Materials Science and Engineering, R20(1997)37-124 R Reports: A Review Journal Rapid vapor-phase densification of refractory composites AlliedSignal, Inc, Mail Stop CTC-1, 101 Columbia Road, Morristown, NJ 07962, USA Received 5 March 1996; accepted 3 January 1997 Abstract The status of vapor-phase routes for the rapid densification of high-tem primarily ccramic-matrix composites, is revicwed. Conventional densification of composites such as carbon- carbon and Sic-Sic is accomplished by isothermal, isobaric chemical vapor infiltration(CVi), either alone or in combination with liquid resin impregnation and thermal annealing These are multi-step processes which ake from several hundred to thousands of hours at high temperature. In this paper we review approaches designed to significantly reduce the processing time and the number of steps required for densification, while producing materials with the desired properties. We describe techniques such as inductively-heated thermal- gradient isobaric CVI, radiantly-heated isothermal and thermal-gradient forced-fow CVI, liquid-immersion thermal-gradient CVI and plasma-enhanced CVi. Different heating methods, such as radiative and inductive, and both hot-wall and cold-wall reactors are compared. Available material properties of composites produced by these techniques are given. o 1997 Elsevier Science S.A. filtration: Fiber; Forced flow; High-temperature composites; Inductive heating; Isobaric Isothermal fication: SiC. 地二 Microwave; Plasma; Preform; Pulsed pressure; Radiant heating; Rapid densi- 1. Preface Composite materials, such as carbon-carbon(C-C), offer advantages of light weight and excel lent mechanical and thermal properties, especially for high-temperature applications, e. g. aircraft brake pads, uncooled engine and other airplane parts and leading-edge sections used in rockets [1-4]. Other applications which are receiving attention are in heat-conducting substrates for electronic chips and in sports equipment, for example for tennis, golf and bicycling. Composites generally consist of fibers surrounded by a matrix. Compared to their monolithic counterparts, composites are much tougher mechanically and allow tailoring of and thermal conductivity to a much larger extent Composites also have a much more forgiving failure mode under stress than monolithics, as illustrated in Fig. 1. There are generally three categories of composite materials: polymer-matrix composites, metal-matrix composites and ceramic-matrix com- posites. In this review, we are concerned with ceramic-matrix composites(CMCs), fabricated mainly around carbon and SiC fibers; examples are C-C and Sic-SiC composites. These materials can ithstand the highest use temperatures. Other fibers studied or used to different extents include glass silica, alumina, alumino-silicates, silica-titania, silicon nitride, zirconia, yttrium-aluminum garnets boron-coated tungsten and boron nitride [5]. One of the most common fabrication methods of such osite structures is densification of a porous body, the preform, having the desired shape and isting solely or principally of fibers. The fibers may be continuous or chopped. The preform may Corresponding author. Tel:(201)455-4938 Fax:(201)455-3008 or (201)455-3942. E-mail: GOLECKIGRESEARCH COM 927-796x/97/$17.00 e 1997 Elsevier Science S.A. All rights reserved PS0927-796X(97)00003X
Materials Science ana’ Engineering, R20 (1997) 37-124 Rapid vapor-phase densification of refractory composites I. Golecki * AlliedSignal, Inc., Mail Stop CTC-I, IO1 Columbia Road, Morristown, NJ 07962, USA Received 5 March 1996; accepted 3 January 1997 Abstract The status of vapor-phase routes for the rapid densification of high-temperature composite materials, primarily ceramic-matrix composites, is reviewed. Conventional densification of composites such as carboncarbon and Sic-Sic is accomplished by isothermal, isobaric chemical vapor infiltration (CVI), either alone or in combination with liquid resin impregnation and thermal annealing. These are multi-step processes which take from several hundred to thousands of hours at high temperature. In this paper we review approaches designed to significantly reduce the processing time and the number of steps required for densification, while producing materials with the desired properties. We describe techniques such as inductively-heated thermalgradient isobaric CVI, radiantly-heated isothermal and thermal-gradient forced-flow CVI, liquid-immersion thermal-gradient CVI and plasma-enhanced CVI. Different heating methods, such as radiative and inductive, and both hot-wall and cold-wall reactors are compared. Available material properties of composites produced by these techniques are given. 0 1997 Elsevier Science S.A. Keywords: Carbon-carbon composites; Ceramic-matrix composites; Chemical vapor deposition; Chemical vapor infiltration; Composites; Densification; Fiber; Forced flow; High-temperature composites; Inductive heating; Isobaric; Isothermal; Liquid immersion; Matrix: Microwave; Plasma; Preform; Pulsed pressure; Radiant heating; Rapid densification; Sic-Sic; Thermal gradient 1. Preface Composite materials, such as carbon-carbon (C-C), offer advantages of light weight and excellent mechanical and thermal properties, especially for high-temperature applications, e.g. aircraft brake pads, uncooled engine and other airplane parts and leading-edge sections used in rockets [ l-41. Other applications which are receiving attention are in heat-conducting substrates for electronic chips and in sports equipment, for example for tennis, golf and bicycling. Composites generally consist of fibers surrounded by a matrix. Compared to their monolithic counterparts, composites are much tougher mechanically and allow tailoring of the thermal properties, such as the thermal expansion coefficient and thermal conductivity to a much larger extent. Composites also have a much more forgiving failure mode under stress than monolithics, as illustrated in Fig. 1. There are generally three categories of composite materials: polymer-matrix composites, metal-matrix composites and ceramic-matrix composites. In this review, we are concerned with ceramic-matrix composites (CMCs) , fabricated mainly around carbon and Sic fibers: examples are C-C and Sic-Sic composites. These materials can withstand the highest use temperatures. Other fibers studied or used to different extents include glass, silica, alumina, alumino-silicates, silica-titania, silicon nitride, zirconia, yttrium-aluminum garnets, boron-coated tungsten and boron nitride [ 51. One of the most common fabrication methods of such composite structures is densification of a porous body, the preform, having the desired shape and consisting solely or principally of fibers. The fibers may be continuous or chopped. The preform may * Corresponding author. Tel: (201) 4554938. Fax: (201) 455-3008 or (201) 455-3942. E-mail: GOLECKIQRESEARCH. ALLLED.COM. 0927-796X/97/$17.00 8 1997 Elsevier Science S.A. All rights reserved. PZZSO927-796X(97)00003-X
Monolithic Fiber-reinforced ceramIc Displacement or Surai ig. 1. Typical stress-strain curves for a monolithic ceramic, exhibiting brittle fracture, and a tougher fiber-reinforced composite ceramic, exhibiting more extensive displacement due to fiber pullout bc fabricated using techniques such as weaving of continuous fibers, needle-punching of fibrous mats, or mixing of chopped fibers with resins and powders, followed by thermal treatment in the 200-1000 C range lo evaporate organic binders or residues [1]. The geometrical density of such preforms varies widely in the range 10-80% of the theoretical value. In this review we describe the densification of such porous preform structures by means of chemical vapor deposition(CvD) and chemical vapor infiltration(CVI). In particular, we concentrate on those vapor infiltration methods which result in a significantly shorter densification time than achieved by conventional routes. Cvd and Cvi involve flowing one or several streams of precursor vapors containing the desired element or compound over and around the porous part, while keeping that part at a temperature sufficient to decompose the precursor. Temperatures can vary, for example, in the 600-1500'C range, depending on the particular hemistry and system. Total pressures are generally in the range 10to 10 Torr. Under the appropriate conditions, the vapor decomposes to produce the desired element or compound in the desired micros- tructure within the pores of the part, thus increasing its density. Minimum final density values are he desired mechanical and thermal compared to other densification methods, such as multiple cycles of liquid resin impregnation and high-temperature treatment. CVi allows penetration of the desired atoms or molecules into the smallest pores of the preform and does not require post-densification treatment to remove organics. CVI produces uniform and conformal coatings around each accessible fiber and surface in the preform. The final shape of a part densified by CVi is closest to the desired shape(so-called near-net or net shape) so that only minimal or no post-densification machining is required. On the other hand, liquid impreg nation usually results in major shrinkage and microcracking during the required thermal annealing (curing and pyrolysis) cycles. After densification of the composite part by any of the above methods, additional heat or surface treatments may be required, for example, to improve the physical properties of the part or its resistance to environmental attack(e.g. oxidation), depending upon the specific applications. These additional treatments are not covered in this review In CVI, the deposition rate usually increases moderately with increasing precursor partial pressure, p and exponentially with increasing substrate temperature, T, as p"exp(-AH/kr), where n is the pressure exponent, AH is the activation energy (24eV molecule" for carbon)andk=1.3805x10-16 ergk-l=8.614X10-Sevk-is Boltzmann's constant Pressure in this context signifies thepressure in the reactor chamber. A common application of Cvi involves densification of porous carbon sub- strates, thicker than 2.5 cm, where a large number of such substrates may be placed in an enclosure uniformly heated to a temperature in the range 1000-1100"C and exposed to a reactant gas, e.g methane atp-5-100 Torr [3]. This approach is known as hot-wall CVI and its major drawback is an extremely long CVi time of 600-2000 h to achieve the desired density. Furthermore, the process must usually be interrupted several times to permit grinding of the exterior surfaces of the substrates in order to open the pores and allow further infiltration. For practical reasons, it is desirable to reduce the
38 I. Golecki / Rapid vapor-phase densijication of refractory composites Displacement or Strain Fig. 1. Typical stress-strain curves for a monolithic ceramic, exhibiting brittle fracture, and a tougher fiber-reinforced composite ceramic, exhibiting more extensive displacement due to fiber pullout. be fabricated using techniques such as weaving of continuous fibers, needle-punching of fibrous mats, or mixing of chopped fibers with resins and powders, followed by thermal treatment in the 200-1000 “C range to evaporate organic binders or residues [ 11. The geometrical density of such preforms varies widely in the range lO-80% of the theoretical value. In this review we describe the densification of such porous preform structures by means of chemical vapor deposition (CVD) and chemical vapor infiltration (CVI) . In particular, we concentrate on those vapor infiltration methods which result in a significantly shorter densification time than achieved by conventional routes. CVD and CVI involve flowing one or several streams of precursor vapors containing the desired element or compound over and around the porous part, while keeping that part at a temperature sufficient to decompose the precursor. Temperatures can vary, for example, in the 600-l 500 “C range, depending on the particular chemistry and system. Total pressures are generally in the range 10m3 to lo3 Torr. Under the appropriate conditions, the vapor decomposes to produce the desired element or compound in the desired microstructure within the pores of the part, thus increasing its density. Minimum final density values are necessary for achieving the desired mechanical and thermal properties. CVI has several advantages compared to other densification methods, such as multiple cycles of liquid resin impregnation and high-temperature treatment. CVI allows penetration of the desired atoms or molecules into the smallest pores of the preform and does not require post-densification treatment to remove organics. CVI produces uniform and conformal coatings around each accessible fiber and surface in the preform. The final shape of a part densified by CVI is closest to the desired shape (so-called near-net or net shape), so that only minimal or no post-densification machining is required. On the other hand, liquid impregnation usually results in major shrinkage and microcracking during the required thermal annealing (curing and pyrolysis) cycles. After densification of the composite part by any of the above methods, additional heat or surface treatments may be required, for example, to improve the physical properties of the part or its resistance to environmental attack (e.g. oxidation), depending upon the specific applications. These additional treatments are not covered in this review. In CVI, the deposition rate usually increases moderately with increasing precursor partial pressure, p and exponentially with increasing substrate temperature, T, as p” exp( - AH/kT), where n is the pressure exponent, AH is the activation energy (2-4 eV molecule -‘forcarbon) andk= 1.3805X lo-l6 erg K - ’ = 8.614 X 10m5 eV K- ’ is Boltzmann’s constant. Pressure in this context signifies the pressure in the reactor chamber. A common application of CVI involves densitication of porous carbon substrates, thicker than 2.5 cm, where a large number of such substrates may be placed in an enclosure uniformly heated to a temperature in the range 1000-l 100 “C and exposed to a reactant gas, e.g. methane at p = 5-l 00 Torr [ 31. This approach is known as hot-wall CVI and its major drawback is an extremely long CVI time of 6UO-2OUO h to achieve the desired density. Furthermore, the process must usually be interrupted several times to permit grinding of the exterior surfaces of the substrates in order to open the pores and allow further infiltration. For practical reasons, it is desirable to reduce the
I Golecki/Ropid vapor-phase densification of refractory composites processing time. However, increasing the precursor pressure and/or temperature beyond certain ranges may produce deleterious effects, such as: (1) homogeneous nucleation of powders(soot)in the gas phase instead of carbon deposition inside the pores of the substrate,(2) surface crusting and pore plugging before the desired density is reached, and (3)undesirable microstructure of the material Furthermore, in previous studies, the progress of the densification was not readily measurable, except by direct weighing of the parts in small-scale laboratory reactors. The lack of in-situ monitoring may result in non-optimal time and other process conditions In this review, the present status of vapor-phase routes to the rapid densification of high-temp ature composite materials is described. Approaches are reviewed for reducing the processing time by a factor of up to 1000 and the number of densification cycles to one, while producing materials with the desired properties. Techniques are described such as inductively-heated thermal-gradient isobaric CVI, radiantly- heated isothermal and thermal-gradient forced-fiow CVi, liquid-immersion thermal dient CVI, and plasma-enhanced CVI. Different heating methods, for example, radiative and inductive, and both hot-wall and cold-wall reactors are compared. Emphasis is placed on those tech iques which have demonstrated rapid densification of functional components or that show potential for the same. The different types of densification reactors are described and available material properties of composites produced by these techniques are given. This review focuses primarily on experimental results and the reader is directed to listed references for modeling and simulation studies. First, we provide a brief overview of the techniques of CVD and CVI 2. Introduction to chemical vapor deposition <6, The basic concept of forming a thin solid coating on a substrate by chemical vapor deposition P] is illustrated in Fig. 2 [8, 7, 9-11]. A fat Si substrate is placed on a resistance heater in a vacuum chamber and heated by means of infrared (r) radiation emanating from the heater. The walls of the chamber are water cooled and are therefore at or only slightly above room temperature, Gases, such as methylsilane, Si(CH3)H3(the precursor) and hydrogen, H2, are made to flow over and around the Si substrate. The flow rates are usually controlled by means of electronic mass flow controllers for each gas line and the pressure can be controlled independently by means of a throttle valve, which varies the flow conductance to the pumps. If the surface of the substrate is above the decomposition temperature of the precursor gas, a solid coating will be deposited on the substrate surface. Under Quartz Tube Pyrometer Grid Heater Throttle valve Fig. 2. A cold-wall, plasma-enhanced chemical vapor deposition reactor [7]
I. Golecki / Rapid vapor-phase densijcation of refractory composites 39 processing time. However, increasing the precursor pressure and/or temperature beyond certain ranges may produce deleterious effects, such as: ( 1) homogeneous nucleation of powders (soot) in the gas phase instead of carbon deposition inside the pores of the substrate, (2) surface crusting and pore plugging before the desired density is reached, and (3) undesirable microstructure of the material. Furthermore, in previous studies, the progress of the densification was not readily measurable, except by direct weighing of the parts in small-scale laboratory reactors. The lack of in-situ monitoring may result in non-optimal time and other process conditions. In this review, the present status of vapor-phase routes to the rapid densification of high-temperature composite materials is described. Approaches are reviewed for reducing the processing time by a factor of up to 1000 and the number of densification cycles to one, while producing materials with the desired properties. Techniques are described such as inductively-heated thermal-gradient isobaric CVI, radiantly-heated isothermal and thermal-gradient forced-flow CVI, liquid-immersion thermalgradient CVI, and plasma-enhanced CVI. Different heating methods, for example, radiative and inductive, and both hot-wall and cold-wall reactors are compared. Emphasis is placed on those techniques which have demonstrated rapid densification of functional components or that show potential for the same. The different types of densification reactors are described and available material properties of composites produced by these techniques are given. This review focuses primarily on experimental results and the reader is directed to listed references for modeling and simulation studies. First, we provide a brief overview of the techniques of CVD and CVI. 2. Introduction to chemical vapor deposition The basic concept of forming a thin solid coating on a substrate by chemical vapor deposition [ 61 is illustrated in Fig. 2 [8,7,9-l 11. A flat Si substrate is placed on a resistance heater in a vacuum chamber and heated by means of infrared (IR) radiation emanating from the heater. The walls of the chamber are water cooled and are therefore at or only slightly above room temperature. Gases, such as methylsilane, Si(CH3)H3 (the precursor) and hydrogen, Hz, are made to flow over and around the Si substrate. The flow rates are usually controlled by means of electronic mass flow controllers for each gas line and the pressure can be controlled independently by means of a throttle valve, which varies the flow conductance to the pumps. If the surface of the substrate is above the decomposition temperature of the precursor gas, a solid coating will be deposited on the substrate surface. Under . Quartz Tube Fig. 2. A cold-wall, plasma-enhanced chemical vapor deposition reactor [7]
. Golecki/Rapid vapor-phase densification of refractory composites Excitation Coil Glow Discharge Mass Flow Regulator R.F. Generator Fig 3. A hot-wall, a c plasma-enhanced, carbon chemical vapor deposition and infiltration reactor [14] appropriate temperature(e. g. 750-900C), pressure and flow-rate conditions, single-crystalline, epitaxial Sic thin films are obtained on(100)-oriented, single-crystalline Si substrates [8, 7,9-11.A coating will generally also be deposited on the heater surface. This particular configuration is known as cold-wall CVD. Other means of heating a substrate in a cold-wall reactor include:(a) placing the substrate on an electrically conducting susceptor hcated by a water-cooled induction coil and which radiates IR energy onto the substrate;(b) direct Joule heating of the substrate by time-alternatin induced currents if the substrate is sufficiently electrically conducting;(c) direct Joule heating of the substrate by contacting it to a direct-current (d.c. )or an alternating-current(ac. power supply and applying a voltage across it(used, for example, in the coating of fibers [12]); and(d)direct optical (e.g. IR) heating of the substrate by illuminating it through a properly cooled window with high- intensity tungsten-halogen lamps placed outside the vacuum chamber [13]. Cold-wall CVD can be used to coat one substrate at a time or multiple substrates simultaneously. An advantage of cold-wall reactors is reduced autodoping (i.e. impurity incorporation) from heated parts other than the substrates Another common configuration is the hot-wall CVd reactor, illustrated in Fig. 3[14, 15]. Here, generally several substrates are located in a uniform temperature(isothermal) zone of a furmace. A cylindrical tube furmace or a rectangular fumace is generally used. The interior furnace walls are heated by resistance heaters and emit iR radiation into the central region of the furnace. In another arrangement, a water-cooled coil driven by an a c, power supply may be used to inductively heat a hollow cylindrical (i.e. annular) graphite or other electrically conducting susceptor, which is located inside the coil and which surrounds the center of and emits IR radiation into the furnace. Historically, hot-wall CVD has been used to coat a very large number of substrates in the same run. The most important quantity describing a Cvd process is the deposition rate of the coating, measured, for example in um h. Values of deposition rates may vary from =10-to 10 um h The most important parameter influencing the deposition rate for a given material system is the substrate temperature. Fig. 4(a)shows SiC deposition rates on Si measured in the chamber of Fig. 2 [9, 11] Under these conditions, the deposition rate increases approximately exponentially with inverse absolute temperature. Over a wider temperature range, most cvd processes exhibit a dependence of the deposition rate on temperature as shown in Fig. 4(b). The concept of the gas-phase boundary layer Fig. 5), although a simplification of the actual process, is useful in understanding the behavior of deposition rate in CVD. Simply stated, to produce a solid coating on the heated substrate:(1)the aneous precursor has to be transported from the center of the gas stream to the boundary layer, (2) the precursor then diffuses across the boundary layer to reach the surface of the substrate, and(3)the precursor decomposes on the surface of the substrate to form the solid coating. The last step involves additional sub-processes, including adsorption of precursor-derived moieties on the surface, desorption
40 I. Golecki / Rapid vapor-phase densijcation of refractory composites Furnace /Mass Flow Regnlatorll R.F. Generator 1 7 Fig. 3. A hot-wall, ax. plasma-enhanced, carbon chemical vapor deposition and infiltration reactor [ 141, appropriate temperature (e.g. 750-900 “C), pressure and flow-rate conditions, single-crystalline, epitaxial Sic thin films are obtained on ( lOO)-oriented, single-crystalline Si substrates [8,7,9-l 11. A coating will generally also be deposited on the heater surface. This particular configuration is known as cold-wall CVD. Other means of heating a substrate in a cold-wail reactor include: (a) placing the substrate on an electrically conducting susceptor heated by a water-cooled induction coil and which radiates lR energy onto the substrate; (b) direct Joule heating of the substrate by time-alternating induced currents if the substrate is sufficiently electrically conducting; (c) direct Joule heating of the substrate by contacting it to a direct-current (dc.) or an alternating-current (a.c.) power supply and applying a voltage across it (used, for example, in the coating of fibers [ 121) ; and (d) direct optical (e.g. JR) heating of the substrate by illuminating it through a properly cooled window with highintensity tungsten-halogen lamps placed outside the vacuum chamber ] 131. Cold-wall CVD can be used to coat one substrate at a time or multiple substrates simultaneously. An advantage of cold-wall reactors is reduced autodoping (i.e. impurity incorporation) from heated parts other than the substrates. Another common configuration is the hot-wall CVD reactor, illustrated in Fig. 3 [ 14,151. Here, generally several substrates are located in a uniform temperature (isothermal) zone of a furnace. A cylindrical tube furnace or a rectangular furnace is generally used. The interior furnace walls are heated by resistance heaters and emit IR radiation into the central region of the furnace. In another arrangement, a water-cooled coil driven by an a.c. power supply may be used to inductively heat a hollow cylindrical (i.e. annular) graphite or other electrically conducting susceptor, which is located inside the coil and which surrounds the center of and emits JR radiation into the furnace. Historically, hot-wall CVD has been used to coat a very large number of substrates in the same run. The most important quantity describing a CVD process is the deposition rate of the coating, measured, for example in pm h- ‘. Values of deposition rates may vary from = 10m2 to lo3 p,m h-‘. The most important parameter influencing the deposition rate for a given material system is the substrate temperature. Fig. 4(a) shows Sic deposition rates on Si measured in the chamber of Fig. 2 [9,11]. Under these conditions, the deposition rate increases approximately exponentially with inverse absolute temperature. Over a wider temperature range, most CVD processes exhibit a dependence of the deposition rate on temperature as shown in Fig. 4(b). The concept of the gas-phase boundary layer (Fig. 5)) although a simplification of the actual process, is useful in understanding the behavior of deposition rate in CVD. Simply stated, to produce a solid coating on the heated substrate: (1) the gaseous precursor has to be transported from the center of the gas stream to the boundary layer, (2) the precursor then diffuses across the boundary layer to reach the surface of the substrate, and (3) the precursor decomposes on the surface of the substrate to form the solid coating. The last step involves additional sub-processes, including adsorption of precursor-derived moieties on the surface, desorption
. Golecki/ Rapid vapor-phase densification of refractory composites T(C) 900 800 700 Pn-=100W 0.g 1000/T(1/K) Gas-phase i Gas-phase Surface nucleation imass transport chemical kinetics 1/T(K Fig. 4.(a)Dependence of Sic deposition rate from methylsilane and hydrogen on temperature and plasma power [9].(b)Three regimes in chemical vapor deposition Axis as Flow d Boundary Layer N Fig. 5. Boundary layer fow in chemical vapor deposition of other moieties from the surface, surface diffusion and chemical reactions. The deposition rate, rof a film on a substrate can be expressed [6] by means of Eq. (1) r=CE[k。hg/(k。+hg】/N where Cg is the precursor concentration in the gas phase, ks is the rate constant for heterogeneous decomposition of the precursor into the film on the surface of the substrate, hg is the gas-phase mass transfcr cocfficicnt of precursor to the substrate and Ns is a normalizing constant. The deposition rate is proportional to the concentration of the precursor in the gas phase; the precursor may be, but often is not, the input chemical introduced into the reactor. The gas-phase mass-transfer coefficient can be expressed as hg=D/db, where D is the gas-phase diffusivity and dD is the thickness of the boundary
I. Golecki/ Rapid vapor-phase densijication of refractory composites 41 (a) T (“C) 900 800 700 t’ ’ I I I I 4 P,=looW 1.24 eV 0.8 0.9 1.0 1000/T (I/K) Gas-phase i Gas-phase f Surface nucleation i mass transports chemical kinetics (b> l/T (K-l) Fig. 4. (a) Dependence of Sic deposition rate from methylsilane and hydrogen on temperature and plasma power [9]. (b) Three regimes in chemical vapor deposition. Axis __-._.-_-.-.-.-.---_-.-~-~-. z - Gas Flow t Boundary Layer Substrate Fig. 5. Boundary layer flow in chemical vapor deposition. of other moieties from the surface, surface diffusion and chemical reactions. The deposition rate, Y of a film on a substrate can be expressed [ 61 by means of Eq. ( 1) : r=C,[k,h,l(k,+h,)]lN, (1) where C, is the precursor concentration in the gas phase, k, is the rate constant for heterogeneous decomposition of the precursor into the film on the surface of the substrate, h, is the gas-phase masstransfer coefficient of precursor to the substrate and iV, is a normalizing constant. The deposition rate is proportional to the concentration of the precursor in the gas phase; the precursor may be, but often is not, the input chemical introduced into the reactor. The gas-phase mass-transfer coefficient can be expressed as h, = D/d,, where D is the gas-phase diffusivity and &, is the thickness of the boundary