Stelios Katsanevakis
email: stelios@katsanevakis.com

Yiannis Issaris
email: yissaris@gmail.com

Dimitris Poursanidis
email: dpoursanidis@gmail.com

Maria Thessalou- Legaki

Vulnerability of marine habitats to the invasive green alga Caulerpa racemosa var. cylindracea within a marine protected area

Keywords: benthic ecology, biological pollutants, Caulerpa racemosa, invasive macrophytes, phytobenthos, seagrass, vulnerability, Mediterranean, MPA, Greece

The relative vulnerability of various habitat types to Caulerpa racemosa var. cylindracea invasion was investigated in the National Marine Park of Zakynthos (Ionian Sea, Greece). The density of C. racemosa fronds was modelled with generalized additive models for location, scale and shape (GAMLSS), based on an information theory approach. The species was present in as much as 33% of 748 randomly placed quadrats, which documents its aggressive establishment in the area.

The probability of presence of the alga within randomly placed 20 x 20 cm quadrats was 83% on 'matte morte' (zones of fibrous remnants of a former P. oceanica bed), 69% on rocky bottoms, 86% along the margins of Posidonia oceanica meadows, 10% on sandy/muddy substrates, and 6% within P. oceanica meadows. The high frond density on 'matte morte' and rocky bottoms indicates their high vulnerability. The lowest frond density was observed within P. oceanica meadows. However, on the margins of P. oceanica meadows and within gaps in fragmented meadows relative high C. racemosa densities were observed. Such gaps within meadows represent spots of high vulnerability to C. racemosa invasion.

Introduction

More than 900 introduced species have been reported so far from the Mediterranean Sea, of which ~500 are considered as established. Among them, there are more than 100 established alien macrophytes [START_REF] Zenetos | Additions to the annotated list of marine alien biota in the Mediterranean with special emphasis on Foraminifera and Parasites[END_REF]. Approximately one tenth of these species can be considered as invasive, i.e. they have overcome biotic and abiotic barriers, are able to disseminate away from their area of initial introduction, and have a noticeable ecological and economic impact. Decline of biodiversity, effects on phytobenthic, fish and invertebrate biota, threat to the ecological stability of invaded ecosystems, and pressures on economic activities are among the impacts of invasive macrophytes [START_REF] Boudouresque | Biological pollution in the Mediterranean Sea: invasive versus introduced macrophytes[END_REF][START_REF] Zenetos | Annotated list of marine alien species in the Mediterranean with records of the worst invasive species[END_REF][START_REF] Schaffelke | Impacts of introduced seaweeds[END_REF][START_REF] Piazzi | Invasion of alien macroalgae in different Mediterranean habitats[END_REF].

The green alga Caulerpa racemosa (Forsskål) J. Agardh sensu lato (Chlorophyta: Bryopsidales) is widely distributed in tropical and warm temperate seas [START_REF] Verlaque | On the identity and origin of the Mediterranean invasive Caulerpa racemosa (Caulerpales, Chlorophyta)[END_REF]. Three intraspecific taxa of C. racemosa have been identified in the Mediterranean Sea, among which C. racemosa var. cylindracea (Sonder) Verlaque, Huisman, and Boudouresque [START_REF] Verlaque | On the identity and origin of the Mediterranean invasive Caulerpa racemosa (Caulerpales, Chlorophyta)[END_REF][START_REF] Klein | The Caulerpa racemosa invasion: A critical review[END_REF] (hereafter C. racemosa) is one of the most aggressive invaders in the basin [START_REF] Streftaris | Alien marine species in the Mediterranean -the 100 'worst invasives' and their impact[END_REF][START_REF] Verlaque | Blitzkrieg in a marine invasion: Caulerpa racemosa var. cylindracea (Bryopsidales, Chlorophyta) reaches the Canary Islands (North-East Atlantic)[END_REF][START_REF] Klein | The Caulerpa racemosa invasion: A critical review[END_REF]. This variety was first found in the Mediterranean Sea along the Libyan coast in 1990 [START_REF] Nizamuddin | The green marine algae of Libya[END_REF], being introduced from south-western Australia [START_REF] Famà | High levels of intra-and inter-individual polymorphism in the rDNS ITS1 of Caulerpa racemosa (Chlorophyta)[END_REF][START_REF] Verlaque | On the identity and origin of the Mediterranean invasive Caulerpa racemosa (Caulerpales, Chlorophyta)[END_REF]. It has rapidly spread throughout the entire basin, reaching as far as the Canary Islands in the Eastern Atlantic Ocean [START_REF] Verlaque | Blitzkrieg in a marine invasion: Caulerpa racemosa var. cylindracea (Bryopsidales, Chlorophyta) reaches the Canary Islands (North-East Atlantic)[END_REF]. The species has been reported from all kind of substrata with a variety of benthic assemblages,

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4 between 0 and 70 m depth, in both polluted and unpolluted areas, and has demonstrated an excessive rate of proliferation and high adaptability to physical stressors [START_REF] Piazzi | Interactive effect of sedimentation and Caulerpa racemosa var. cylindracea invasion on macroalgal assemblages in the Mediterranean Sea[END_REF][START_REF] Streftaris | Alien marine species in the Mediterranean -the 100 'worst invasives' and their impact[END_REF][START_REF] Klein | The Caulerpa racemosa invasion: A critical review[END_REF][START_REF] Tsiamis | Alien marine macrophytes in Greece: a review[END_REF]. Its stolons can quickly elongate and easily overgrow other macroalgal [START_REF] Piazzi | Threat to macroalgal diversity: effects of the introduced green alga Caulerpa racemosa in the Mediterranean[END_REF][START_REF] Piazzi | Persistence of biological invasion effects: Recovery of macroalgal assemblages after removal of Caulerpa racemosa var. cylindracea[END_REF] or invertebrate species [START_REF] Kružić | The highly invasive alga Caulerpa racemosa var. cylindracea poses a new threat to the banks of the coral Cladocora caespitosa in the Adriatic Sea[END_REF][START_REF] Baldacconi | Effects of the spread of the alga Caulerpa racemosa var. cylindracea on the sponge assemblage from coralligenous concretions of the Apulian coast (Ionian Sea, Italy)[END_REF]. Quite often C. racemosa forms compact meadows in the Mediterranean Sea [START_REF] Argyrou | Expansion of the macroalga Caulerpa racemosa and changes in soft bottom macrofaunal assemblages in Moni Bay, Cyprus[END_REF][START_REF] Ruitton | Seasonal changes of the introduced Caulerpa racemosa var. cylindracea (Caulerpales, Chlorophyta) at the northwest limit of its Mediterranean range[END_REF][START_REF] Klein | The Caulerpa racemosa invasion: A critical review[END_REF][START_REF] Katsanevakis | Spatial distribution, abundance and habitat use of the protected fan mussel Pinna nobilis in Souda Bay, Crete[END_REF], contrary to its native range in south-western Australia, where it does not form monospecific stands but grows intermixed with other algae [START_REF] Verlaque | On the identity and origin of the Mediterranean invasive Caulerpa racemosa (Caulerpales, Chlorophyta)[END_REF]. It has a high potential for dispersal by fragmentation and re-establishment of stolons [START_REF] Ceccherelli | Dispersal of Caulerpa racemosa fragments in the Mediterranean: Lack of detachment time effect on establishment[END_REF], formation of propagules [START_REF] Renoncourt | Formation of propagules on an invasive strain of Caulerpa racemosa (Chlorophyta) in the Mediterranean Sea[END_REF], and sexual reproduction [START_REF] Panayotidis | Sexual reproduction of the invasive green alga Caulerpa racemosa var. occidentalis in the Mediterranean Sea[END_REF].

C. racemosa was initially reported in Greece from Zakynthos Island and Pylos Bay (Ionian Sea) in 1993 [START_REF] Panayotidis | Caulerpa racemosa (Chlorophyta) on the Greek coasts[END_REF]. Since then, the species has been spreading extremely rapidly along Greek coasts and it is now found on the majority of the Greek coastal areas [START_REF] Tsiamis | Updated records and range expansion of alien marine macrophytes in Greece (2009)[END_REF][START_REF] Salomidi | Invasion of the green alga Caulerpa racemosa in the Hellenic coasts[END_REF].

There are a few studies in the literature comparing invasion success of C. racemosa on different substrates such as varying shoot densities of Posidonia oceanica (L.) Delile meadows [START_REF] Ceccherelli | Response of the non-indigenous Caulerpa racemosa (Forsskål) J. Agardh to the native seagrass Posidonia oceanica (L.) Delile: effect of density of shoots and orientation of edges of meadows[END_REF] or different types (encrusting, filamentous, and erect) of vegetation layers [START_REF] Ceccherelli | Spread of introduced Caulerpa species in macroalgal habitats[END_REF]. However, the extent to which biotic and abiotic interactions influence the spread of the alga is not fully understood, although there are some relevant efforts (e.g., [START_REF] Bulleri | Facilitation of the introduced green alga Caulerpa racemosa by resident algal turfs: experimental evaluation of underlying mechanisms[END_REF]. So C. racemosa, in one of the sites of its first appearance in the early 90's (Laganas Bay, Zakynthos Island, Ionian Sea). Special focus was given on various substrate types within fragmented Posidonia oceanica meadows. Hereafter, vulnerability is defined as the susceptibility to the introduction, establishment and spatial spread of the invader, i.e. how easily it may attain high densities.

Until lately Marine Protected Areas (MPAs) were established to protect biodiversity via the complete removal of human exploitation and other activities, and thus were intended to be "islands" of nature and tranquility in a sea of hostile environment [START_REF] Simberloff | No reserve is an island: marine reserves and nonindigenous species[END_REF][START_REF] Boudouresque | Marine portected areas: a tool for coastal areas management[END_REF]. However, in many cases MPAs have not been able to stop or limit the expansion of invasive alien species. For example, the boundaries of many northwestern Mediterranean marine parks offered no protection to the invasion of the green alga Caulerpa taxifolia, once it was present along neighboring coasts [START_REF] Meinesz | Killer algae[END_REF]. International shipping is usually prohibited in MPAs but recreational or small fishing boats could equally well be the vectors of alien species introductions. Indeed an assessment of the importance of recreational boating carried out in 100 yachts in Scotland has confirmed that boating must be considered as a high risk vector for non-native species [START_REF] Ashton | Risk assessment of hull fouling as a vector for marine non-natives in Scotland[END_REF]). The spreading of both Caulerpa racemosa and C. taxifolia in the western Mediterranean was facilitated by fishing activity, in particular by bottom trawlers and trammel nets [START_REF] Relini | The role of fishing gear in the spreading of allochthonous species: the case of Caulerpa taxifolia in the Ligurian Sea[END_REF][START_REF] Magri | La présence de Caulerpa racemosa le long des côtes septentrionales de la Toscane et les conséquences possibles sur l'activité de pêche[END_REF] For all the points above we aimed to quantify the potential differences among habitats/substrate types and management zones by appropriately modelling the frond density of C. racemosa.

Materials and Methods

Study area

The survey was conducted in the National Marine Park of Zakynthos (NMPZ) (Fig. 1). The bay of Laganas in Zakynthos Isl. hosts the most important known nesting aggregation of the endangered loggerhead sea turtle, Caretta caretta (Linnaeus,

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), in the Mediterranean. NMPZ was founded in 1999, with the protection of the critical nesting habitat of the loggerhead sea turtle being its main objective.

Within the marine area of the NMPZ there are three main management zones (A, B, C) in Laganas Bay and a buffer zone (Ia) (Fig. 1). In Zone A no recreational boat activity is allowed, in Zone B boats are allowed to enter with a speed limit of 6 knots but anchoring is forbidden, and in Zone C boats are allowed to travel with a speed limit of 6 knots and also to anchor. Commercial fishing using mobile gear and recreational fishing is prohibited in all zones (including the buffer zone) and commercial fishing using static gears is prohibited in Zone A from the beginning of May until the end of October. The management zones do not conform to some specific features of the marine environment, but they were established almost exclusively on criteria regarding undisturbed approach of turtles to the nesting beaches. The three management zones and the buffer zone provide a quasiexperimental setting of varying level of protection (from very low in Ia to high in zone A), which was exploited in this study to test our third hypothesis. (3) narrow range of surveyed depths. The latter requirement was to avoid adding depth as an extra covariate, as there is an important bathymetry-related variation of C. racemosa density [START_REF] Salomidi | Invasion of the green alga Caulerpa racemosa in the Hellenic coasts[END_REF], and also for logistical reasons for diving. Thus, all measurements were conducted at depths between 5 and 13 m. The above restrictions caused a non-uniform spatial distribution of the sampling sites in the study area. Specifically, the central regions of zones B and Ia were not sampled as they were generally deep (reaching 70 m in zone Ia), while the shallow regions (north part of zone B) had exclusively sandy bottoms (all other habitat/substrate types were absent).

Field work

The line transects were defined with a nylon line, marked with water-resistant numbered signs every five meters, that was deployed using a diving reel. After deploying each line transect, the number of C. racemosa fronds within a 20x20 cm quadrat was counted at each sign, i.e. eleven measurements were conducted along each line. For each quadrat, the habitat type was also logged. Five different habitat types were identified: unvegetated sandy bottoms (S); Posidonia oceanica meadows (PO); 'matte morte' (MM); margins of P. oceanica meadows, usually with sandy

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9 bottoms or 'matte morte' (PM); and rocky bottoms (R). Although, PM is strictly not a separate 'habitat type' than PO, we catachrestically use this terminology hereafter.

In the second survey, the number of C. racemosa fronds within 20x20 cm quadrats was measured in a total of 44 patches distributed in 11 sites (Fig. 1), where fragmented P. oceanica meadows was the dominant ecotope. In each site, a 200-m line was deployed defining a straight line transect. When the line crossed a patch, frond measurements were conducted in six locations along the line, representing three different substrate types: at the two intersections of the patch margin and the line (PM), within the P. oceanica meadow at a 2 m distance from the margin of the patch (PO), and in two randomly chosen spots within the patch (Pa) (Fig. 2). The size of the patch, defined as its maximum dimension (in m), was also recorded.

Modelling frond counts

Some distinctive features of data collected on the abundance of organisms (count data) are: (i) the tendency to contain many zero values (zero-inflated dataset); (ii) overdispersion, defined as the extra variation, which is not explained by the Poisson distribution alone; (iii) heterogeneity of variances (where the variance of the distribution of the response variable changes with explanatory variables). Failure to account for these issues will cause bias in parameter estimations and the associated measures of uncertainty, can reduce the capability to detect relationships, or even lead to incorrect inference [START_REF] Martin | Zero tolerance ecology: improving ecological inference by modelling the source of zero observations[END_REF][START_REF] Stasinopoulos | Generalized additive models for location, scale, and shape (GAMLSS) in R[END_REF].

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10 Herein, generalized additive models for location, scale and shape (GAMLSS; [START_REF] Rigby | Generalized additive models for location, scale and shape[END_REF] were applied to effectively confront the abovementioned issues. GAMLSS are regression type models that overcome many of the limitations associated with the popular generalized linear models (GLM; McCullagh and Nedler, 1989) and generalized additive models (GAM; [START_REF] Hastie | Generalized Additive Models[END_REF]. In GAMLSS the exponential family distribution assumption for the response variable is relaxed and replaced by a general distribution family, and the systematic part of the model is expanded to allow modelling not only of the mean (or location) but also other parameters of the distribution [START_REF] Stasinopoulos | Generalized additive models for location, scale, and shape (GAMLSS) in R[END_REF]. For fitting the models, the gamlss package [START_REF] Stasinopoulos | Generalized additive models for location, scale, and shape (GAMLSS) in R[END_REF] was used in R v.2.8.0 (R Development Core Team 2008).

Inference was based on the information theory approach [START_REF] Burnham | Model selection and multi-model inference[END_REF], i.e. an integrated process of a priori specification of a set of candidate models, model selection based on the principle of parsimony according to Akaike's information criterion (AIC; [START_REF] Akaike | Information theory as an extension of the maximum likelihood principle[END_REF], and the estimation of parameters and their precision.

For data collected in the first survey, two categorical variables, habitat type (habitat) and the corresponding management zone of the NMPZ (zone) were used as potential predictor variables of the C. racemosa frond counts. Seven different GAMLSS, g i , i = 0 to 6, were fitted (Table 1). Either only the mean response (models g 1 -g 3 ) or both the mean response and the standard deviation (models g 4 -g 6 ) were modelled, with one or both predictor variables, i.e. in models g 1 -g 3 the standard deviation was assumed M A N U S C R I P T A C C E P T E D constant, while in g 4 -g 6 dependent on the predictor variables. Model g 0 was the null model, with no predictor variable, while g 6 was the full model.

To identify the best underlying distribution of the response variable, six potential distributions were compared: Poisson, negative binomial [START_REF] Anscombe | Sampling theory of the negative binomial and logarithmic series approximations[END_REF], zeroinflated Poisson [START_REF] Lambert | Zero-inflated Poisson regression with an application to defects in manufacturing[END_REF], Poisson-inverse Gaussian [START_REF] Dean | A mixed Poisson-inverse Gaussian regression model[END_REF], Sichel [START_REF] Rigby | A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution[END_REF], and Delaporte [START_REF] Rigby | A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution[END_REF]. Comparisons among different error distributions were based on the full model (g 6 ), and selection was based on AIC. The AIC differences,

min i AIC AIC - = ∆ i
, were computed over all candidate models.

Models with ∆ i > 10 had essentially no support and could be omitted from further consideration, while all models with ∆ i < 2 had substantial support [START_REF] Burnham | Model selection and multi-model inference[END_REF]. To quantify the plausibility of each model, given the data and set of six models, the 'Akaike weight' w i of each model was calculated, where

∑ ∆ - ∆ - = j j i i w ) 5 . 0 exp( ) 5 . 0 exp(
. The 'Akaike weight' is considered as the weight of evidence in favour of model i being the actual best model of the available set of models [START_REF] Akaike | Information measures and model selection[END_REF][START_REF] Buckland | Model selection: An integral part of inference[END_REF][START_REF] Burnham | Model selection and multi-model inference[END_REF].

Estimates of the relative importance of predictor variables j was made by summing the Akaike weights across all the models in the set where each variable occurs [w + (j)]; the larger the sum of Akaike weights the more important that variable is, relative to the other variables [START_REF] Burnham | Model selection and multi-model inference[END_REF].

For data collected in patches within P. oceanica meadows (second survey), two potential predictor variables were included in the candidate models, patch size (size),

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12 which was a continuous variable, and the location of the sampled quadrat in the region of the patch (location). Location was a categorical variable with three levels (Pa, PM, and PO) (Fig. 2). Seven different GAMLSS, h i , i = 0 to 6, were fitted (Table 2). Either only the mean response (models h 1 -h 3 ) or both the mean response and the standard deviation (models h 4 -h 6 ) were modelled, with one or both predictor variables. Model h 0 was the null model, with no predictor variable, while h 6 was the full model. The same approach as with the data of the first survey was followed for the identification of the best underlying statistical distribution and for model selection.

Results

Vulnerability of habitat types (Survey 1)

Among the six tested distributions, negative binomial gave the lowest AIC and was selected as the underlying distribution for modeling the counts of C. racemosa fronds (Table 3). The Delaporte distribution also had substantial support by the data with only 1.9 units higher AIC score, while all other distributional assumptions were not supported (∆ i >>10).

The best models of the expected number of C. racemosa fronds were g 4 (w 4 = 49.7%) and g 6 (w 6 = 50.3%) with almost equal support, while all other models had essentially no support (∆ i >>10). The habitat type was more important as predictor variable of C. racemosa frond counts [w + (habitat) = 1.00] than zone [w + (zone) = 0.50]. Model g 4 was nested within g 6 (full model), and their only difference was in the inclusion of

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13 zone in g 6 . In relation to habitat type (habitat) the output of the two models did not differ, and thus only g 6 is presented herein (Fig. 3). Models that considered constant variance (g 0 -g 3 ) had essentially no support (∆ i >>10) indicating that heterogeneity of variances could not be ignored in our case. As assessed by normal QQ-plots (plots of the quantiles of the normal distribution against the distribution of the residuals), the normalized residuals of model g 6 behaved well, while in model g 3 , which is the same as g 6 for the mean response but did not include standard deviation modelling, the residuals deviated from the normal distribution (Figure 4).

C. racemosa was found in 245 of the 748 quadrats that were sampled (33%). The probability of presence within a randomly placed 20 x 20 cm quadrat was 83% in MM, 69% in R, 86% in PM, 10% in S, and 6% in PO. The frond density was the highest on matte morte and rocky bottoms (mostly rocky reefs with photophilic macroalgal assemblages, dominated by Cystoseira spp.) and the lowest within P. oceanica meadows (Fig. 3; Table 4). C. racemosa was completely absent in unfragmented robust P. oceanica beds, and was only recorded in a few spots of low P. oceanica density in fragmented meadows. On the margins of P. oceanica meadows, intermediate C. racemosa densities were observed. On sandy bottoms C. racemosa density was generally low and highly variable (Fig. 3). Similar values of frond density were observed in the three management zones of the NMPZ (A, B, and C), while a slightly lower mean value was observed in the buffer zone Ia (Fig. 3; Table 4).

Vulnerability of fragmented P. oceanica meadows (Survey 2)

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ACCEPTED MANUSCRIPT 14 Among the six tested distributions, negative binomial gave the lowest AIC and was selected as the underlying distribution for modeling the counts of C. racemosa fronds (Table 3). The Delaporte distribution also had substantial support, the Sichel distribution had substantially less support, while all other distributional assumptions were not supported (∆ i >>10).

The best model of the expected number of C. racemosa fronds in the region of patches within P.oceanica meadows was h 4 (w 4 = 59.6%) (Table 2). Model h 6 also had substantial support (w 6 = 23.0%), and models h 1 and h 3 had essentially less support. The other models were not supported by the data (∆ i >>10). Location was more important as a predictor variable of C. racemosa frond counts [w + (location) = 1.00] than patch size [w + (size) = 0.33]. Only model h 4 is presented herein; in h 6 size was included as an additional variable but it was not significant either for the mean (p = 0.22) or for the standard deviation (p = 0.72), thus h 6 did not provide any additional information and was not investigated further. Models that assumed constant variance (g 0 -g 3 ) had less support [w + ( g 0 -g 3 ) = 17.4%] than models assuming heterogeneous variances [w + ( g 4 -g 6 ) = 82.6%]. Model h 4 provided a good absolute fit, as assessed by the Q-Q plot and the distribution of normalized residuals.

C. racemosa was present in 50% of the quadrats in PM, 19.3% of those in Pa, and only 4.5% of those in PO. The frequency of presence of the alga in PM was significantly higher than the frequency of presence in Pa (chi-square test, p < 0.001).

The frond density in Pa (2.0-8.5 fronds in 20 x 20 cm quadrats; 95% CI) was not significantly different (p = 0.19) than the frond density in PM (4.7-11.3 fronds) but was significantly higher (p < 0.001) than the frond density in PO (0.03-0.32 fronds)

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ACCEPTED MANUSCRIPT 15 (Fig. 5). Conditional on presence, the counts of C. racemosa fronds did not differ between Pa (7.8-12.9 fronds) and PM (9.4-13.3 fronds) but were significantly lower in PO (1.1-3.3 fronds) (95% Tukey test, based on log-transformed counts to homogenize variances).

Discussion

The habitat types investigated were found to have markedly different vulnerabilities to the Caulerpa racemosa invasion. The highest frond densities were observed in matte morte and rocky habitat types, indicating their high vulnerability. On sandy/muddy bottoms the frond density was substantially lower indicating lower vulnerability of this habitat type, especially in unstable soft substrata. Similar qualitative observations have been reported elsewhere, e.g. at Marseilles, dense C. racemosa meadows have colonized matte morte areas and coarse sediment but the sandy bottoms were only poorly colonized by few stolons [START_REF] Ruitton | Seasonal changes of the introduced Caulerpa racemosa var. cylindracea (Caulerpales, Chlorophyta) at the northwest limit of its Mediterranean range[END_REF]. Under certain conditions, C. racemosa may form compact dense mats on soft bottoms but such mats seem to mostly occur in matte morte zones [START_REF] Argyrou | Expansion of the macroalga Caulerpa racemosa and changes in soft bottom macrofaunal assemblages in Moni Bay, Cyprus[END_REF][START_REF] Ruitton | Seasonal changes of the introduced Caulerpa racemosa var. cylindracea (Caulerpales, Chlorophyta) at the northwest limit of its Mediterranean range[END_REF] or areas where native meadows pre-existed, e.g. in Caulerpa prolifera beds (S.K., pers. obs.), and not in previously unvegetated sandy/muddy bottoms. Probability of C. racemosa presence in P. oceanica meadows was low and the alga was mostly restricted in the narrow margins of the seagrass meadows and in gaps within fragmented meadows. As demonstrated by this study, C. racemosa average density was high in the meadow margins and in patches, while it abruptly declined to

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16 practically zero within the meadow at a short distance from the margin. Thus, in fragmented seagrass meadows, the sandy or matte morte patches represent the spots of high vulnerability to C. racemosa invasion. As indicated by the significantly higher probability of C. racemosa presence in PM than in Pa locations, the margins of patches represent the preferable location within a fragmented meadow for the successful establishment of the alga. Other studies have indicated that both the vertical and horizontal algal growth is negatively influenced by seagrass density, and dense and healthy P. oceanica meadows represent a barrier to the colonization of C. racemosa [START_REF] Ceccherelli | Response of the non-indigenous Caulerpa racemosa (Forsskål) J. Agardh to the native seagrass Posidonia oceanica (L.) Delile: effect of density of shoots and orientation of edges of meadows[END_REF][START_REF] Klein | The Caulerpa racemosa invasion: A critical review[END_REF].

The above results seem to be partly due to the dispersal mechanism of the species by fragmentation, which is one of the main reasons for the fast spread of C. racemosa in the Mediterranean. Fragments of the species, produced by either natural or anthropogenic (fishing gear, anchoring) processes, have to stabilize for some time on the substrate in order to get firmly anchored with their rhizoids and become resistant to hydrodynamic forces. Once anchored, the stolons will elongate and the clone will quickly expand locally. Matte morte, rocky habitats, and the margins of P. oceanica meadows provide appropriate substrates with many overhangs and other spots of easy attachment of algal fragments, contrary to sandy/muddy substrates. For example, fragments drifting by waves or currents will easily be brought to a halt when they reach the margin of a sea grass meadow as they will get entangled to sea grass rhizomes or leaves. Similarly, on sandy/muddy bottoms it is not uncommon to observe C. racemosa fragments attached to small 'hard' objects such as marine litter or sponge colonies, while the alga is completely absent from everywhere else in the surrounding soft bottom (SK, pers. obs.). Such hard objects probably act as anchoring

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points of drifting C. racemosa fragments. We assume that this mechanism of attachment of algal fragments is the cause of a substantial part of the variation of C. racemosa frond density among habitat/substrate types.

In seagrass beds that are not in a good state (high fragmentation, large areas of matte morte, low shoot density) vulnerability to C. racemosa invasion would be higher.

Scars created by anchoring in seagrass beds will also favour the distributional expansion of the alga. It has been demonstrated that boat anchoring has a destructive impact on seagrass meadows destroying many shoots and creating scars that can take a long-time to be re-colonized, especially in meadows of slow growth seagrass species such as P. oceanica [START_REF] Francour | Effects of boat anchoring in Posidonia oceanica seagrass beds in the Port-Cros National Park (north-western Mediterranean Sea)[END_REF][START_REF] Milazzo | Boat anchoring on Posidonia oceanica beds in a marine protected area (Italy, western Mediterranean): effect of anchor types in different anchoring stages[END_REF][START_REF] Ceccherelli | Short-term response of the slow growing seagrass Posidonia oceanica to simulated anchor impact[END_REF][START_REF] Montefalcone | BACI design reveals the decline of the seagrass Posidonia oceanica induced by anchoring[END_REF]. Thus, anchoring in P. oceanica beds creates favourable microhabitats for the settlement and expansion of C. racemosa. In this respect, a ban of anchoring in seagrass beds, apart from being a management measure for the protection of the meadow, is a measure against the further distributional expansion of C. racemosa.

Fishing, especially with mobile gears, such as otter trawls and boat seines, but also with trammel nets, substantially contributes to the spread of Caulerpa spp [START_REF] Relini | The role of fishing gear in the spreading of allochthonous species: the case of Caulerpa taxifolia in the Ligurian Sea[END_REF][START_REF] Magri | La présence de Caulerpa racemosa le long des côtes septentrionales de la Toscane et les conséquences possibles sur l'activité de pêche[END_REF]. Fragments of the species are collected by fishing gear and then discarded by fishermen together with all by-catch and debris, sometimes far from the fishing site. In a similar way recreational boats may assist the dispersion of C. racemosa by fragmenting its stolons during anchoring and even transferring such fragments to the surface seawater layers from where longer dispersion is likely.

Currents may further disperse the fragments created by fishing gear or anchoring and Although the vulnerability of seagrass meadows to C. racemosa has been sufficiently investigated, there is a lack of knowledge on the vulnerability of rocky bottoms. The type of macroalgal assemblages (turf, encrusting, and erect) and the number of algal species seem to affect the C. racemosa invasion, with substrates dominated by turfing algae being more vulnerable [START_REF] Ceccherelli | Spread of introduced Caulerpa species in macroalgal habitats[END_REF]. Turfs probably physically assist the settlement of C. racemosa by entrapment of fragments of the alga and/or enhancing the anchoring of stolons [START_REF] Bulleri | Facilitation of the introduced green alga Caulerpa racemosa by resident algal turfs: experimental evaluation of underlying mechanisms[END_REF]. However, the extent of the invasion of C. racemosa on rocky bottoms, the invasion rates, impacts to and competition with the benthic fauna, and the factors that favour successful settlement and high densities are issues that need further investigation. As found in this study, but also based on our personal observations in other areas of the Ionian and the Aegean Sea, the extent of C. racemosa invasion on rocky habitats is dramatic and the consequences largely unknown.

In conclusion: (1) we verified that vulnerability to C. racemosa invasion changes with habitat and the most vulnerable habitat/substrate types are matte morte areas, the margins of P. oceanica meadows, and rocky bottoms; (2) within fragmented Posidonia oceanica meadows vulnerability differs among structural substrate types and is higher along the margins of patches with no apparent dependence on patch size;

  is no comparative study investigating the relative vulnerability of the main Mediterranean habitat types to C. racemosa invasion. To provide such insight, in the present study we investigated the relative vulnerability of a variety of habitat types to

Field

  work was conducted during the first fortnight of October 2009. Two independent surveys were conducted. The aim of the first survey was to assess vulnerability of all main habitat types of the survey area, while the second survey focused on Posidonia oceanica beds, specifically on the relative vulnerability of different substrate types within fragmented meadows. Fragmented meadows are defined as those where the continuity of the meadow is interrupted by gaps of various P. oceanica (mentioned as 'patches' hereafter), which could be either sandy or 'matte morte' areas, i.e. zones of fibrous remnants (dead rhizomes) of a former P. oceanica bed.In the first survey, three 50-m line transects were surveyed with SCUBA diving in each of 23 sites. The selection of the 23 sites was based on the following requirements: (1) good spatial coverage of the study area; (2) representative samples on all major habitat types of the study area;

  settle and get established in a very distant location. Even in zones of absolute protection within MPAs, where both fishing and anchoring are prohibited, such fragments may arrive with currents from neighbouring sites with no such restrictions. In addition, the ability of C. racemosa to reproduce sexually renders any protection measures in MPAs inefficient to prevent the introduction of the species from other locations.Commercial fishing with nets is allowed in all management zones of NMPZ and in all neighbouring areas with only a six-month ban in zone A; anchoring is permitted within zone C. Due to the existence of the marine park and the developed eco-touristic industry focusing on turtle-watching, there is an increased number of recreational boats sailing in zones B and C and anchoring in zone C. As NMPZ is a semi-closed bay, C. racemosa fragments that are produced within any of the protection zones are likely to remain within the boundaries of NMPZ. Thus, the role of fishing gear and recreational boats to the further spread of the species in NMPZ cannot be avoided by the existing protection measures. The operation of NMPZ for the last eleven years did not seem to decelerate the spread of C. racemosa in Laganas Bay. On the contrary, the species is found in the NMPZ with very high probabilities of presence within randomly placed 20 x 20 cm quadrats on matte morte areas, rocky bottoms, and the margins of P. oceanica meadows. For the reasons stated above (role of fishing gear and anchoring, ability of sexual reproduction, further expansion of existing stolons) no direct effect of the protection measures to the spread of the species is anticipated in the future, unless the protection measures will prove effective to improve the status of P. oceanica meadows within NMPZ (mainly because of the ban of anchoring and use of mobile fishing gear) and thus reduce favourable microhabitats, such as patches However, the process of sea grass recovery is very slow and no short-term effects are likely. If the management measures prove ineffective to protect the P. oceanica meadows in the park and fragmentation increases in the future, C. racemosa will further spread covering all new matte morte areas.

Fig. 1 :

 1 Figure captions

Fig. 2 :

 2 Fig. 2: Location (representing three different substrate types) of C. racemosa frond

Fig 3 :

 3 Fig 3: Estimated values of the two categorical variables (habitat type and zone) for

Fig. 4 :

 4 Fig. 4: Normal QQ plots for models g 6 (full model) and g 3 .

Fig. 5 :

 5 Fig. 5: Estimated mean response and standard deviation of C. racemosa frond counts

  . Restrictive measures on the activities of recreational and fishing boats within MPAs might have a positive effect by

	ACCEPTED MANUSCRIPT
	decelerating the invasion rates of some alien macrophytes. Another aim of this study
	was to investigate whether protection measures within an MPA (the National Marine
	Park of Zakynthos) had any effect on C. racemosa spread.
	M Specifically, we tested the following hypotheses: 1. Various habitat types have different vulnerability to the invasive alga C. racemosa. 2. Within fragmented Posidonia oceanica meadows vulnerability differs among structural substrate types such as the meadow margin and gaps of various sizes without alive P. oceanica. 3. Protection measures in MPAs, especially restrictions on recreational boating A N U S C R I P T and fishing activities, effectively restrain the expansion of C. racemosa.
	A C C E P T E D
	6

Table 1 :

 1 Survey 1: Summary of the parameterisation of the seven candidate models g i of the counts of C. racemosa fronds in 20 x 20 cm plots.For each model, the Akaike Information Criterion AIC, Akaike differences ∆i, Akaike weights wi, and degrees of freedom (df) are given. Models with substantial support (∆i < 2) are given in bold. µ, σ are the mean and the standard deviation of the distribution of the response variable.

	Model	Modeled distribution parameters	Predictor variables	df	AIC	∆ i	w i (%)
	g 0	-	Null	2	2748.19	462.82	0.0
	g 1	µ	habitat	6	2440.25	154.88	0.0
	g 2	µ	zone	5	2744.40	459.03	0.0
	g 3	µ	habitat + zone	9	2441.58	156.21	0.0
	g 4	µ, σ	habitat	10	2285.39	0.03	49.7
	g 5	µ, σ	zone	8	2712.07	426.70	0.0
	g 6	µ, σ	habitat + zone	16	2285.37	0.00	50.3

Table 2 :

 2 Survey 2: Summary of the parameterisation of the seven candidate models hi of the counts of C. racemosa fronds in 20 x 20 cm plots in the second survey (patches in P. oceanica meadows).For each model, the Akaike Information Criterion AIC, Akaike differences ∆i, Akaike weights wi, and degrees of freedom (df) are given. Models with substantial support (∆i < 2) are given in bold. µ, σ are the mean and the standard deviation of the distribution of the response variable.

	Model	Modeled distribution parameters	Predictor variables	df	AIC	∆ i	w i (%)
	h 0	-	Null	2	825.01	56.21	0.0
	h 1	µ	location	4	772.92	4.12	7.6
	h 2	µ	size	3	826.80	58.00	0.0
	h 3	µ	location + size 5	772.41	3.61	9.8
	h 4	µ, σ	location	6	768.80	0.00	59.6
	h 5	µ, σ	size	4	828.21	59.41	0.0
	h 6	µ, σ	location + size 8	770.71	1.91	23.0

Acknowledgements

This study was part of the project "Survey of the Marine Benthic Fauna in the National Marine Park of Zakynthos", funded by EU and national sources (coordinator: M.T.-L.). We thank the park authorities and staff for their valuable assistance during the survey. We are grateful to three anonymous reviewers who made valuable comments and suggestions that improved the quality of this paper.


John Price Hirth

Robert Charles Pond
email: r.c.pond@exeter.ac.uk

Strains and Rotations in Thin Deposited Films

Keywords: dislocation theory, epitaxy, thin films, interfacial structures, strained layers, topological theory of defects

come

Introduction

Thin deposited films are widely exploited in modern technology. These films are often partially coherent, exhibiting arrays of dislocations at the interface between the misfitting film and its substrate. Such dislocations may accommodate coherency strains to some extent, but may also give rise to crystal rotations. Both these features affect the physical properties of a film, and a great deal of research has been directed at elucidating this. Understanding the role of dislocations in mitigating coherency strains was pioneered by Frank and van der Merwe in 1949 [1], and several reviews summarize the current position [START_REF] Beanland | Handbook on Semiconductors[END_REF][START_REF] Jain | [END_REF][4][START_REF] Freund | Thin Film Materials[END_REF]. Rigid body rotations of hetero-epitaxial films away from the nominal orientation relationship were first reported by Igarishi in 1971 [6]: he attributed the rotations to dislocations in the interface with Burgers vector components inclined to the interface, thereby producing a small-angle tilt wall superposed on an underlying misfit array. Since that time numerous observations of rotated films have been reported and their consequences discussed [7][8][9][10][11][12][START_REF] Bai | Materials Research Society Symposium Proceedings[END_REF][START_REF] Du | [END_REF][15]. This paper focuses on two aspects of coherency strains and rotations in a thin film, B, deposited on a substrate A: characteristic distances and partitioning. In the immediate vicinity of an interface (parallel to the plane x,y ) the strain field has a complex form. However, at distances greater than m z any residual misfit strain (i.e. coherency strain uncompensated after the introduction of misfit dislocations) becomes constant.

Moreover, these strains are partitioned along y into A Here, we compare the values of m z and φ z ; in general these depend on the elastic properties of A and B [START_REF] Dundurs | Mathematical Theory of Dislocations[END_REF], but we consider only the isotropic homogeneous case for illustration [START_REF] Hirth | [END_REF]. The partitioned values A yy ε and B yy ε are known to depend on the relative thicknesses of A and B, A d and B d [18]. However, in general, the partitioning into A φ and B φ does not depend on A d and B d , as has become clear only recently from studies of interfaces in phase transformations [19,[START_REF] Pond | Dislocations in Solids[END_REF]. Finally, we do not consider the plastic deformation that may lead to the interfaces we describe. Instead we deal with the properties of static interfaces once they have formed.

Coherent precipitates or martensite plates are subjected to biaxial stresses and strains arising from coherency. In addition, matrix compatibility constraints often lead to added stresses acting normal to the interface. However in the topological model [19,[START_REF] Pond | Dislocations in Solids[END_REF] or the closely related phenomenological theory of phase transformations [START_REF] Wechsler | [END_REF]22], accommodation effects are treated separately and the models are applied to an infinite bicrystal with free surfaces normal to the interface. Hence, the models yield plane stress solutions that are directly applicable to the thin film case treated here. The details of rotational partitioning have only recently been considered in phase transformations and have not been treated in terms of characteristic distances for thin films, the basis for the present work. For simplicity, we treat the case of isotropic, homogeneous elasticity. The results are qualitatively similar to those for the anisotropic elastic case, or the nonlinear elastic case in atomistic simulations.

There have been many analyses of the stresses and properties of planar coherent interfaces and their relaxation by misfit dislocations, reviewed in [START_REF] Beanland | Handbook on Semiconductors[END_REF][START_REF] Jain | [END_REF][4][START_REF] Freund | Thin Film Materials[END_REF]. .Our focus here is on the narrower issue of interfaces with coherent terraces and with steps or disconnections relieving misfit, which have received little attention. The planar interface is treated briefly, but as a reference basis for the other cases considered.

Interface Defects

Evidently, the nature of defects at a particular interface is a key factor governing the resulting characteristic distances and partitioning. The two principal categories of line defects are dislocations and disconnections [START_REF] Pond | Dislocations in Solids[END_REF][START_REF] Hirth | [END_REF]: the former are line-defects with Burgers vector b, whereas the latter have dislocation and step character, b and h.

Many observations of dislocations have been reported [25][26][27][28][29][30], some with ] 0 , , [ . Figure 1(a) is a high-resolution TEM image showing the two types in an epitaxial interface between a deposited GaAs film and a Si substrate [31]. [15]. For example, films are often deposited on vicinal substrate surfaces comprising an array of equally spaced steps. When overgrown, these steps may become disconnections, but this array is unlikely to fully accommodate the misfit.

(b,0) a) b) c) (b,h) (b,0) 0 , 0 = = φ θ 0 , 0 ≠ = φ θ 0 , 0 ≠ ≠ φ θ
Consequently, additional dislocations may be introduced as in figure 2 3. Elastic distortions.

Misfit without rotation (

0 = φ ).
As an example of the simple misfit case, figure 2(a), we consider one parallel array of dislocations, with an attendant plane strain elastic field. Reference states for the definition of Burgers vectors are summarized in Appendix A. The general result for strain partitioning is given in Appendix B. For most purposes the simple engineering strain of equations (B5) and (B6), which also implies equal partitioning of the difference in lattice parameters of the two crystals in the small strain limit, suffices. The results for one array are qualitatively similar to the biaxial case with intersecting arrays: for the latter the form of Hooke's law differs and there are Poisson fields of one array superposed on the other [START_REF] Jain | [END_REF]32,33,[START_REF] Hirth | Theory of Dislocations[END_REF]. For an array of misfit dislocations with spacing 0 λ , figure 3 (where the perfect crystal atom spacings in the y direction are Near the misfit dislocations, r << λ 0 , their elastic fields approach the r ln field present at a distance r from an isolated dislocation. For r ~ 0.1 0 λ , the field becomes the hyperbolic sinh/cosh field of parallel arrays of dislocations [START_REF] Jain | [END_REF]33,[START_REF] Hirth | Theory of Dislocations[END_REF]. For . At free surfaces normal to y, there are also varying surface forces and torques to satisfy the free surface boundary condition, but the accompanying strains and twists are confined to a distance 0 λ ≈ from the free surface [START_REF] Shinohara | [END_REF] and need not be considered here. Beyond this distance, image strains become uniform. For a thin bilayer, these lateral strains would produce bending in a free bilayer. These are eliminated for a thin central member of a trilayer, or a thin layer in a multilayer.

/ / )] 2 / tan( [ λ λ φ ε y z y c yy b b b ≈ - = - . (2)
The angle φ is given by the Frank relation

0 0 1 / ) 2 / ( sin 2 λ λ φ z z b b ≈ = - , (3)
where the approximate forms are accurate in the limit of smallφ . Analogous to the nonlinear treatment of phase transformations [36], equations ( 2) and (3) are solved for As shown in equation C-5 in Appendix C and as depicted in figure 7 there are local distortion fields

j i x u ∂ ∂ /
, near the dislocations, where the i u are displacements.

These comprise both strains,

2 / ] / / [ i j j i x u x u ∂ ∂ + ∂ ∂
, and rotations,

2 / ] / / [ i j j i x u x u ∂ ∂ - ∂ ∂
. As indicated in figure 4, the strains rapidly converge to zero in the fully misfit relieved case for e z z > , leaving only the rotations at long-range. For e z z > the strains vanish, so force equilibrium is satisfied. Hence, unlike the misfit

Misfit accommodation by disconnections (

0 , 0 ≠ ≠ φ θ ).
For the offcut disconnection case, figure 2(c), the nucleation mechanism [38,39] tends to favor the incorporation of a large misfit component in the disconnection, as in figure 1(c), in turn leading to a component b z ' in the habit plane. For the special case of cube on cube deposition, the component b z ' can be very small [40]: we do not consider such a case here. For the offcut case or when some like-sign disconnections occur in a local region during growth, when b z ' ≠ 0, the imposed disconnection density may not completely remove misfit and added misfit dislocations might be needed.

We first consider the disconnections, depicted in figure 8. The added feature is that the step height h equals the smaller of the complementary free-surface steps and not necessarily the vicinal step height. The steps cause an interface rotation, θ . Now, the misfit balance and the tilt rotations are performed in the rotated coordinates ' shown in the figure . A nonlinear treatment, analogous to that for misfit dislocations, is available for the disconnection case [36]. The equilibrium disconnection spacing that produces a long-range dislocation strain equal and opposite to the coherency strain is

0 0 ] tan [ ] 2 / tan( [ tan )] 2 / tan( [ λ θ λ φ θ θ φ θ ε z y z y z y c yy b b b b h b b + ≈ - + = - + = - . (4)
The angle ϕ is given by

          - - - - - - - = - 2 ) 2 / cos( ) 2 / sin( 2 tan ) 2 / cos( )] 2 / sin( ) 2 / cos( [ sin 2 1 φ θ φ θ ε θ φ θ φ θ φ θ φ c yy y z h b b 0 1 2 cos sin 2 sin ] sin cos [ sin 2 λ θ θ ε θ θ θ z c yy y z b h b b ≈           - - ≅ - . (5)
Here, the successive approximations become accurate in the limit of small φ and smallθ , respectively. Evidently, when θ is zero, equations ( 4) and ( 5) reduce to [START_REF] Beanland | Handbook on Semiconductors[END_REF] and (3), respectively. Just as for the misfit dislocation array cases (figure 2a When 0 λ is increasingly large, the angle φ becomes so small as to be negligible. In general the disconnections from vicinal steps do not remove all the coherency strain. The disconnection spacing λ 0 is known so equation ( 2) gives the portion of the strain removed (or added), cv yy ε . The difference between this value and c yy ε is the portion of the strain to be removed by misfit dislocations, either in-plane or inclined. The nominal spacing of these defects can be determined from equation ( 2)

and ( 4) in the usual manner. Actually, while the strain portions are known, and hence the numbers of the two types of defects, the spacings λ 0 do not remain constant. The spacings rearrange in a somewhat nonuniform manner so that the total strain energy of the defects is minimized. The consequence is that the m z distance where the strain becomes uniform is somewhat smaller than the smaller of the two individual uniform spacings. There could be a similar mixture of the two types of misfit dislocations, although in that case only the dislocations with in-plane Burgers vectors would rearrange to minimize strain energy: The inclined dislocations would be sessile for inplane motion. The determination of the effective value of φ z for such superposed arrays becomes more complicated.

The convergence of the strains to zero is less rapid, reflecting the variation in spacings of the two types of defects, but is still of the order of the average spacing of the defects. Hence, φ z will still be much less than the characteristic distance, say d B , for the superposed arrays. In principle, the disconnections could even increase misfit strain, although this is not commonly expected. The major role of the vicinal steps is to augment island (disc or spherical segment) nucleation there during deposition [38,39]. Elastic energy has a role (not considered in detail in the original papers) but surface energy is a major factor. The latter factor may dominate, allowing the unfavorable elastic energy nucleus to form.

Hence, there typically should be no effect of relative layer thicknesses on the partitioning of rotations, provided that the layer thickness analogous to the misfit strains considered here, but these effects vanish in the isotropic elastic case. When φ z d B < , i.e. for widely spaced misfit defects (inclined dislocations or disconnections), the local rotations become a complicated function of position. The net rotation may be less than in the large z limit. Of course the net rotation becomes small when the misfit defects are widely spaced, so this limit does not represent a very important case.

Applications

One example of each of the three interface types in figure 2 Du and Flynn used MBE to grow epitaxial films of hcp and fcc materials on bcc substrates [START_REF] Du | [END_REF]. Very thin films were observed by X-ray diffraction to be homogeneously strained to be commensurate with the substrate, but this coherency strain was relieved in thicker deposits. The thicker films were also tilted by values up to several degrees: for example, ( 110

≠ φ θ

Disconnection arrays have been observed using TEM in parent-martensite interfaces [19,[START_REF] Pond | Dislocations in Solids[END_REF]. In this case, the disconnections are mobile, and form arrays as in figure 8 with spacings close to the equilibrium value, e λ . (In general, a second array of dislocations, inclined to the disconnection array, is present in the interface for complete misfit accommodation). Figure 10 shows a disconnection array in Ti-10wt.%Mo [44], and we compute the value of θ andφ using equations ( 4) and ( 5).

Equation. ( 4) can be written

0 tan ] ) 2 / tan( ) [( tan ] ) 2 / tan( [ 1 1 2 = + + - + + + + c yy i y i z c yy i z i y h b b h b b ε θ φ ε θ φ . (6)
One can start with the linear result . Evidently, φ is small in this case and the approximate (linear) forms of equations ( 4) and ( 5 Table : Step height ratios and habit plane parameters for "Ti" obtained by the linear and non-linear forms of equations ( 4) and [START_REF] Freund | Thin Film Materials[END_REF]. All angles are expressed in degrees. A special case occurs for some offcut crystals where disconnections form at the former free surface steps on the substrate, but no added misfit dislocations form.

Hence the coherency strain is not completely relaxed. The above derivations are still applicable for this case, but λ replaces λ 0 in equation ( 4) and the corresponding strain is only a portion of the equilibrium strain that would completely remove misfit. Al 2 O 3 [48].

the substrate terrace planes, they do not contribute to film tilting in the first approximation. The CdTe:Al 2 O 3 case is similar, although the misfit dislocation array (thought to be a triangular network of in-plane

CdTe > < 0 1 1 2 / 1
dislocations) was not imaged directly. Thus, the observed tilts,φ , are produced by the z b components of the disconnection array, and Aindow and Pond [15] correlated these with measurements of the vicinal angle of the substrates, figure 12. Vicinal angles can be expressed

as v A v h λ φ / - =
, where v λ is the spacing of the vicinal steps (the minus sign is necessary for consistency with the sense ofφ defined in Appendix C), and A h is the height of the substrate surface steps. Since

v A B v z h h b λ λ φ / ) ( / - = ≈ , we can write v A B h h φ φ ) / 1 ( - ≈
. Thus the slope of a φ -versus-v φ plot is negative when

A B h h >
and positive otherwise. The dashed lines in figure 12 represent the ideal relation associated with particular disconnections, and these show excellent agreement with the least squares fit to the data, solid line. , where h is the smaller of A h or B h , and

φ ω ω = -A B
. Thus, for CdTe, B, on alumina, A,

v B A B h h λ ω 2 / ) ( + = and v B A A h h λ ω 2 / ) 3 ( + - =
, and for Si, B, on alumina, A,

v B A B h h λ ω 2 / ) 3 ( - = and v B A A h h λ ω 2 / ) ( + = .

Discussion

In the foregoing we have identified the characteristic lengths arising from interfacial dislocation and disconnection arrays. When residual strain persists the elastic misfit and coherency strains are partitioned according to equation ( 1 We have presented the above analysis for the plane strain, single-array case.

However the characteristic lengths, the focus of the present treatment, would be the same for the biaxial strain, intersecting-array case as well. The characteristic lengths would also be similar for an imbedded layer or for a layer within a multilayer. The results also have implications for the phase transformation case. For lath type transformation products or for plate-shaped products with large aspect ratios, the rotations produced by interface defects should also be fully partitioned when the plate thickness is greater than the defect spacing in the interface.

While the presence of rotations associated with tilt components of dislocations and disconnections have long been known, the present results provide explicit results for expected rotations and their partitioning. Particularly significant is the finding that the critical length scale for achievement of long-range rotation partitioning is distinct from that characterizing the partitioning of residual long-range elastic strains. From a practical point of view, the dislocation and disconnection sites should provide local charged defect centers that can act as traps for electrons, holes and ionic defects such as vacancies while the inhomogeneous strain fields also give weak scattering of the electronic defects. The disconnections have smaller Burgers vectors than the misfit dislocations in general and thus represent weaker trapping sites, with concordant implications for electronic properties.

However, the disconnections and inclined-b dislocations with a misfit component carry rotations with them, also giving electron scattering. Indeed the deleterious effect of tilt wall components and accompanying rotations has been demonstrated for a high T c superconductor [49]. Understanding the source and magnitudes of the rotations provides an opportunity in thin film design.

As an example of a method to control the degree of rotation, research on films of Ni on Cu agree with the above trends, but with differences in whether the observed misfit dislocations are Lomer dislocations with ½ <110> Burgers vectors lying in the interface [25,27,28] or with in-plane and out-of-plane Burgers vectors inclined to the interface [26,29]. Recent work [50] has shown that near the critical thickness, the dislocations are mainly inclined dislocations of this type and that with increasing thickness these convert increasingly to Lomer dislocations. Consistent with this finding, in-plane bs predominate in high-misfit cases: e.g. they form at island edges, whereas out-of-plane bs arise in 2-D growth films where misfit is lower [START_REF] Beanland | Handbook on Semiconductors[END_REF]. Thus selecting the degree of misfit is a way to control the degree of rotation at an interface in such systems.

Both the nonlinear pair of equations ( 4) and ( 5), derived from the topological model for phase transformations [36], and equations ( 2) and (3) for the misfit case involve simultaneous transcendental equations that must be solved by iteration. Yet in many cases, the linear forms of the equations can be used. Then ϕ in equations ( 3) and ( 5) is linearly related to 1/λ 0 and after substitution into equations ( 2) and (4), the latter can be solved directly for λ 0 . This is the basis of the initial linear model for the topological model [19,[START_REF] Pond | Dislocations in Solids[END_REF], where for the phase transformation case one first solves While we consider the static interface here, the defect content of the interface can also be determined from the lattice transformation between the two crystals. The defect content per unit length, c b , of a coherent interface with respect to the natural reference can be calculated from the Frank-Bilby equation as follows

v A B b ) ( 1 1 - -- = n c n c c , ( A1
)
where v is a unit length probe vector along the coherent terrace, the matrix n c A represents the homogeneous deformation that transforms the natural Alattice into the coherent A-lattice, and similarly for n c B .

Appendix B. Coherency strain

The partitioning of coherency strains arises from the condition that the net Newtonian force must vanish at equilibrium on a plane normal to the interface. This corresponds to a balance of stresses. In turn the stresses in general would be given by an expression containing first order and higher order displacement derivatives, with coefficients corresponding to higher order elastic constants. In the small strain limit only the Hooke's law, linear term is significant. The stress is then linear in true strain so that the stress balance gives the strain balance (B5)

The difference between the magnitudes of these is of order 2 ) ( a ∆ and is negligible to linear elastic order. Hence to linear elastic order, strain partitioning gives partitioning of a ∆ , so that one can use the simpler strain definition

> < ∆ ≈ = - a a B yy A yy 2 / ε ε . (B6)
To summarize, when all anisotropic and nonlinear elastic effects are absent as well as geometric effects associated with differing layer thicknesses, the coherent lattice parameter at the interface is always given by the geometric mean, An alternative approximation is to set the engineering strains equal, i.e.

/

, i.e. the square of the geometric mean divided by the arithmetic mean [START_REF] Pond | Dislocations in Solids[END_REF]. Thus, there will always be a slight inequality of lattice parameter changes when there is equal (true or engineering) strain partitioning, i.e. ) ( ) (

B i i A a a a a - ≠ -
). However, in the small strain limit the difference between these alternatives is negligible, and the simpler arithmetic mean can be used; this is the usual assumption in the literature, e.g. [19].

Appendix C. Crystal rotations

In order to simplify the following equations, which are understood to apply to

  yy ε and B yy ε . (Superscripts A and B refer to values of strain or rotation partitioned to the A and B crystals). Similarly, rotational distortions adopt constant values at distances greater than φ z , and are partitioned to the values A φ and B φ . The characteristic distances m z and φ z , as well as the partitioning into A yy ε and B yy ε and A φ and B φ , arise from the superposition of the elastic fields of the defects at the interface on the initial coherency strain in the film.

Figure 1 (

 1 b) is a schematic illustration of the extra {111} planes terminating at the cores of these dislocations[START_REF] Beanland | Handbook on Semiconductors[END_REF].

Figure 1 (Figure 1

 11 Figure 1(c) is a schematic illustration of the formation of a disconnection when a surface step on a ) 2 1 01 (

Figure 2

 2 Figure 2 Schematic illustration of three principal interface topologies.

  (a) or (b). The characteristic distances and partitioning for these possibilities are addressed in sections 3.1 and 3.2 where elastic distortions with 0

  , the coherency strain can be envisioned as arising from a continuous distribution of infinitesimal dislocations. The integral of this distributionper unit length, l , is l / c y b, and this is numerically equal to the total coherency strain, c yy ε (see Appendix A). For the present case this representation of the distribution leads to no loss in rigor. The zero long-range strain condition is satisfied for the equilibrium spacing of misfit dislocations, e λ . Then the Burgers vector of the misfit dislocations,

Figure 3 .

 3 Figure 3. Misfit dislocations spaced by 0 λ at the interface between A and B. The curved line represents a constant value of m yy ε ; the strains tend to become constant for m z z > .

  with St. Venant's principle. The zero longrange condition is defined when the misfit strain is equal and opposite to the interface

Figure 4 Figure 4 .

 44 Figure 4 is a plot of m yy ε as a function of z on the same size scale as figure 3.

0λFigure 5 .Figure 6

 56 Figure 5. Crystal dislocations impinge on the interface to relieve misfit. Their Burgers vectors are inclined to the interface, having both misfit, m y b , and tilt, m z b , components.

Figure 7 .

 7 Figure 7. The local distortions of a tilt wall are represented by the curved lines. For m z z > these distortions rapidly converge to pure rotations, partitioned to 2 / φ in each

  and b), the lattice rotations in the disconnection case partition equally when e z z > , in accord with equation C-5 in Appendix C, and there is no dependence on the ratio A B d d / .

Figure 8

 8 Figure 8 Schematic illustration showing the disconnection content of an interface, with Burgers vector components resolved in the terrace (upper) and habit plane (lower) frames[19,[START_REF] Pond | Dislocations in Solids[END_REF]. The terrace plane is inclined at an angle θ to the horizontal

  φ z d B > . Other effects, such as differing elastic constants, would partition rotations at the larger size scale,

.≈Figure 9 .

 9 Figure 9. Weak beam TEM image showing one array of an orthogonal network of 90° ½<110> dislocations in (001) GaAs:Si [31].

φ

  depends only weakly on i φ .For this example h α = 0.2556 nm, h β = 0.2678 nm and 038

  ) are satisfactory. In fact, this is the case for most martensitic transformations (which involve modest volume changes). The calculated values are consistent with experimental observations, though the uncertainties in the latter are of the order °±1 and °± 25 . 0 for ω and φ respectively in ideal conditions. Only in exceptional cases would the non-linear corrections be needed; an example would be the martensitic transformation in a hypothetical Ti alloy with the same value of c yy ε as above but with α β h h / = 1.15, as shown in the Table. A further real example arises in Pu-Ga alloys, where °≈ 6 φ and α β h h / = 1.2 [45]. The rotation produced in individual disconnections can be discerned in TEM images of defects with large values of z b , such as those observed in Ti:TiH interfaces [46]. For Ti10wt%Mo, the calculated value of e λ for disconnections with the smallest b is 1.26nm [20]. However, the disconnections observed in figure 10 are 'double' disconnections, (2b, 2h) with double spacing; thus the LID is considered to be individual twinning dislocations with b parallel to [ ] α . If these defects accumulate into micro-twins, as is observed experimentally [47], the resulting value of φ z z m = increases to become equal to the average twin spacing.

Figure 10 .

 10 Figure 10. Transmission electron micrograph of a parent-martensite interface in Ti-10wt.%Mo [44]. The electron beam is oriented parallel to the disconnection array, as in figure 8.

4. 4 Figure 11 .

 411 Figure 11. Weak beam TEM image of misfit dislocation array in ) 001 ( Si deposited

Figure 12 . 3 ..

 123 Figure 12. Correlation of film tilt angle,φ , with surface vicinal angle, v φ , for (a) ) 111 ( CdTe on ) 0001 ( Al 2 O 3 and (b) ) 001 ( Si on ) 2 1 01 ( Al 2 O 3 . All angles are in units of minutes of arc. Solid lines represent the least-squares best-fit to the data, and the dashed lines are the ideal relations for specific disconnections [15].

.

  ), the characteristic distance for this partitioning being the smaller of the layer thicknesses, here B d . For the relaxed, zero long-range strain case, the two strains are equal and opposite when for all three interface types illustrated in figure 2. For a "mixed" case, such as ) dislocation spacing, and φ z by the vicinal step spacing, and The strain partitioning depends on d B but the rotation partitioning is independent of d B provided that d B > z m ≈ z ϕ ≈ λ 0 .

  θ assuming ϕ to be zero and then superposes the tilt wall array with the attendant partitioned rotation ϕ. is the union when the lattices are uniaxially strained into coherence on the terraces, and the Burgers vectors and step heights h of disconnections and misfit dislocations are defined in this reference state.

coordinates of figure 7 ,

 7 we drop the primes. For a single tilt dislocation z b in the array spaced by λ , at the origin in the coordinates of figure7, the displacement fields as given by equation in[START_REF] Dundurs | Mathematical Theory of Dislocations[END_REF], are dislocation in the array, Y becomes n Y + and one can sum the rotations over the n dislocations from -Z the second term in brackets vanishes and the first term becomes unity.Hence, the rotation φ properly partitions:

  This is the standard relation between the geometric and arithmetic means. In the

	small strain limit
	ε	A yy	=	ln(	a	i	/	a	A	)	=	1	/	2	ln(	a	B			/	a	A	)	=	1	/	2	1 ln[	-	(	a ∆	/	a	A	)]	≈	-	a ∆	/	2	a	A	, (B4)
	and
																	ε	B yy	≈	-	a 2 / ∆	a	B	.
		F o r
								P
												e
															e r
																												R
																																			e
																																								v i e
																																										w
								-	ε	A yy	=	-	ln(	a	i		/	a	A	)		=	ε		B yy	=	ln(	a	i	/	a	B	)	, O (B1) n l
	which yields for the interface strain												y
																	i a =	(	a	A	a	B	1 )	/	2	.	(B2)
	Thus i a is the geometric mean of A a and B a . Since	<	a	>=	(	a	A	+	a	B	)	/	2	is the
	arithmetic mean and		a ∆	=	a	A	-	a	B	, we have
													<	a		>	2	=	(	a	i	)	2	+	(	a ∆	)	2	/	4	.	(B3)

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Summary

The strains and rotations arising in thin films deposited on misfitting and/or offcut substrates have been quantified for the case of elastically isotropic materials. These distortions become homogeneous at distances greater than a characteristic distance related to the spacing of interfacial defects,

. Strains are partitioned in a manner dependent on the ratio

when the interfacial defects present do not fully compensate the coherency strain. On the other hand, rotations may be finite even in the fully compensated case, and their partitioning does not depend on . Case (iv) is unlike the others in the sense that two characteristic distances arise, associated with the misfit dislocation and disconnection spacings. Non-linear treatments are presented, and shown to correspond with simpler linear versions for small values of θ andφ . The analysis is applied to one example of each of the four cases, and excellent agreement is obtained between calculated and experimentally observed values.

Appendix A. Reference states

The reference states are defined by the dichromatic patterns of interfacial symmetry theory [START_REF] Pond | Dislocations in Solids[END_REF]. The natural dichromatic pattern is created by the union of the two crystal lattices with terrace planes aligned to be parallel. The Burgers vectors of the interface coherency dislocations are defined with respect to this reference. The

where ν is Poisson's ratio. This gives the rotation