How to model this mixed model I have following data:

Each column represents a specific manufacturer (that's why I skipped the title) and from each one I have several sensors. Now I would like to perform a mixed model. I thought about using the manufacturer and the sensors as random effects but I'm not quite sure about that.
In r resp. lme4 the syntax would be:
lmer( R ~ c + (Manufacturer | sensor), data = data)

Is this correct?
What you can see here is the response of the resistance of the sensors against several gas concentrations.
edit: More information:
There are three manufacturers with two of 5 sensors each and one of 10 sensors. Basically it is:
Manufacturer A: SensorA1, SensorA2, SensorA3, SensorA4, SensorA5
Manufacturer B: SensorB1, SensorB2, SensorB3, SensorB4, SensorB5
Manufacturer C: SensorC1, SensorC2, SensorC3, SensorC4, SensorC5, SensorC6, SensorC7, SensorC8, SensorC9, SensorC10

All the sensors are housed in the same experimental unit and measured in parallel under varying conditions. Here in this case it is the applied concentration varying from 50 to 500 ppm.
So all I did in the plot is to group these sensors according to their manufacturer. The colours represent the different concentrations (it is hardly visible in the index, I have to admit..), so basically the colours are not necessary as the axis already provides this information.
My goal now is to figure out how the sensors react to the different gases individually and how do the manufacturers.
Plots:
The formula 
lmer(R~ poly(c, 1) + Manufacturer + (c | Manufacturer/Sensor), data = data)

results in

respectively in

and the formula 
lmer(R~ poly(c, 1) + Manufacturer:c + (c | Sensor), data = data)

results in 

I think the blue fit for the third group is not bad while the other two need an improvement. I'm playing a bit around but I'm not really able to enhance the situation. What I wonder is: Without any knowledge I would draw an almost straight line through the data points, separately for each Manufacturer (color). Why are above formulas not able to do so?
editedit: 
Matteo's suggestion lmer( I(log(R)) ~ I(log(c)) + Manufacturer + I(log(c)):Manufacturer + ( I(log(c)) | sensor), data = data) results in 

I can't get my head around it why the fit behaves as it does. Why are they so off all the time?
data: https://www.file-upload.net/download-13853720/data.csv.html
my code:
lmer_model<- lmer(log10(R) ~ log10(c) + Manufacturer + log10(c):Manufacturer + 
                    (log10(c) | Sensor), data = data)
ggplot(data, aes(x = c, y = R, colour = Manufacturer)) +
  geom_point(alpha = 0.5) +
  scale_x_continuous(
    trans='log10'
    , breaks = round(seq(min(data$c), max(data$c), by = 150),500)) +
  scale_y_continuous(
    trans='log10'
    , labels=format_si()
    , breaks = round(seq(min(data$R), max(data$R), by = 500000), 1000000)) +
  theme_classic() +
  geom_line(data = cbind(data, pred = predict(lmer_model)), aes(y = pred), size = 1) + 
  ylab(TeX("R / $ \\Omega $ ")) + xlab(TeX("c / ppm")) + 
  theme(
    legend.position = "none",
    panel.spacing = unit(2, "lines"),
    legend.text=element_text(size=7.5),
    legend.box.margin=margin(-13,-13,-13,-13)
  )

 A: 
In r resp. lme4 the syntax would be:
lmer( R ~ c + (Manufacturer | sensor), data = data)

Is this correct?

No. The above model specifies that observations are clustered within sensor (which is correct) but it also says that the  effect of Manufacturer varies within each level of sensor (ie there are random slopes for Manufacturer), which is incorrect, since sensor is nested within Manufacturer. Also, you do not specify Manufacturer as a fixed effect which means that the random slopes vary around zero.
You could start with the following model:
lmer( R ~ c + (1 | Manufacturer/sensor), data = data)

which is equivalent to
lmer( R ~ c + (1 | Manufacturer) + (1 | Manufacturer:sensor), data = data)

and assuming that sensor is coded uniquely as it seems to be in the information you added, it is also equivalent to:
lmer( R ~ c + (1 | Manufacturer) + (1 | sensor), data = data)

These models fit fixed effects for concentration and random intercepts for Manufacturer and sensor. The main issue I see with this is that there are only 3 levels of Manufacturer so you are unlikely to get a good estimate of it's variance. So an alternative is to specify Manufacturer as a fixed effect:
lmer( R ~ c + Manufacturer + (1 | sensor), data = data)

You might also want to allow the effect of concentration to vary across sensors by specifying random slopes for concentration:
lmer( R ~ c + Manufacturer + (c | sensor), data = data)

A: I think the model you want in lme4 syntax should be written as
lmer( R ~ c + Manufacturer + c:Manufacturer + ( c | sensor), data = data)

This model will have sensor-specific random intercept and slopes, and manufacturer specific (fixed) intercept and slopes, so I think it address your goal of estimating how individual sensors and manufactures respond to gas concentration.
I think that conceptually also Manufacturer should be a random effects, but because you have only three of them it is not very practical and not much to gain in trying to estimate the between-manufacturers variance. 
One final note, by looking at your plot it seems that the relationship is linear when both variables (concentration and response) are expressed in logarithmic units. So I think it would be appropriate to log-transform the variables before entering them in the model. Note that this changes the interpretation of the coefficients which would code for relative difference instead of absolute differences as in standard linear regression. In a nutshell, a slope coefficient of $\beta$ would indicate that a $x\%$ increase in the predictor correspond approximately to $x\beta\,\%$ increase in the response, or more precisely to a change in the response equivalent to multiplying it by $e^{\beta\log\left(\frac{100+x}{100}\right) }$.

EDIT
I had a look at the data you posted, and fit the model I suggested above. The model seems to fits quite well, so there must be an issue in the way you computed or plotted the model predictions. I paste below the code and the resulting plot.
d <- read.table("data.csv",header=T,sep="\t")
library(ggplot2)
library(lme4)

# log transform variables
d$log_c <- log(d$concentration)
d$log_R <- log(d$Resistance)

# fit mixed model
m0 <- lmer( log_R ~ log_c + Manufacturer + log_c:Manufacturer + ( log_c | Sensor), data = d)

# not converging, try some more iterations
ss <- getME(m0, c("theta","fixef"))
m0 <- update(m0, start=ss, control=lmerControl(optCtrl=list(maxfun=2e4))) # OK

# make a new data frame for calculating model predictions
# (note that I am calculating them using only the fixed effects)
nd <- expand.grid(Manufacturer = unique(d$Manufacturer), log_c = seq(min(d$log_c),max(d$log_c), length.out=100))
nd$log_R <- predict(m0, newdata=nd, re.form=NA)
nd$concentration <- exp(nd$log_c)
nd$Resistance <- exp(nd$log_R)

# plotting
ggplot(data=d,aes(x=concentration, y=Resistance, color=Manufacturer))+geom_point()+ scale_x_continuous(trans = 'log10') +scale_y_continuous(trans = 'log10')+facet_grid(.~Manufacturer)+geom_line(data=nd)


