derived from the Cauchy plot Items Numbers C UV 1.155 C IR [45 ] 0.892 UV (rad/s) 2.08E+16 IR (rad/s) [45 ] 8.67E+13
49 Figure 2 4 The Cauchy plot for n alkanes. The numbers next to the lines represent the n umber of carbon in the chains. copyright 2017 Elsevier T able 2 5 T he C UV C IR UV IR 0 data collection for n alkanes Materials 0 C UV UV (rad/s) C IR IR (rad/s) O ctane [47 ] 1.948 0.925 1.86E+16 2.30E 2 5.54E+14 D ecane [47 ] 1.991 0.965 1.87E+16 2.60E 2 5.54E+14 D odecane [47 ] 2.014 0.991 1.88E+16 2.30E 2 5.54E+14 T etradecane [47 ] 1.01 1.85E+16 2.50E 2 5.54E+14 H exa de cane [47 ] 1.03 1.85E+16 2.50E 2 5.54E+14
50 Figure 2 5 The Cauchy plot of cellulose at various thicknesses. Tr iangle represents 100 layers. The w hite circle represents 120 layers and the s olid circle represents 140 layers. copyright 2017 Springer Table 2 6 The average optical constants of cellulose derived from Cauchy plots Items Numbers C UV [48 ] 1.24 C IR [48 ] 2.52 UV (rad/s) [48 ] 1.29E+16 IR (rad/s) [48 ] 2.14E+14
51 Figure 2 6 The dielectric response function on the real frequency for the silica alkane cellulose particle system. The green cross mark represents the response function of silica substrate. The red cube represents the response function of cellulose part icle. The dark blue, light blue red, purple and yellow plus signs represent the decane (C10), dodecane (C12), tetradecane (C14), hexadecane (C16) and the octane (C8) intervening coating. Table 2 7 The calculated Hamaker constants and Lifshitz van der Waals forces in silica alkane cellulose particle systems with equal separation distance. Systems (Substrate Thin Film X) Average Thickness of the Self Assembled Monolayer (nm) Ham a ker constants (zJ) Force Per Unit Area (p N /nm 2 ) Silica C8 Cellulose 1.0 0.73 0.038 Silica C10 Cellulose 1.0 0.19 0.01 Silica C12 Cellulose 1.0 0.068 3.61E 3 Silica C14 Cellulose 1.0 0. 22 0.012 Silica C16 Cellulose 1.0 0. 34 0.018
52 Figure 2 7 The calculated Hamaker constants in silica alkane cellulose systems. Figure 2 8 The dielectric response function with respect to the real frequency for the silica alkane cellulose particle system with sqrab line involved. The green cross represents the response function of silica substrate. The red square represents the response function of cellulose part icle. The dark blue, light blue red, purple and yellow plus signs represent the decane (C10), dodecane ( C12), tetradecane (C14), hexadecane (C16) and the octane (C8) intervening coating. The cyan line represents the sqrab curve (sqr ab) in the system.
53 Figure 2 9 The calculated van der Waals forces in silica X cellulose systems with fixing the separation distance as 1nm. The black cube represents the forces in silica alkane cellulose systems. The red sphere represents the maximum repulsion in silica X cellulose system. Figure 2 10 The calculated Lifshitz van der Waals forces in silica alkane cellulose system with varied separation distances. The black cube represents the silica octane cellulose system. The red sphere represents the silica decane cellulose system. The blue triangle represents the s ilica dodecane cellulose system. The green triangle represents the silica tetradecane cellulose system. The purple triangle represents the silica hexadecane cellulose system
54 Table 2 8 The NP constants for the material s of silico n, silica and the magnetite. Materials T/C 0 C UV UV (rad/s) C IR IR (rad/s) Silica RT 3.82 1.155 2.08 E+16 0.829 8.67E+13 Silicon  RT 11.9 10.65 6.24E+15 0.25 1.13 E+14 Magnetite  RT 20 2.4 4.0E+15 16.6 1.1 E+14 Figure 2 11 The dielectric response functions of the materials in SSM system with respect t o the frequency. The light green circle represents silicon. The green cross represe nts the silica. The purple diamond represents the magnetite a nd the cyan triangle represents sqrab curve in the system Table 2 9 The optical constants collection for the materials in Martian dust repelling systems Materials T/C 0 C UV UV (rad/s) C IR IR (rad/s) Silica RT 3.82 1.13 2.03 E+16 0.829 8.67E+13 Alumina  RT 10.1 2.072 2.0E+16 7.03 1.0 E+14 Magnetite  RT 20 2.4 4.0E+15 16.6 1.1 E+14 Maghemite  RT 20 3.6 4.4E+15 1.54E+1 1.20E+14 Hematite  RT 12 4.2 5.4E+15 6.8 1.0E+14
55 Figure 2 12 The dielectric response functions of the materials in silica alumina magnetite system with respect to the frequency. The orange circle represents alumina The green cross represents the silica The purple diamond represents the magnetite and the cyan cu rve represents sqrab curve in the system Figure 2 13 The dielectric response functions of the materials in the silica alumina magnetite system with respect to the frequency. The orange circle represents alumina The green cross represents the silica The dark red diamond represents the maghemite and the cyan curve represents sqrab curve in the system
56 Figure 2 14 The dielectric response functions of the materials in the silica alumina magneti te system with respect to the frequency. The orange circle represents alumina The green cross represents the silica The dark diamond represents the hematite and the cyan curve represents sqrab curve in the system Table 2 10 The calculated Hamaker constants A in alumina silica iron compounds systems Systems A (zJ) A maximum repulsion (zJ) Alumina silica magnetite 14.4 18.6 Alumina silica maghemite 8.94 17.5 Alumina silica hematite 0.91 12.9
57 CHAPTER 3 DISCUSSION AND SUMMARY Tenacious adhesion of the dust to the surface can be minimized by applying the Lifshitz van der Waals construction which intrinsically pre vent s the dust from sticking This particular structure, different from other dust repellent systems that call for additiona l energy supply, generates the repulsive van der Waals forces between the substrate and the dust through the electromagnetic interaction. The physical foundation behind the appearance owes the repulsion to the dielectric properties instead of the separation distance The analysis of the separation distance dependent van der Waals forces in silica alk ane cellulose systems concluded that the increasing distance only reduced the absolute value of the forces whereas make no difference for the attraction repulsion transition. The theoretical derivation of the van der Waals force shows the strong dependence of the character of the interaction on the dielectric response f unction. O nly specific combination of the functions can lead to negative Hamaker constants as well as the repulsive interaction. The obsolete standard to pick up the repulsive systems restrict s in the robust comparison s It just stressed every data point at specific frequency should have their dielectric response value s of the intervening material in between the substrate and the dust. ( a > c > b ) However, this will cause contradiction when the dielectric response functions cross over at some point. Considering that, the new method to solve the problem was developed in the thesis. Since the sqrab curve represents the mathematically minimum value for the Hama ker constant, t he system consisting of substrate sqrab dust will give rise to the maximum repulsion to achieve. By matching dielectric response function of th e intervening material with this theoretically calculated curve, the repulsive systems are much easier to establish The correspondence of the curve coupling to the derived Hamaker const ants
58 or van der Waals forces is further demonstrated in the attraction repulsion transition in the silica alkane cellulose system, which in turn, exhibits t he probability of using the Lifshitz van der Waals system to repel organic dust. The physical significance not only reside in the mitigation of the organic dust it also fits in the designs to remove the inorganic dust. The multiple choices for the repell ent systems containing the iron compounds bring the variety for the application of the Martian dust repulsion. The deviation of the Hamaker constants value from the theoretical minimum is compatible to other systems published in literatures. This illuminat es the fundamental power to unravel the mechanism of dust surface adhesion and provide revolutionary perspective for the surface science.
59 APPENDIX A REPULS I VE VAN DER WAALS SYSTEMS IN LIQUID AND GASEOUS PHASES Except for the solid system, there are also other systems in liquid and gaseous phase that displayed repulsive interaction between the materials. These kinds of the systems not only presented the negative Hamaker constants in the Lifshitz model, but also could contain varied phases of the materials as the components. Silica Diiodomethane Magnetite (SDM) System In the system, the silica served as the substrate materials and the magnetite as the top layer. They are supposed to interact across the media of the diiodomethane, whi ch, in contrast, was in the liquid phase. The NP constants of each material were also derived from the Cauchy plot by plugging in the refractive index with respect to the wavelengths. The calculated constants in the SDM system were listed in table A 1   Thus, with applying the first 1531 data points to the equation 2 20, the dielectric response function of each material can be yielded. The function curves were exhibiting in the figure A 1 In the figure A 1 the crossover of the silica and the magnetite curves occurred at the frequency around 8E+15 rad/s. Even though in the range smaller than this frequency, it did not satisfy the premises of the repulsive van der Waals forces. The supplement s in the high frequency range offset the positivity tendency in the lower one so that the system ended up with repulsive interaction over the entire frequency range. In the solid system, despite of the repulsive interaction energy, the molecules are still remarkable displacement between the interfaces. However, in the liquid system, the molecules in the intervening media have larger mobility to converted the relative distance between the two sol id/liquid interfaces. Thus, the repulsion in the system triggered the broadening of the diiodomethane layer so that it separated
60 the silica and magnetite even further. If the magnetite was fixed at the end of the AFM cantilever, the separation can be preci sely measured by measuring the cantilever distortion during the approaching and relaxing process. Silicon Silica Air (SSA) System repulsive system if the air was sti ll in the intervening media. However, this intrinsic property of the air can provide a series of the reduction in response function if the air served as the top phase in the system. Therefore, the system consisting of the silicon, silica and air can yield repulsive interaction and might bring significant influence for the growth of the silica on the silicon substrate. The constants of silicon, silica and air were also obtained by refractive index measurement and subsequent extraction in the Cauchy equati on. The constants were list in t able A 2   and the related response function of each material were presented in the figure A 2 In the figure A 2 since the dielectric response function curve of the silica was in between calculated Hamaker constant was negative, which indicated the repulsive interaction. The affe ction of the repulsive force intended to separate the air from the silicon substrate and thickening the intervening layer. But due to the fluidity of the air molecules, the effects caused by the repulsion could result in polarization rearrangement of the b ulk of air and then restrict the growth of the silica coating.
61 Table A 1 The NP constants for the material of silica, diiodomethane and the magnetite. Materials T/C 0 C UV UV (rad/s) C IR IR (rad/s) Silica RT 3.82 1.155 2.08 E+16 0.829 8.67E+13 Diiodomethane  RT 5.32 1.88 1.09E+16 2.46 5.54 E+14 Magnetite  RT 20 2.4 4.0E+15 16.6 1.1 E+14 Figure A 1 The dielectric response functions of the materials in S D M system with respect to the frequency. The green cross represents silica The yellow star represents the diiodomethane The purple diamond represents the magnetite and the cyan curve represents sqrab curve in the system Table A 2 The NP constants for the material s of silicon, silica and the air Materials T/C 0 C UV UV (rad/s) C IR IR (rad/s) Silica RT 3.82 1.13 2.03 E+16 0.829 8.67E+13 Silicon  RT 11.9 10.65 6.24E+15 0.25 1.13 E+14 Air  RT 1.00 5.54E 4 2.46E+16 4.27E 4 2.14 E+14
62 Figure A 2 The dielectric response functions of the materials in S SA system with respect to the frequency. The light green circle represents silicon The green cross represents the silica The red dot represents the air and the cyan curve represents sqrab curve in the system
63 APPENDIX B MULTIPLE LAYERS IN LIFSHITZ VAN DER WAALS MODEL As discussed, the separation of the solid materials in the plate plate LW systems will give rise to the split of the i nterface. The surrounding air will flood into the intervening media and convert the solid solid interface to two solid air interfaces. In this case, the multiple layer Lifshitz van der Waals system is reconstructed. During this process, the interaction ene rgy of the initial triple layer structure and the multiple layer reconstruction is different. To pay for the energy gap, additional amount of energy should be applied in the transition. Previously, the derivation of the interaction energy in plate plate sy stem has been thoroughly described. In this section the calculation of the interaction in the multiple layer system will be introduced. Thus, it provided the theoretical quantitation for the energy consumption in the systems conversion. A s figure B 1 shows, in the four layer LW model, there are 4 phases and 3 interfaces. Since the Lifshitz theory mainly considered the interaction between interfaces, there are generally 3 pairs of the interfaces to be concerned: (th e sign of the interface was presented by the two adjacent phases, for example, the interface between C and m was presented as Cm) 1. Cm/B1m interaction 2. Cm/BB1 interaction 3. B1m/BB1 interaction However, since the last pair did not contribute to the relative mo tion of B and C, it should pairs in this model. For the first pair, the equation to address the interaction was similar to the equation 4 4. The only difference is one of the materials was replaced by the coating material B1. As a result, the expression of the interaction free energy per unit area should be
64 ( 2 30) ( 2 31) where A is the Hamaker constant, l is the separation distance and G is the free energy of response function for the two involved interfaces. R n is the factor of the retardance For the second pair, the Hamaker constant stil l contained two parts of the dielectric functions multiplying by each other but at this time, they were corresponded to the two interfaces (Cm and BB1) separated by two phases in this system. Therefore, the distance between the two interfaces included both the separation distance and the coating thickness. ( 2 32) ( 2 33) where A is the Hamaker constant, G is the interaction free energy per unit area, l is separation response function for th e two involved interfaces. R n is the retarded factor. To wind up the final expression, just make a summation of the two pairs of the interaction. Thus, the final expression of the parallel plates interaction per unit area for the four layer LW model was ( 2 34) Assuming the material B and C were initially interacting across the solid film B1, they are then forced apart by applying the additional energy. Finally, during the transition, the air involved and stabilized at the four layer system. To calculate the transition energy from the
65 plate plate system to the four layer system, the su btraction between the eq uation 2 7 and 2 33 was made and t he derived energy gap was presenting as followed. ( 2 35) Figure B 1 T he parallel plates model for four layer lifshitz van der Waals interaction. B is the substrate material. B1 is the extrinsic coating on the substrate of B. C is the contacting material. m is the intervening medium, l is the separation distance and d is the thickness of the coating.
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70 BIOGRAPHICAL SKETCH Long Ma was born in Henan, China and spent his early time studying and living there while growing up. After graduation from Qingdao University of Science and Technology, He attended the Univ ersity of Florida pursuing the m as ter degree in the Department of Material s Science and E ngineering During the time, he was mainly working on the repellent surface and the calculation of the Lifshitz model. In this period, he attended the conference of the Soiling Mitigation for Solar E nergy and Innovations in Concentrating Solar Power where he gave a poster presentation.