MECHANISM OF LIGHT TRANSMISSION THROUGH WET POLYMER FILMS

Transcription

1 MECHANISM OF LIGHT TRANSMISSION THROUGH WET POLYMER FILMS A. Jaffrin, S. Makhlouf I.N.R.A. URIH Sophia Antipolis; route des Colles; BIOT-France. Abstract. The mechanism responsible for the reduction of the light transmission of polymer films in the presence of water deposits is studied from classical optics concepts. The hemi-spherical dioptres created by condensation droplets on non wettable polymers can be responsible for a 40 % reduction of the optical transmittance of a film at normal incidence, and slightly less for obligue rays. On the contrary, the complete wettability obtained by the inclusion of chemical agents in the polymer resin makes wetted films to be even more transparent than dry films. Numerical simulations based on these optical concepts accurately reproduce thermal effects resulting from a loss of wettability of a film used for soil solarisation. 1. Introduction. Polymere resins, when transformed into films or rigid plates could not, until now, offer the same optical transmittance as mineral glass in the presence of water condensates. This different behavior is related to the presence of hydrophobic radicals on the surface of polymers, contrary to glass where the hydrophilic caracter is due to silice. The only polymer with surface property somehow closer to that of glass is poly-metacrylate. This non wettable character is particularly sensible on the polyethylene (PE) or ethyl-vinyl-acetate (EVA) resins used to produce film for agricultural purposes. Such films, placed in a humid environment, can be covered with condensation droplets and lose part of their transparency. This defect had immediately been noticed by growers who used transparent mulches and needed to have a clear observation of the plant development stage (this is the case of asparagus for instance); resin manufacturers started to produce anti-drop mulches long ago. The incidence of the drop-like condensation on the thermal or photosynthetic performance of a greenhouse is also very important but has long been underestimated: little attempt has been made up to now to have long-lasting antidrop films. The same effect occurs inside the air channels of double layer rigid plastic transparents pahnels obtained by extrusion process (polycarbonate or acrylic pannels). Even if the pannels are sealed on their ends, the high permeability of such plastics to air and water vapor allows Acta Horticulturae 281, 1990 Greenhouse Construction, Design 11

2 for water transfers and condensation inside the channels. This is commonly observed in sealed 6 mm polycarbonate pannels used as tunnel greenhouse cover. The present study intends to analyse the detailed optical paths of solar beams incident through polymer films with water condensation on their inner surface: the presence of water can take the form of half-spherical droplets, if the polymer is non-wettable, or the form of a continuous layer, if the polymer contains an additive which reduces the surface tension of the water. Optical transmission coefficients are calculated and actual consequences from a thermal point of view are considered. Soil solarisation experiments under under various films are reported and numerical simulations of soil temperatures under wetted or non wetted films are compared with observed values. 2. Optical transmission of parallel dioptres. To simplify the notations, we shall consider optical coefficients averaged over the whole range of solar wave lengths, so that no wave length dépendance appears any more in the coefficients. In addition, all optical media are supposed to have a négligeable optical absorption: only reflexion and transmission are assumed to take place Single dioptric plane. A plane dioptre separating 2 optical media (i) and (j) with refraction indices n. and n., is characterized by a reflexion coefficient r.. àffectirlg a radiation beam of incidence angle ^ (accoràlng to Fresnel's law): r ±j = 1/2 sin(0 i -0 j )/sin(0 i +0 j ) 2 + 1/2 tg(0 i -0 j )/tg( i +0 j ) 2 where the angle of refraction 0^ is given by Descart's law: n. sin0. = n. sin D D The transmission coefficient, in the absence of absorption, is : t.. = 1 - r.. ID ID 2.2. Two parallel dioptric planes. A polymere sheet (plate or film), a medium denoted by (2), placed between two optical media (1) and (3), is submitted to multiple reflexions, like shown in Fig. 1. The transmission coefficient T 1?, can be expressed as a series of single dioptric coefficients r^ and t^j: 2 2 T 123 = fc 12 fc 23 + r 12 r 23 + r 12 r

3 a series which is easily summed up into: T 123 = t 1 " r 12^1 " ~ r 12 r 2 3 ^ The resulting reflexion coefficient writes: R 123 = [r 12 + r 23 " 2 r 12 r 23 3/[1 " r 12 r 23 ] 2.3. Three parallel dioptric planes. This is the situation of perfectly wetted plastic films or glass sheets. The formulas are written in the general case of 4 successive optical media 1, 2,3 and 4. The problem is easily related to the previous case, as suggested by Fig. 2. The global transmission coefficient, denoted by T, is obtained as = T 123 fc 34 [1 + R 123 r 34 + R 123 r 34 + ] an infinite series which can be summed up as: (1 - r 12 )(l - r 23 )(l - r 34 ) T = C 1 - r 12 r 23 " r 34 ( r 12 + r 23-2 r 12 r 23> J In usual circomstances, optical media 1 and 4 are identical (the air); if we denote by n the refraction index of polymer relative to air, by n ^that of water relative to air and by r, r and r the w reflexion coefficients for each of the ^successive dioptric planes, we can write the transmission coefficient of a perfectly wetted polymer as: T = < 1 - r p )(1 - r pw )(1 ' V f 1 * r p r pw " r w ( r p + r pw " 2 ^ V 5 ] and that of a dry polymer as: T = (1 - r p ) 2 /(l - r p 2 ) = (1 - r p )/(l + r p ) For most incidence angles, the reflexion coefficients r. are small compared to unity, and the second order terms can be neglected; this enables to express the ratio T /T into the approximate form: T/T=(l-r v w pw' )(1 v - r w ' )(1 v + r p' ) = 1 + r - r -r p pw w This shows that it is possible, when r > r + r, to have a transmission coefficient larger fop a 8etted w film than for a dry film. The results of a complete calculation, using the exact expressions of the reflexion coefficients involved in the formulas, confirms this statement (see 13

4 chap. 5. below). 3. Optical transmission of a hemi-spherical lens. A condensation droplet hanging on a polymer surface is assumed to have a strictly hemi-spherical shape for simplicity, although other situations (like shown in Fig. 3) are possible. On Fig. 4 are drawn various optical paths of incident radiation beams, depending on their position with respect to the optical center of the lens: those away from the center undergo successive total reflexions; those closer to the center are transmitted after some deviation. This last property is at the origin of the concept of "hortical glass" which was long used for greenhouses: smooth and shallow corrugations with approximate spherical shape were created on one side of the glass sheets (the inner side) to help diffuse the solar radiation in various directions and prevent the "burning effect" of direct light on the plants. This diffuseness effect is. obtained at the cost of some additional reflexion losses at large incidence angles as shown below Hemi-spherical lens under normal incidence. The limiting angle for total reflexion is given by: sin 0 lim 1/n where n is the index of water relatively to the air (n is close lo 1.33). The cercle delimiting the inner region, allowing light transmission, has a radius R terms of the drop radius R by: lim given in lim R sin 0,. = R/n lim w The fraction of the lens disk free giving rise to light transmisión is therefore: z = 1/n w (i.e. roughly 57 %) Hemi-spherical lens under oblique incidence. The Fig. 5 shows how to relate the optical path of an oblique ray to that obtained under normal incidence: this is done by tilting the plane of the lens by the angle related to 0. by: D sin 0. = n sin 0. i : In that virtual plane, the inner region with optical transmission is the same as before; this corresponds, in the actual plane of the lens, to an inner ellipse obtained by ortogonal projection of the inner circle. The ellipse area is given by: U

5 S = n R 2 /(n 2 COS0.) w 3 = n R 2 /[n w /(n w 2 -sin 2 0 i )] and depends of course on the angle of incidence 0.. It is remarkable that the resulting transmission coefficient r z = [nv(n 2 -sin 2 0)] _1 (where the notations have been simplified by omitting suffixes), reaches its maximum for tangential incidence 0=90. It is then roughly egual to 86 %. This conclusion only holds for an hemi-spherical lens: for smaller portions of spheres, like the shallow corrugations of horticole glass, the fraction of the disk giving rise to transmission can be as large as 100 % at normal incidence, but it then decreases for larger incidences, to a minimum of 50 % for tangential directions, as illustrated in Fig Application of lens optics to non-wettable polymers. An assumption on the distribution of condensation droplets on the inner surface of a non-wettable polymer is now necessary. One could imagine various types of disks arrangements, along a sguared or hexagonal mesh, to pave the plane of the film with identical droplets. But direct observations show that drops grow by mutual capture and retract themselves under the effect of surface tension. They tend to form erratic pattern involving every possible sizes of droplets, arranged in a seemingly dense fashion like suggested in Fig. 7. This gives a sound basis for adopting the simplest hypothesis: that of a fractalassembly of droplets leaving no free space between themselves, and thus having the same effect as a unigue drop covering the entire surface. Within this hypothesis, the transmission coefficient r computed above is valid for the entire film. The second effect to take into account is the partial reflexion taking place on the film sides, a process which was discussed in 2.2. A third effect is the partial reflexion affecting the radiation rays incident on the inner surface of the droplet dioptric sphere: it is more difficult to treat exactly this term because of the widely varying incidence angles involved in the multiple reflexion processes. To get an estimate of this effect, a mean incident angle 0 = 30 (situated in the middle of the 0-50 effective range) was considered for these partial reflexions. By including all these effects, one can obtain a reasonable approximation of the optical transmission coefficient of a non-wettable polymer covered with 15

6 condensation droplets under the form: [1 - r (0)][1 - r (0)1[1 - r (0 )] T _ p v ' pw v ' J 1 w v o' 1 [1 - r (0)r ( ) ] n [n 2 - sin 2 ] 55 L P pw v ' 1 w L J w where n w is the refraction index of water with respect to the air, and r, r and r are the reflexion coefficients of the various p diop??ic systems (air-polymer, polymer-water and water-air). 5. Results of the analytical modelisation. Replacing the Fresnel expressions of the reflexion coefficients r.(0) in the previous formula for T gives the final analytical expression for the optical transmission coefficient of a non-wettable polymer film with condensation on its lower side. Fig.8 shows the comparison between a single dioptric plane (curve a), a dry film (curve b), a perfectly wetted film (curve c) and a film with droplet condensation (curve d): the wetted film is endeed more transparent than the dry film for incidence angles less than 60. It loses its advantage for large incidence angles. But the non-wettable film transmits very poorly the incident radiation for incidences up to 60. It starts recovering a better transparency after 70, when the inner elliptical regions of the drops extend to a larger fraction of the surface. 6. Thermal consequences of polymer light transmittance. The implication of a better film transmittance in a humid environment should translate itself into better plant growth and higher quality for crops obtained under plastic greenhouses. For the time being, as wettable (often called "anti-drop") greenhouse films are just begining to appear on the market and still do not keep their surface properties for a long period, the effect on photosynthesis has yet to be proved on the long run; but thermal consequences are simpler and can be checked on shorter time scales. Soil solarisation experiments performed in Antibes INRA Center have indeed shown [1 ] that temperatures achieved under an anti-drop EVA plastic film were much higher than temperatures obtained under the same film after it had lost its wetting agent. This is what appears in Fig. 9 a & b, which report actual temperatures measured in these two situations. This is conforted by the curves of Fig. 10 a & b which accuratly reproduce the experimental values by a numerical simulation applied to both situations. 7. Conclusion. In this study, the optical transmission of a polymer film in the presence of water condensates was considered by analysing reflexion losses. A more complete approach would 16

7 need to include absorption by the polymer itself. But it can already be concluded that the dominant factor is the wettability of the polymer: only wettable polymers should be used as an alternative to glass in humid environments when light transmittance is important. It is hoped that further progress will soon be made to increase the lifetime of the wetting agents included into the polymer resins. Reference : 1- Jaffrin A., Makhlouf S., Scotto La Massese C., Bettachini A. Voisin R.; Effet de la mouillabilitè d'un film polymère sur les températures et l'action nêmaticide obtenues en solarisation d'un sol de culture. Rev. Agronomie Sept. 89. * 17

8 R tv. 2.2 t r r t 2 rv 3 t r r' < Ht r 2 r ' 2 ^ t r 3 r' 3 t r 1.. t r r 2.3 t r r ' t t' 2 2 t t'r r' t t'r r 1 Figure 1: Transmission through 2 parallel dioptric T planes T 2 r T r R T r R 2 2' * 33 T r R T r R T r R T r \ /T r R \ / T r R ^ T t AT t r R ^ T t r 2 R 2 "A.. T Figure 2: Transmission through 3 parallel dioptric planes. 18

9 Figure 3: Various droplet shapes on a polyme 19

10 Figure 5: Reflexive cross section for a hemi-spherical droplet.

11 Figure 6: idem for a cap-shaped droplet.

12 Figure 7: Random distribution of droplets. Figure 8: Film optical transmittance under various conditions (see text). 22

13 Figure 9: Soil temperatures observed under a film; (a): wetted film ; (b) film with droplets. 23

14 Figure 10: Results of a numerical simulation of soil temperatures under a film with a continuous layer of water condensates (a), and with condensates forming a random distribution of droplets (b). 24